One sample z-test and t-test
|
|
- Jesse Fitzgerald
- 5 years ago
- Views:
Transcription
1 One sample z-test and t-test January 30, 2017 psych10.stanford.edu
2 Announcements / Action Items Install ISI package (instructions in Getting Started with R) Assessment Problem Set #3 due Tu 1/31 at 7 PM R Lab in class on W 2/1 Navarro CH 8 Survey Quiz
3 from Tintle et al. Congratulations! we ve now introduced most of the complicated ideas (and most terminology) and now will revisit these ideas by exploring different types of data
4 Last time We can calculate mean, variance, and standard deviation for a process We can predict descriptive characteristics of distributions of hypothetical variables using: rules of linear transformation rules of summing variables the central limit theorem
5 This time Recap: how do we describe our newest sampling distribution, a distribution of sample means? How can we use distributions of sample means to make inferences? What s the difference between a z-test and a t-test? How do we measure effect size for single means?
6 This time Recap: how do we describe our newest sampling distribution, a distribution of sample means? How can we use distributions of sample means to make inferences? What s the difference between a z-test and a t-test? How do we measure effect size for single means?
7 Inferences about means Do herbal supplements change performance on a standardized test of memory? Scores on the test are known to be normally distributed with μ = 50 and σ = 12. A sample of n = 16 participants take herbal supplements for 90 days and then take the test. They get a mean score of x = 54. We want to know if the mean score in the population that takes herbal supplements, μ herbal, is different from 50. Is x = 54 a plausible sample mean if herbal supplements actually have no effect on mean memory performance? State two hypotheses H 0 : herbal supplements do not change mean performance, μ herbal = 50 H A : herbal supplements do change mean performance, μ herbal 50 Think about the distribution of sample means we could get if μ = 50 (and σ = 12) Determine the probability of getting a sample mean as or more extreme as x = 54 if μ = 50 (i.e., a p-value)
8 Reminder: proportions Population / probability distribution describing the population or process (the distribution that we want to know about) summarized by π, our population proportion Proportion Y N Sample distribution describing our sample data summarized by p, our sample proportion Sampling distribution (distribution of sample proportions) describing all of the possible samples of size n that we could have gotten from a hypothesized population Proportion Probability Y N Proportion Successes
9 Distribution of sample means One population, described by µ and σ and shape Many potential samples of size n, each described by x and s and shape } } } } x 1 x 2 x 3 x distribution of sample means: distribution of all of these possible sample means, described by µx and σx and shape
10 Distribution of sample means Distribution of sample means: collection of sample means for all of the possible random samples of a particular size (n) that can be drawn from a particular population population distribution sample distribution (n = 2) M M M M M M M M M M M M M M M M distribution of sample means describing all of the possible samples of size n that we could have gotten from a population
11 Distribution of sample means How can we describe the central tendency, variability, and shape of a distribution of sample means? Simulation Theoretical approach central limit theorem
12 Central limit theorem (CLT) For any population with a mean μ and standard deviation σ, the distribution of sample means from that population for sample size n will: 1. have a mean of μ written E[M], E[X ], μ M, µ X 2. have a standard deviation of σ / n written SD[M], SD[X ], σ M, σ X, standard error of the mean, SEM increases as σ gets large decreases as n gets large 3. approach a normal distribution as n approaches infinity in fact, distribution is almost perfectly normal if either: a. the population follows a normal distribution b. n 30
13 Distribution of sample means normal bimodal skewed prop population distribution distributions of sample means as n increases
14 Central limit theorem (CLT) For any population with a mean μ and standard deviation σ, the distribution of sample means from that population for sample size n will: 1. have a mean of μ written E[M], E[X ], μ M, µ X 2. have a standard deviation of σ / n written SD[M], SD[X ], σ M, σ X, standard error of the mean, SEM increases as σ gets large decreases as n gets large 3. approach a normal distribution as n approaches infinity in fact, distribution is almost perfectly normal if either: a. the population follows a normal distribution b. n 30
15 Careful! Standard deviation Standard error of the mean σ = σ 2 = (SS / N) σx = σ / n = (σ 2 / n) s = s 2 = (SS / (n - 1)) sx = s / n = (s 2 / n) typical distance between a single score and the population mean typical distance between a sample mean and the population mean used to convert single scores to z-scores filllllllllllllllllllllllllllllllllllllllller used to convert sample means to z-scores, often called z-statistics
16 Applying the central limit theorem A population distribution is normally distributed and has a mean of μ = 10 and a standard deviation of σ = 20. What are the mean, standard deviation, and shape of a distribution of sample means for n = 16? μ X = μ = 10 σ X = σ / n = 20 / 16 = 5 shape is normal because population distribution is normal A population distribution has a mean of μ = 400 and a standard deviation of σ = 100. What are the mean, standard deviation, and shape of a distribution of sample means for n = 64? μ X = μ = 400 σ X = σ / n = 100 / 64 = 12.5 shape is normal because n 30 A population distribution has a mean of μ = 50 and a standard deviation of σ = 10. What are the mean, standard deviation, and shape of a distribution of sample means for n = 4? μ X = μ = 50 σ X = σ / n = 10 / 4 = 5 careful, we re not sure about the shape!
17 This time Recap: how do we describe our newest sampling distribution, a distribution of sample means? How can we use distributions of sample means to make inferences? What s the difference between a z-test and a t-test? How do we measure effect size for single means?
18 Inferences about means Do herbal supplements change performance on a standardized test of memory? Scores on the test are known to be normally distributed with μ = 50 and σ = 12. A sample of n = 16 participants take herbal supplements for 90 days and then take the test. They get a mean score of x = 54. We want to know if the mean score in the population that takes herbal supplements, μ herbal, is different from 50. Is x = 54 a plausible sample mean if herbal supplements actually have no effect on mean memory performance? State two opposing hypotheses H 0 : herbal supplements do not change mean performance, μ herbal = 50 H A : herbal supplements do change mean performance, μ herbal 50 Think about the distribution of sample means we could get if μ = 50 (and σ = 12) Determine the probability of getting a sample mean as or more extreme as x = 54 if μ = 50 (i.e., a p-value)
19 Distribution of sample means the central limit theorem tells us that the distribution of sample means will: have a mean of μ X = μ = 50 have a standard deviation of σ X = σ / n = 12 / 16 = 3 be approximately normally distributed why? 3 x = 54 Is our sample mean, x, likely or unlikely if the null hypothesis is true? 50 What types of sample means would we expect to see if the null hypothesis was true (µ = 50, σ = 12)?
