Confidence Intervals for the Mean. When σ is known

Transcription

1 Confidence Intervals for the Mean When σ is known

2 Objective Find the confidence interval for the mean when s is known.

3 Intro Suppose a college president wishes to estimate the average age of students attending classes this semester. The president could select a random sample of 100 students and find the average age of these students, say, 22.3 years. From the sample mean, the president could infer that the average age of all the students is 22.3 years. This type of estimate is called a.

4 Determine why other measures of central tendency, such as the median and mode, are not used to estimate the population mean. Sample measures (i.e., statistics) are used to estimate population measures (i.e., parameters). These statistics are called.

5 A Good Estimator The estimator should be an estimator. That is, the expected value or the mean of the estimates obtained from samples of a given size is equal to the parameter being estimated. The estimator should be ; i.e., as sample size increases, the value of the estimator approaches the value of the parameter estimated. The estimator should be a estimator. That is, of all the statistics that can be used to estimate a parameter, the relatively efficient estimator has the smallest variance.

6 Confidence Intervals An of a parameter is an interval or a range of values used to estimate the parameter. This estimate may or may not contain the value of the parameter being estimated. The parameter is specified as being between two values. For example, an interval estimate for the average age of all students might be 26.9 m 27.7, or years.

7 Confidence Intervals The of an interval estimate of a parameter is the probability that the interval estimate will contain the parameter, assuming that a large number of samples are selected and that the estimation process on the same parameter is repeated. A is a specific interval estimate of a parameter determined by using data obtained from a sample and by using the specific confidence level of the estimate.

8 Confidence Intervals Three common confidence intervals are used: the 90, the 95, and the 99% confidence intervals. The central limit theorem states that when the sample size is large, approximately 95% of the sample means taken from a population and same sample size will fall within 1.96 standard errors of the population mean, that is,

9 The is the maximum likely difference between the point estimate of a parameter and the actual value of the parameter. Rounding Rule for a Confidence