20 Inferences about means Do herbal supplements change performance on a standardized test of memory? Scores on the test are known to be normally distributed with μ = 50 and σ = 12. A sample of n = 16 participants take herbal supplements for 90 days and then take the test. They get a mean score of x = 54. We want to know if the mean score in the population that takes herbal supplements, μ herbal, is different from 50. Is x = 54 a plausible sample mean if herbal supplements actually have no effect on mean memory performance? State two opposing hypotheses H 0 : herbal supplements do not change mean performance, μ herbal = 50 H A : herbal supplements do change mean performance, μ herbal 50 Think about the distribution of sample means we could get if μ = 50 (and σ = 12) Determine the probability of getting a sample mean as or more extreme as x = 54 if μ = 50 (i.e., a p-value)
21 Normal distribution Know one-to-one mapping between every z-score and corresponding quantile z = quantile z = quantile ~68% of observations fall within 1 standard deviation of mean Density ~95% of observations fall within 2 standard deviations of mean z
22 Distribution of sample means the central limit theorem tells us that the distribution of sample means will: have a mean of μ X = μ = 50 have a standard deviation of σ X = σ / n = 12 / 16 = 3 be approximately normally distributed why? 50 3 x = 54 Convert x to a z-score: z = (x - μx ) / σx z = (x - μ) / (σ / n) z = (54-50) / 3 z = 4/3 = 1.33 What types of sample means would we expect to see if the null hypothesis was true (µ = 50, σ = 12)?
23 Careful! Standard deviation Standard error of the mean σ = σ 2 = (SS / N) σx = σ / n = (σ 2 / n) s = s 2 = (SS / (n - 1)) sx = s / n = (s 2 / n) typical distance between a single score and the population mean typical distance between a sample mean and the population mean used to convert single scores to z-scores filllllllllllllllllllllllllllllllllllllllller used to convert sample means to z-scores, often called z-statistics
24 From z to p-value What is the the probability of observing a sample statistic (x z) as or more extreme as our sample statistic (x = 54 z = 1.33) if the null hypothesis was true? as or more extreme: z < or z > 1.33 p(z < -1.33) + p(z > 1.33) [mutually exclusive] z = x = 50 z = 0 x = 54 z = 1.33
25 R: the norm() family of functions pnorm(q, mean = 0, sd = 1, lower.tail = TRUE) by default, the pnorm() function returns the cumulative distribution function at quantile q, i.e., the cumulative relative frequency* at q, i.e., the probability of getting a value less than or equal to q by default, q corresponds to a z-score in a normal distribution with µ = 0, σ = 1 > pnorm(-1.96) [1] > pnorm(1.96) [1] > pnorm(1.96, lower.tail = FALSE) [1] *equivalent to 1 - pnorm(q) z = z = 0 1 z = 0 1 z = 0 z = z = * note, I have been a bit sloppy in referring to this as cumulative frequency
26 From z to p-value What is the the probability of observing a sample statistic (x z) as or more extreme as our sample statistic (x = 54 z = 1.33) if the null hypothesis was true? p =.18 as or more extreme: z < or z > 1.33 p(z < -1.33) + p(z > 1.33) [mutually exclusive] x = 54 > pnorm(-1.33) [1] z = z = 1.33 > pnorm(1.33, lower.tail=false) [1] x = 50 > pnorm(-1.33) * 2 [1] z = 0
27 Distribution of sample means the central limit theorem tells us that the distribution of sample means will: have a mean of μ X = μ = 50 have a standard deviation of σ X = σ / n = 12 / 16 = 3 be approximately normally distributed why? 50 3 x = 54 Is our sample mean, x, likely or unlikely if the null hypothesis is true? p >.05, so plausible What types of sample means would we expect to see if the null hypothesis was true (µ = 50, σ = 12)?
28 Inferences about means Do herbal supplements change performance on a standardized test of memory? Scores on the test are known to be normally distributed with μ = 50 and σ = 12. A sample of n = 16 participants take herbal supplements for 90 days and then take the test. They get a mean score of x = 54. We want to know if the mean score in the population that takes herbal supplements, μ herbal, is different from 50. Is x = 54 a plausible sample mean if herbal supplements actually have no effect on mean memory performance? State two opposing hypotheses H 0 : herbal supplements do not change mean performance, μ herbal = 50 H A : herbal supplements do change mean performance, μ herbal 50 Think about the distribution of sample means we could get if μ = 50 (and σ = 12) the probability of getting a sample mean as or more extreme as x = 54 if μ = 50 is p =.18 fail to reject H 0 and conclude we do not have evidence that herbal supplements change mean performance on the test of memory
29 Case study: quality assurance Company standards at a fast food chain require that each franchise has a mean wait time of less than µ = 10 minutes. It is already known that wait times have a standard deviation of σ = 2. To test whether a franchise is in compliance, a manager looks at data from a recent customer service survey with a sample of n = 100 customers and wants to see if there is evidence that wait times are greater than 10 minutes for the population of all customers, using α =.05. She finds that the franchise has a mean wait time of of x = 10.5 minutes. State two opposing hypotheses: H0: μ = 10 (μ 10) HA: μ > 10 Is our sample mean likely if H0 is true?
30 Case study: quality assurance What types of sample means would we get if μ = 10, σ = 2, n = 100? the central limit theorem tells us that the distribution of sample means will: have a mean of μ X = μ = 10 have a standard deviation of σ X = σ / n = 2 / 100 = 0.20 be approximately normally distributed why? What is the probability of getting a sample mean as extreme as our sample mean of x = 10.5, if H 0 is true? (1) convert x to a z-score z = (x - μ M ) / σ M = (x - μ) / (σ / n) = ( ) / (2 / 100) =.5 /.2 = 2.5 (2) use pnorm() to find (one-tailed) p-value > pnorm(2.5, lower.tail=false) [1] z = 0 1 z = +2.5
31 Case study: quality assurance Company standards at a fast food chain require that each franchise has a mean wait time of less than µ = 10 minutes. It is already known that wait times have a standard deviation of σ = 2. To test whether a franchise is in compliance, a manager looks at data from a recent customer service survey with a sample of n = 100 customers and wants to see if there is evidence that wait times are greater than 10 minutes for the population of all customers, using α =.05. She finds that the franchise has a mean wait time of of x = 10.5 minutes. State two opposing hypotheses: H0: μ = 10 (μ 10) HA: μ > 10 Is our sample mean likely if H0 is true? No, p < α ( <.05)
32 Case study: quality assurance Company standards at a fast food chain require that each franchise has a mean wait time of less than µ = 10 minutes. It is already known that wait times have a standard deviation of σ = 2. To test whether a franchise is in compliance, a manager looks at data from a recent customer service survey with a sample of n = 100 customers and wants to see if there is evidence that wait times are greater than 10 minutes for the population of all customers, using α =.05. She finds that the franchise has a mean wait time of of x = 10.5 minutes. Some considerations about sampling How did they recruit a representative group of customers (e.g., who arrived at the franchise at representative times of day)? Were some customers more likely to respond than others?
33 We can also use pnorm() to get exact p-values when using the normal approximation for a distribution of sample proportions (Which, as a reminder, follows directly from combining expected value and variance of a process with the central limit theorem!)
34 Case study: focus group A user experience researcher wants to know if all users prefer one website layout (Layout A) over another (Layout B). He samples n = 64 customers and asks them which layout they prefer. He finds a sample proportion of p =.68 prefer Layout A, and wants to know if this is evidence that there is a preference in the population of all users, using α =.01. State two opposing hypotheses: H0: π =.50 HA: π.50 Is our sample proportion likely if H0 is true?
35 Case study: focus group What types of sample means would we get if π =.50, n = 64? the central limit theorem tells us that the distribution of sample proportions will: have a mean of μ M = π =.50 have a standard deviation of σ M = (π*(1-π) / n) = (.5*.5/64) =.0625 be approximately normally distributed why? What is the probability of getting a sample proportion as extreme as our sample mean of p =.68, if H 0 is true? (1) convert x to a z-score z = (x - μ M ) / σ M = ( ) /.0625 =.18 /.0625 = 2.88 (2) use pnorm() to find (two-tailed) p-value > pnorm(-2.88) * 2 z = z = [1] z = 0
36 Case study: focus group A user experience researcher wants to know if all users prefer one website layout (Layout A) over another (Layout B). He samples n = 64 customers and asks them which layout they prefer. He finds a sample proportion of p =.68 prefer Layout A, and wants to know if this is evidence that there is a preference in the population of all users, using α =.01. State two opposing hypotheses: H0: π =.50 HA: π.50 Is our sample proportion likely if H0 is true? No, p <.01
37 This time Recap: how do we describe our newest sampling distribution, a distribution of sample means? How can we use distributions of sample means to make inferences? What s the difference between a z-test and a t-test? How do we measure effect size for single means?
38 Case study: vertical-horizontal illusion The vertical line is 2 inches. How long is the horizontal line? They are the same length! After learning that the vertical line is 2 in., a sample of n = 25 participants estimate the length the horizontal line, and give a mean estimate of x = 1.7. Do we have evidence that, in the entire population, people misjudge the relative lengths of the two lines? State two hypotheses: H0: people judge the lines to be of equal length, μ = 2 HA: people do not judge the lines to be of equal length, μ 2
39 t-statistic (vs. z-statistic) z = (x - μx ) / σx t = (x - μx ) / sx z = (x - μ) / (σ / n) t = (x - μ) / (s / n) z = (M - μm) / (σ 2 / n) t = (M - μm) / (s 2 / n) problem: z-statistic requires population standard deviation for H0 solution: get our best estimate of the population standard deviation
40 t-statistic (vs. z-statistic) z = (x - μx ) / σx t = (x - μx ) / sx z = (x - μ) / (σ / n) t = (x - μ) / (s / n) z = (M - μm) / (σ 2 / n) t = (M - μm) / (s 2 / n) problem: z-statistic requires population standard deviation for H0 solution: get our best estimate of the population standard deviation
41 What determines whether we calculate a z-statistic or a t-statistic? Do we know the population standard deviation for H0? yes z-statistic no t-statistic Note: in the case of a proportion, both the mean and variance are determined by π we always know the population standard deviation for H0 always use z-statistic
42 Careful! Standard deviation Standard error of the mean σ = σ 2 = (SS / N) σx = σ / n = (σ 2 / n) s = s 2 = (SS / (n - 1)) sx = s / n = (s 2 / n) typical distance between a single score and the population mean typical distance between a sample mean and the population mean used to convert single scores to z-scores filllllllllllllllllllllllllllllllllllllllller used to convert sample means to z-scores, often called z-statistics
43 Case study: vertical-horizontal illusion The vertical line is 2 inches. How long is the horizontal line? They are the same length! After learning that the vertical line is 2 in., a sample of n = 25 participants estimate the length the horizontal line, and give a mean estimate of x = 1.7 with a standard deviation of s =.5. Do we have evidence that, in the entire population, people misjudge the relative lengths of the two lines, using α =.05? State two hypotheses: H0: people judge the lines to be of equal length, μ = 2 HA: people do not judge the lines to be of equal length, μ 2
44 Case study: vertical-horizontal illusion What is the probability of getting a sample mean as extreme as our sample mean of x = 1.7, if H 0 is true (if μ = 2)? (1) convert x to a t-statistic t = (x - μ M ) / s M = (x - μ) / (s / n) = (1.7-2) / (.5 / 25) = -.3 /.1 = -3.0 (2) use pt() to find (two-tailed) p-value > pt(-3.0, df = 24) * 2 [1]
45 Distributions of t-statistics z = (x - μx ) / σx t = (x - μx ) / sx z = (x - μ) / (σ / n) t = (x - μ) / (s / n) Consider z-and t-statistics across different samples Numerator (same): x will vary across samples (sampling error) Denominator (different): z: σ does not vary across samples (defined by population) t: s does vary across samples (sampling error) added source of variability in t-statistic t-statistics are more variable than z-statistics t-statistics are not quite normally distributed heavier tails
46 Distributions of t-statistics Student s t-distribution a family of distributions Consider z-and t-statistics across different samples Numerator (same): x will vary across samples (sampling error) Denominator (different): z: σ does not vary across samples (defined by population) t: s does vary across samples (sampling error) added source of variability in t-statistic t-statistics are more variable than z-statistics t-statistics are not quite normally distributed heavier tails
47 Distributions of t-statistics Student s t-distribution a family of distributions Consider z-and t-statistics across different samples Numerator (same): x will vary across samples (sampling error) Denominator (different): z: σ does not vary across samples (defined by population) t: s does vary across samples (sampling error) added source of variability in t-statistic t-statistics are more variable than z-statistics t-statistics are not quite normally distributed heavier tails
48 Distributions of t-statistics what will determine variability of s? larger sample size s becomes less variable (variability of sampling distribution of s decreases) (law of large numbers) Consider z-and t-statistics across different samples Numerator (same): x will vary across samples (sampling error) Denominator (different): z: σ does not vary across samples (defined by population) t: s does vary across samples (sampling error) added source of variability in t-statistic t-statistics are more variable than z-statistics t-statistics are not quite normally distributed heavier tails
49 Distributions of t-statistics t-distribution(s) mean of 0 symmetrical bell-shaped more probability in the tails ( heavier tails ) scores z-distribution (normal) t-distribution (small sample size) t-distribution (medium sample size) t-distribution (large sample size) with larger sample size approaches a normal distribution
50 Distributions of t-statistics t-distribution(s) mean of 0 symmetrical bell-shaped more probability in the tails ( heavier tails ) scores z-distribution (normal) t-distribution (small sample size) t-distribution (medium sample size) t-distribution (large sample size) with larger sample size approaches a normal distribution
51 Distributions of t-statistics t-distribution(s) mean of 0 symmetrical bell-shaped more probability in the tails ( heavier tails ) scores z-distribution (normal) t-distribution (small sample size) t-distribution (medium sample size) t-distribution (large sample size) with larger sample size approaches a normal distribution
52 Degrees of freedom Rather than defining t-distributions by sample size, we define them by degrees of freedom (df) the number of values in the sample that are free to vary Suppose we have a sample of n = 3 with x = 5 sample 1 sample 2 sample 3 x1 x2 x3
53 Degrees of freedom Rather than defining t-distributions by sample size, we define them by degrees of freedom (df) the number of values in the sample that are free to vary Suppose we have a sample of n = 3 with x = 5 sample 1 sample 2 sample 3 x1 2 x2 6 x3?
54 Degrees of freedom Rather than defining t-distributions by sample size, we define them by degrees of freedom (df) the number of values in the sample that are free to vary Suppose we have a sample of n = 3 with x = 5 sample 1 sample 2 sample 3 x1 2 x2 6 x3 7
55 Degrees of freedom Rather than defining t-distributions by sample size, we define them by degrees of freedom (df) the number of values in the sample that are free to vary Suppose we have a sample of n = 3 with x = 5 sample 1 sample 2 sample 3 x1 2 1 x2 6 3 x3 7?
56 Degrees of freedom Rather than defining t-distributions by sample size, we define them by degrees of freedom (df) the number of values in the sample that are free to vary Suppose we have a sample of n = 3 with x = 5 sample 1 sample 2 sample 3 x1 2 1 x2 6 3 x3 7 11
57 Degrees of freedom Rather than defining t-distributions by sample size, we define them by degrees of freedom (df) the number of values in the sample that are free to vary Suppose we have a sample of n = 3 with x = 5 sample 1 sample 2 sample 3 x x x3 7 11?
58 Degrees of freedom Rather than defining t-distributions by sample size, we define them by degrees of freedom (df) the number of values in the sample that are free to vary Suppose we have a sample of n = 3 with x = 5 sample 1 sample 2 sample 3 x x x
59 Degrees of freedom Rather than defining t-distributions by sample size, we define them by degrees of freedom (df) the number of values in the sample that are free to vary Suppose we have a sample of n = 3 with x = 5 sample 1 sample 2 sample 3 x x x free to vary not free to vary
60 Degrees of freedom Rather than defining t-distributions by sample size, we define them by degrees of freedom (df) the number of values in the sample that are free to vary Suppose we have a sample of n = 3 with x = 5 sample 1 sample 2 sample 3 x x x free to vary not free to vary For a single sample of n scores, df = n - 1
61 Degrees of freedom Rather than defining t-distributions by sample size, we define them by degrees of freedom (df) the number of values in the sample that are free to vary Suppose we have a sample of n = 3 with x = 5 sample 1 sample 2 sample 3 x x x free to vary not free to vary Can think of s 2 = SS / (n-1) as s 2 = SS / df
62 Distributions of t-statistics t-distribution(s) mean of 0 symmetrical bell-shaped more probability in the tails ( heavier tails ) scores z-distribution (normal) t-distribution (df = 2) t-distribution (df = 5) t-distribution (df = 50) with larger sample size approaches a normal distribution
63 R: pt() pt(q, df, ncp, lower.tail = TRUE, log.p=false) by default, the pt() function returns the cumulative distribution function at quantile q, i.e., the cumulative relative frequency* at q, i.e., the probability of getting a value less than or equal to q q corresponds to a t-statistic in a t distribution, defined by df > pt(-1.96, df = 10) [1] > pt(1.96, df = 10) [1] t = t = 0 t = 0 t = > pt(1.96, df = 10, lower.tail = FALSE) [1] *equivalent to 1 - pt(1.96, df = 10) * note, I have been a bit sloppy in referring to this as cumulative frequency t = 0 t = +1.96
64 Case study: vertical-horizontal illusion The vertical line is 2 inches. How long is the horizontal line? They are the same length! After learning that the vertical line is 2 in., a sample of n = 25 participants estimate the length the horizontal line, and give a mean estimate of x = 1.7 with a standard deviation of s =.5. Do we have evidence that, in the entire population, people misjudge the relative lengths of the two lines, using α =.05? State two hypotheses: H0: people judge the lines to be of equal length, μ = 2 HA: people do not judge the lines to be of equal length, μ 2
65 Case study: vertical-horizontal illusion What is the probability of getting a sample mean as extreme as our sample mean of x = 1.7, if H 0 is true (if μ = 2)? (1) convert x to a t-statistic t = (x - μ M ) / s M = (x - μ) / (s / n) = (1.7-2) / (.5 / 25) = -.3 /.1 = -3.0 (2) use pt() to find (two-tailed) p-value > pt(-3.0, df = 24) * 2 [1] t = t = 0 t = p-value is less than alpha reject H0 H0: people judge the lines to be of equal length, μ = 2 HA: people do not judge the lines to be of equal length, μ 2
66 R: t.test() t.test(x, alternative=c( two.sided, less, greater ), mu=0) t.test performs many types of t-tests, for now we ll only focus on a few arguments x is a vector of values mu is the population mean (µ) if H 0 is true alternative specifies whether you are performing a two-tailed or one-tailed test
67 R: t.test() t.test(x, alternative=c( two.sided, less, greater ), mu=0) t.test performs many types of t-tests, for now we ll only focus on a few arguments x is a vector of values mu is the population mean (µ) if H 0 is true alternative specifies whether you are performing a two-tailed or one-tailed test
68 This time Recap: how do we describe our newest sampling distribution, a distribution of sample means? How can we use distributions of sample means to make inferences? What s the difference between a z-test and a t-test? How do we measure effect size for single means?
69 We rejected H 0, now what? We rejected H0 and concluded that μ 2 Statistical significance: if our null hypothesis is true, would it be plausible to observe x in our sample? Effect size: is the difference between x and μ from our null hypothesis (μ0) meaningful in the real world? Why not judge based on p-value, z-statistic, or t-statistic? influenced by difference between x and μ0 influenced by sample size (n)
70 Proportion variance explained (r 2 ) x μ0 x μ0 x μ total variance variance we can explain variance we cannot explain r 2 = variance we can explain / total variance r 2 = t 2 / (t 2 + df) r 2 is a proportion so it ranges from 0 to 1 in our illusion example, r 2 = (-3) 2 / ((-3) ) =.27
71 Recap The central limit theorem can tell us the mean, standard deviation, and shape of a distribution of sample means We can use a distribution of sample means to tell us whether a sample mean is plausible given a hypothesized population mean We use a z-test when we know the population standard deviation and a t-test when we estimate the population standard deviation We ve introduced r 2 as a new measure of effect size
72 Quiz 1 Nice work! median =.91, IQR =.15 mean =.86, sd =.13 Will post solutions / common mistakes soon If you are surprised by, or not happy with, your score, please see us ASAP to start planning for next quiz 10 count Note: left-most bin represents 60 (this is an open-ended distribution) score
73 Questions?
IOP 201-Q (Industrial Psychological Research) Tutorial 5
IOP 201-Q (Industrial Psychological Research) Tutorial 5 TRUE/FALSE [1 point each] Indicate whether the sentence or statement is true or false. 1. To establish a cause-and-effect relation between two variables,
More informationHypothesis Tests: One Sample Mean Cal State Northridge Ψ320 Andrew Ainsworth PhD
Hypothesis Tests: One Sample Mean Cal State Northridge Ψ320 Andrew Ainsworth PhD MAJOR POINTS Sampling distribution of the mean revisited Testing hypotheses: sigma known An example Testing hypotheses:
More informationChapter 11: Inference for Distributions Inference for Means of a Population 11.2 Comparing Two Means
Chapter 11: Inference for Distributions 11.1 Inference for Means of a Population 11.2 Comparing Two Means 1 Population Standard Deviation In the previous chapter, we computed confidence intervals and performed
More informationChapter 7 Sampling Distributions and Point Estimation of Parameters
Chapter 7 Sampling Distributions and Point Estimation of Parameters Part 1: Sampling Distributions, the Central Limit Theorem, Point Estimation & Estimators Sections 7-1 to 7-2 1 / 25 Statistical Inferences
More information1) 3 points Which of the following is NOT a measure of central tendency? a) Median b) Mode c) Mean d) Range
February 19, 2004 EXAM 1 : Page 1 All sections : Geaghan Read Carefully. Give an answer in the form of a number or numeric expression where possible. Show all calculations. Use a value of 0.05 for any
More informationThe Two-Sample Independent Sample t Test
Department of Psychology and Human Development Vanderbilt University 1 Introduction 2 3 The General Formula The Equal-n Formula 4 5 6 Independence Normality Homogeneity of Variances 7 Non-Normality Unequal
More informationReview: Population, sample, and sampling distributions
Review: Population, sample, and sampling distributions A population with mean µ and standard deviation σ For instance, µ = 0, σ = 1 0 1 Sample 1, N=30 Sample 2, N=30 Sample 100000000000 InterquartileRange
More information(# of die rolls that satisfy the criteria) (# of possible die rolls)
BMI 713: Computational Statistics for Biomedical Sciences Assignment 2 1 Random variables and distributions 1. Assume that a die is fair, i.e. if the die is rolled once, the probability of getting each
More informationFinancial Econometrics (FinMetrics04) Time-series Statistics Concepts Exploratory Data Analysis Testing for Normality Empirical VaR
Financial Econometrics (FinMetrics04) Time-series Statistics Concepts Exploratory Data Analysis Testing for Normality Empirical VaR Nelson Mark University of Notre Dame Fall 2017 September 11, 2017 Introduction
More informationμ: ESTIMATES, CONFIDENCE INTERVALS, AND TESTS Business Statistics
μ: ESTIMATES, CONFIDENCE INTERVALS, AND TESTS Business Statistics CONTENTS Estimating parameters The sampling distribution Confidence intervals for μ Hypothesis tests for μ The t-distribution Comparison
More informationSTA 320 Fall Thursday, Dec 5. Sampling Distribution. STA Fall
STA 320 Fall 2013 Thursday, Dec 5 Sampling Distribution STA 320 - Fall 2013-1 Review We cannot tell what will happen in any given individual sample (just as we can not predict a single coin flip in advance).
More informationCH 5 Normal Probability Distributions Properties of the Normal Distribution
Properties of the Normal Distribution Example A friend that is always late. Let X represent the amount of minutes that pass from the moment you are suppose to meet your friend until the moment your friend
More informationECON 214 Elements of Statistics for Economists
ECON 214 Elements of Statistics for Economists Session 7 The Normal Distribution Part 1 Lecturer: Dr. Bernardin Senadza, Dept. of Economics Contact Information: bsenadza@ug.edu.gh College of Education
More informationDistribution. Lecture 34 Section Fri, Oct 31, Hampden-Sydney College. Student s t Distribution. Robb T. Koether.
Lecture 34 Section 10.2 Hampden-Sydney College Fri, Oct 31, 2008 Outline 1 2 3 4 5 6 7 8 Exercise 10.4, page 633. A psychologist is studying the distribution of IQ scores of girls at an alternative high
More informationStatistics and Probability
Statistics and Probability Continuous RVs (Normal); Confidence Intervals Outline Continuous random variables Normal distribution CLT Point estimation Confidence intervals http://www.isrec.isb-sib.ch/~darlene/geneve/
More information2011 Pearson Education, Inc
Statistics for Business and Economics Chapter 4 Random Variables & Probability Distributions Content 1. Two Types of Random Variables 2. Probability Distributions for Discrete Random Variables 3. The Binomial
More informationNormal Probability Distributions
Normal Probability Distributions Properties of Normal Distributions The most important probability distribution in statistics is the normal distribution. Normal curve A normal distribution is a continuous
More informationLecture 8. The Binomial Distribution. Binomial Distribution. Binomial Distribution. Probability Distributions: Normal and Binomial
Lecture 8 The Binomial Distribution Probability Distributions: Normal and Binomial 1 2 Binomial Distribution >A binomial experiment possesses the following properties. The experiment consists of a fixed
More informationOverview/Outline. Moving beyond raw data. PSY 464 Advanced Experimental Design. Describing and Exploring Data The Normal Distribution
PSY 464 Advanced Experimental Design Describing and Exploring Data The Normal Distribution 1 Overview/Outline Questions-problems? Exploring/Describing data Organizing/summarizing data Graphical presentations
More informationExam 2 Spring 2015 Statistics for Applications 4/9/2015
18.443 Exam 2 Spring 2015 Statistics for Applications 4/9/2015 1. True or False (and state why). (a). The significance level of a statistical test is not equal to the probability that the null hypothesis
More informationUnit2: Probabilityanddistributions. 3. Normal distribution
Announcements Unit: Probabilityanddistributions 3 Normal distribution Sta 101 - Spring 015 Duke University, Department of Statistical Science February, 015 Peer evaluation 1 by Friday 11:59pm Office hours:
More informationNORMAL RANDOM VARIABLES (Normal or gaussian distribution)
NORMAL RANDOM VARIABLES (Normal or gaussian distribution) Many variables, as pregnancy lengths, foot sizes etc.. exhibit a normal distribution. The shape of the distribution is a symmetric bell shape.
More informationMeasures of Variation. Section 2-5. Dotplots of Waiting Times. Waiting Times of Bank Customers at Different Banks in minutes. Bank of Providence
Measures of Variation Section -5 1 Waiting Times of Bank Customers at Different Banks in minutes Jefferson Valley Bank 6.5 6.6 6.7 6.8 7.1 7.3 7.4 Bank of Providence 4. 5.4 5.8 6. 6.7 8.5 9.3 10.0 Mean
More informationLecture 5 - Continuous Distributions
Lecture 5 - Continuous Distributions Statistics 102 Colin Rundel January 30, 2013 Announcements Announcements HW1 and Lab 1 have been graded and your scores are posted in Gradebook on Sakai (it is good
More informationStatistics 431 Spring 2007 P. Shaman. Preliminaries
Statistics 4 Spring 007 P. Shaman The Binomial Distribution Preliminaries A binomial experiment is defined by the following conditions: A sequence of n trials is conducted, with each trial having two possible
More informationChapter 5. Continuous Random Variables and Probability Distributions. 5.1 Continuous Random Variables
Chapter 5 Continuous Random Variables and Probability Distributions 5.1 Continuous Random Variables 1 2CHAPTER 5. CONTINUOUS RANDOM VARIABLES AND PROBABILITY DISTRIBUTIONS Probability Distributions Probability
More informationLecture 2. Probability Distributions Theophanis Tsandilas
Lecture 2 Probability Distributions Theophanis Tsandilas Comment on measures of dispersion Why do common measures of dispersion (variance and standard deviation) use sums of squares: nx (x i ˆµ) 2 i=1
More informationNormal distribution Approximating binomial distribution by normal 2.10 Central Limit Theorem
1.1.2 Normal distribution 1.1.3 Approimating binomial distribution by normal 2.1 Central Limit Theorem Prof. Tesler Math 283 Fall 216 Prof. Tesler 1.1.2-3, 2.1 Normal distribution Math 283 / Fall 216 1
More informationCHAPTER 6. ' From the table the z value corresponding to this value Z = 1.96 or Z = 1.96 (d) P(Z >?) =
Solutions to End-of-Section and Chapter Review Problems 225 CHAPTER 6 6.1 (a) P(Z < 1.20) = 0.88493 P(Z > 1.25) = 1 0.89435 = 0.10565 P(1.25 < Z < 1.70) = 0.95543 0.89435 = 0.06108 (d) P(Z < 1.25) or Z
More informationKey Objectives. Module 2: The Logic of Statistical Inference. Z-scores. SGSB Workshop: Using Statistical Data to Make Decisions
SGSB Workshop: Using Statistical Data to Make Decisions Module 2: The Logic of Statistical Inference Dr. Tom Ilvento January 2006 Dr. Mugdim Pašić Key Objectives Understand the logic of statistical inference
More informationSampling Distributions
Sampling Distributions This is an important chapter; it is the bridge from probability and descriptive statistics that we studied in Chapters 3 through 7 to inferential statistics which forms the latter
More informationσ 2 : ESTIMATES, CONFIDENCE INTERVALS, AND TESTS Business Statistics
σ : ESTIMATES, CONFIDENCE INTERVALS, AND TESTS Business Statistics CONTENTS Estimating other parameters besides μ Estimating variance Confidence intervals for σ Hypothesis tests for σ Estimating standard
More informationData Distributions and Normality
Data Distributions and Normality Definition (Non)Parametric Parametric statistics assume that data come from a normal distribution, and make inferences about parameters of that distribution. These statistical
More informationChapter 7. Inferences about Population Variances
Chapter 7. Inferences about Population Variances Introduction () The variability of a population s values is as important as the population mean. Hypothetical distribution of E. coli concentrations from
More informationQuantitative Methods for Economics, Finance and Management (A86050 F86050)
Quantitative Methods for Economics, Finance and Management (A86050 F86050) Matteo Manera matteo.manera@unimib.it Marzio Galeotti marzio.galeotti@unimi.it 1 This material is taken and adapted from Guy Judge
More informationChapter 7: SAMPLING DISTRIBUTIONS & POINT ESTIMATION OF PARAMETERS
Chapter 7: SAMPLING DISTRIBUTIONS & POINT ESTIMATION OF PARAMETERS Part 1: Introduction Sampling Distributions & the Central Limit Theorem Point Estimation & Estimators Sections 7-1 to 7-2 Sample data
More informationStatistics 511 Supplemental Materials
Gaussian (or Normal) Random Variable In this section we introduce the Gaussian Random Variable, which is more commonly referred to as the Normal Random Variable. This is a random variable that has a bellshaped
More informationLecture 8: Single Sample t test
Lecture 8: Single Sample t test Review: single sample z-test Compares the sample (after treatment) to the population (before treatment) You HAVE to know the populational mean & standard deviation to use
More informationECON 214 Elements of Statistics for Economists 2016/2017
ECON 214 Elements of Statistics for Economists 2016/2017 Topic The Normal Distribution Lecturer: Dr. Bernardin Senadza, Dept. of Economics bsenadza@ug.edu.gh College of Education School of Continuing and
More informationChapter 15: Sampling distributions
=true true Chapter 15: Sampling distributions Objective (1) Get "big picture" view on drawing inferences from statistical studies. (2) Understand the concept of sampling distributions & sampling variability.
More informationData Analysis. BCF106 Fundamentals of Cost Analysis
Data Analysis BCF106 Fundamentals of Cost Analysis June 009 Chapter 5 Data Analysis 5.0 Introduction... 3 5.1 Terminology... 3 5. Measures of Central Tendency... 5 5.3 Measures of Dispersion... 7 5.4 Frequency
More informationMath 2311 Bekki George Office Hours: MW 11am to 12:45pm in 639 PGH Online Thursdays 4-5:30pm And by appointment
Math 2311 Bekki George bekki@math.uh.edu Office Hours: MW 11am to 12:45pm in 639 PGH Online Thursdays 4-5:30pm And by appointment Class webpage: http://www.math.uh.edu/~bekki/math2311.html Math 2311 Class
More informationIndependent-Samples t Test
Chapter 14 Aplia week 8 (Two independent samples) Testing hypotheses about means of two populations naturally occurring populations introverts vs. extroverts neuroticism experimentally defined (random
More informationChapter 7. Sampling Distributions
Chapter 7 Sampling Distributions Section 7.1 Sampling Distributions and the Central Limit Theorem Sampling Distributions Sampling distribution The probability distribution of a sample statistic. Formed
More informationBusiness Statistics 41000: Probability 4
Business Statistics 41000: Probability 4 Drew D. Creal University of Chicago, Booth School of Business February 14 and 15, 2014 1 Class information Drew D. Creal Email: dcreal@chicagobooth.edu Office:
More informationA continuous random variable is one that can theoretically take on any value on some line interval. We use f ( x)
Section 6-2 I. Continuous Probability Distributions A continuous random variable is one that can theoretically take on any value on some line interval. We use f ( x) to represent a probability density
More information8.2 The Standard Deviation as a Ruler Chapter 8 The Normal and Other Continuous Distributions 8-1
8.2 The Standard Deviation as a Ruler Chapter 8 The Normal and Other Continuous Distributions For Example: On August 8, 2011, the Dow dropped 634.8 points, sending shock waves through the financial community.
More informationThe normal distribution is a theoretical model derived mathematically and not empirically.
Sociology 541 The Normal Distribution Probability and An Introduction to Inferential Statistics Normal Approximation The normal distribution is a theoretical model derived mathematically and not empirically.
More informationDESCRIBING DATA: MESURES OF LOCATION
DESCRIBING DATA: MESURES OF LOCATION A. Measures of Central Tendency Measures of Central Tendency are used to pinpoint the center or average of a data set which can then be used to represent the typical
More informationHonors Statistics. Daily Agenda
Honors Statistics Daily Agenda 1. Review OTL C6#5 2. Quiz Section 6.1 A-Skip 35, 39, 40 Crickets The length in inches of a cricket chosen at random from a field is a random variable X with mean 1.2 inches
More informationHomework: Due Wed, Feb 20 th. Chapter 8, # 60a + 62a (count together as 1), 74, 82
Announcements: Week 5 quiz begins at 4pm today and ends at 3pm on Wed If you take more than 20 minutes to complete your quiz, you will only receive partial credit. (It doesn t cut you off.) Today: Sections
More information8.1 Estimation of the Mean and Proportion
8.1 Estimation of the Mean and Proportion Statistical inference enables us to make judgments about a population on the basis of sample information. The mean, standard deviation, and proportions of a population
More informationBoth the quizzes and exams are closed book. However, For quizzes: Formulas will be provided with quiz papers if there is any need.
Both the quizzes and exams are closed book. However, For quizzes: Formulas will be provided with quiz papers if there is any need. For exams (MD1, MD2, and Final): You may bring one 8.5 by 11 sheet of
More information10/1/2012. PSY 511: Advanced Statistics for Psychological and Behavioral Research 1
PSY 511: Advanced Statistics for Psychological and Behavioral Research 1 Pivotal subject: distributions of statistics. Foundation linchpin important crucial You need sampling distributions to make inferences:
More informationTWO μs OR MEDIANS: COMPARISONS. Business Statistics
TWO μs OR MEDIANS: COMPARISONS Business Statistics CONTENTS Comparing two samples Comparing two unrelated samples Comparing the means of two unrelated samples Comparing the medians of two unrelated samples
More informationChapter 9: Sampling Distributions
Chapter 9: Sampling Distributions 9. Introduction This chapter connects the material in Chapters 4 through 8 (numerical descriptive statistics, sampling, and probability distributions, in particular) with
More informationSection3-2: Measures of Center
Chapter 3 Section3-: Measures of Center Notation Suppose we are making a series of observations, n of them, to be exact. Then we write x 1, x, x 3,K, x n as the values we observe. Thus n is the total number
More informationLecture Slides. Elementary Statistics Tenth Edition. by Mario F. Triola. and the Triola Statistics Series. Slide 1
Lecture Slides Elementary Statistics Tenth Edition and the Triola Statistics Series by Mario F. Triola Slide 1 Chapter 6 Normal Probability Distributions 6-1 Overview 6-2 The Standard Normal Distribution
More informationLecture 6: Confidence Intervals
Lecture 6: Confidence Intervals Taeyong Park Washington University in St. Louis February 22, 2017 Park (Wash U.) U25 PS323 Intro to Quantitative Methods February 22, 2017 1 / 29 Today... Review of sampling
More informationSTAT Chapter 6: Sampling Distributions
STAT 515 -- Chapter 6: Sampling Distributions Definition: Parameter = a number that characterizes a population (example: population mean ) it s typically unknown. Statistic = a number that characterizes
More informationReview of commonly missed questions on the online quiz. Lecture 7: Random variables] Expected value and standard deviation. Let s bet...
Recap Review of commonly missed questions on the online quiz Lecture 7: ] Statistics 101 Mine Çetinkaya-Rundel OpenIntro quiz 2: questions 4 and 5 September 20, 2011 Statistics 101 (Mine Çetinkaya-Rundel)
More informationHonor Code: By signing my name below, I pledge my honor that I have not violated the Booth Honor Code during this examination.
Name: OUTLINE SOLUTIONS University of Chicago Graduate School of Business Business 41000: Business Statistics Special Notes: 1. This is a closed-book exam. You may use an 8 11 piece of paper for the formulas.
More informationStatistics & Statistical Tests: Assumptions & Conclusions
Degrees of Freedom Statistics & Statistical Tests: Assumptions & Conclusions Kinds of degrees of freedom Kinds of Distributions Kinds of Statistics & assumptions required to perform each Normal Distributions
More informationStatistics for Business and Economics
Statistics for Business and Economics Chapter 5 Continuous Random Variables and Probability Distributions Ch. 5-1 Probability Distributions Probability Distributions Ch. 4 Discrete Continuous Ch. 5 Probability
More informationContinuous Distributions
Quantitative Methods 2013 Continuous Distributions 1 The most important probability distribution in statistics is the normal distribution. Carl Friedrich Gauss (1777 1855) Normal curve A normal distribution
More informationAs you draw random samples of size n, as n increases, the sample means tend to be normally distributed.
The Central Limit Theorem The central limit theorem (clt for short) is one of the most powerful and useful ideas in all of statistics. The clt says that if we collect samples of size n with a "large enough
More informationAMS 7 Sampling Distributions, Central limit theorem, Confidence Intervals Lecture 4
AMS 7 Sampling Distributions, Central limit theorem, Confidence Intervals Lecture 4 Department of Applied Mathematics and Statistics, University of California, Santa Cruz Summer 2014 1 / 26 Sampling Distributions!!!!!!
More information4 Random Variables and Distributions
4 Random Variables and Distributions Random variables A random variable assigns each outcome in a sample space. e.g. called a realization of that variable to Note: We ll usually denote a random variable
More informationStatistics, Their Distributions, and the Central Limit Theorem
Statistics, Their Distributions, and the Central Limit Theorem MATH 3342 Sections 5.3 and 5.4 Sample Means Suppose you sample from a popula0on 10 0mes. You record the following sample means: 10.1 9.5 9.6
More informationThe Normal Probability Distribution
1 The Normal Probability Distribution Key Definitions Probability Density Function: An equation used to compute probabilities for continuous random variables where the output value is greater than zero
More information2 DESCRIPTIVE STATISTICS
Chapter 2 Descriptive Statistics 47 2 DESCRIPTIVE STATISTICS Figure 2.1 When you have large amounts of data, you will need to organize it in a way that makes sense. These ballots from an election are rolled
More informationIntroduction to Statistics I
Introduction to Statistics I Keio University, Faculty of Economics Continuous random variables Simon Clinet (Keio University) Intro to Stats November 1, 2018 1 / 18 Definition (Continuous random variable)
More informationDescribing Data: One Quantitative Variable
STAT 250 Dr. Kari Lock Morgan The Big Picture Describing Data: One Quantitative Variable Population Sampling SECTIONS 2.2, 2.3 One quantitative variable (2.2, 2.3) Statistical Inference Sample Descriptive
More informationProblem Set 4 Answer Key
Economics 31 Menzie D. Chinn Fall 4 Social Sciences 7418 University of Wisconsin-Madison Problem Set 4 Answer Key This problem set is due in lecture on Wednesday, December 1st. No late problem sets will
More informationMath 227 Elementary Statistics. Bluman 5 th edition
Math 227 Elementary Statistics Bluman 5 th edition CHAPTER 6 The Normal Distribution 2 Objectives Identify distributions as symmetrical or skewed. Identify the properties of the normal distribution. Find
More informationHomework: Due Wed, Nov 3 rd Chapter 8, # 48a, 55c and 56 (count as 1), 67a
Homework: Due Wed, Nov 3 rd Chapter 8, # 48a, 55c and 56 (count as 1), 67a Announcements: There are some office hour changes for Nov 5, 8, 9 on website Week 5 quiz begins after class today and ends at
More informationStat 139 Homework 2 Solutions, Fall 2016
Stat 139 Homework 2 Solutions, Fall 2016 Problem 1. The sum of squares of a sample of data is minimized when the sample mean, X = Xi /n, is used as the basis of the calculation. Define g(c) as a function
More informationIt is common in the field of mathematics, for example, geometry, to have theorems or postulates
CHAPTER 5 POPULATION DISTRIBUTIONS It is common in the field of mathematics, for example, geometry, to have theorems or postulates that establish guiding principles for understanding analysis of data.
More information1 Inferential Statistic
1 Inferential Statistic Population versus Sample, parameter versus statistic A population is the set of all individuals the researcher intends to learn about. A sample is a subset of the population and
More informationChapter 6 Continuous Probability Distributions. Learning objectives
Chapter 6 Continuous s Slide 1 Learning objectives 1. Understand continuous probability distributions 2. Understand Uniform distribution 3. Understand Normal distribution 3.1. Understand Standard normal
More informationLESSON 7 INTERVAL ESTIMATION SAMIE L.S. LY
LESSON 7 INTERVAL ESTIMATION SAMIE L.S. LY 1 THIS WEEK S PLAN Part I: Theory + Practice ( Interval Estimation ) Part II: Theory + Practice ( Interval Estimation ) z-based Confidence Intervals for a Population
More informationCommonly Used Distributions
Chapter 4: Commonly Used Distributions 1 Introduction Statistical inference involves drawing a sample from a population and analyzing the sample data to learn about the population. We often have some knowledge
More informationAverages and Variability. Aplia (week 3 Measures of Central Tendency) Measures of central tendency (averages)
Chapter 4 Averages and Variability Aplia (week 3 Measures of Central Tendency) Chapter 5 (omit 5.2, 5.6, 5.8, 5.9) Aplia (week 4 Measures of Variability) Measures of central tendency (averages) Measures
More informationChapter 6. y y. Standardizing with z-scores. Standardizing with z-scores (cont.)
Starter Ch. 6: A z-score Analysis Starter Ch. 6 Your Statistics teacher has announced that the lower of your two tests will be dropped. You got a 90 on test 1 and an 85 on test 2. You re all set to drop
More informationDr. Allen Back. Oct. 28, 2016
Dr. Allen Back Oct. 28, 2016 A coffee vending machine dispenses coffee into a paper cup. You re supposed to get 10 ounces of coffee., but the amount varies slightly from cup to cup. The amounts measured
More informationData Analysis and Statistical Methods Statistics 651
Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Lecture 14 (MWF) The t-distribution Suhasini Subba Rao Review of previous lecture Often the precision
More informationSTA 103: Final Exam. Print clearly on this exam. Only correct solutions that can be read will be given credit.
STA 103: Final Exam June 26, 2008 Name: } {{ } by writing my name i swear by the honor code Read all of the following information before starting the exam: Print clearly on this exam. Only correct solutions
More informationDepartment of Quantitative Methods & Information Systems. Business Statistics. Chapter 6 Normal Probability Distribution QMIS 120. Dr.
Department of Quantitative Methods & Information Systems Business Statistics Chapter 6 Normal Probability Distribution QMIS 120 Dr. Mohammad Zainal Chapter Goals After completing this chapter, you should
More informationLecture 9. Probability Distributions. Outline. Outline
Outline Lecture 9 Probability Distributions 6-1 Introduction 6- Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7- Properties of the Normal Distribution
More informationChapter 7: Point Estimation and Sampling Distributions
Chapter 7: Point Estimation and Sampling Distributions Seungchul Baek Department of Statistics, University of South Carolina STAT 509: Statistics for Engineers 1 / 20 Motivation In chapter 3, we learned
More informationBIOL The Normal Distribution and the Central Limit Theorem
BIOL 300 - The Normal Distribution and the Central Limit Theorem In the first week of the course, we introduced a few measures of center and spread, and discussed how the mean and standard deviation are
More informationOutline. Unit 3: Descriptive Statistics for Continuous Data. Outline. Reminder: the library metaphor
Unit 3: Descriptive Statistics for Continuous Data Statistics for Linguists with R A SIGIL Course Designed by Marco Baroni 1 and Stefan Evert 2 1 Center for Mind/Brain Sciences (CIMeC) University of Trento,
More informationUnit 5: Sampling Distributions of Statistics
Unit 5: Sampling Distributions of Statistics Statistics 571: Statistical Methods Ramón V. León 6/12/2004 Unit 5 - Stat 571 - Ramon V. Leon 1 Definitions and Key Concepts A sample statistic used to estimate
More informationUnit 5: Sampling Distributions of Statistics
Unit 5: Sampling Distributions of Statistics Statistics 571: Statistical Methods Ramón V. León 6/12/2004 Unit 5 - Stat 571 - Ramon V. Leon 1 Definitions and Key Concepts A sample statistic used to estimate
More informationLecture 1: Review and Exploratory Data Analysis (EDA)
Lecture 1: Review and Exploratory Data Analysis (EDA) Ani Manichaikul amanicha@jhsph.edu 16 April 2007 1 / 40 Course Information I Office hours For questions and help When? I ll announce this tomorrow
More informationSTAT 113 Variability
STAT 113 Variability Colin Reimer Dawson Oberlin College September 14, 2017 1 / 48 Outline Last Time: Shape and Center Variability Boxplots and the IQR Variance and Standard Deviaton Transformations 2
More informationAnnouncements. Unit 2: Probability and distributions Lecture 3: Normal distribution. Normal distribution. Heights of males
Announcements Announcements Unit 2: Probability and distributions Lecture 3: Statistics 101 Mine Çetinkaya-Rundel First peer eval due Tues. PS3 posted - will be adding one more question that you need to
More informationWeek 7. Texas A& M University. Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4
Week 7 Oğuz Gezmiş Texas A& M University Department of Mathematics Texas A& M University, College Station Section 3.2, 3.3 and 3.4 Oğuz Gezmiş (TAMU) Topics in Contemporary Mathematics II Week7 1 / 19
More informationLecture 9. Probability Distributions
Lecture 9 Probability Distributions Outline 6-1 Introduction 6-2 Probability Distributions 6-3 Mean, Variance, and Expectation 6-4 The Binomial Distribution Outline 7-2 Properties of the Normal Distribution
More informationData Analysis and Statistical Methods Statistics 651
Review of previous lecture: Why confidence intervals? Data Analysis and Statistical Methods Statistics 651 http://www.stat.tamu.edu/~suhasini/teaching.html Suhasini Subba Rao Suppose you want to know the
More information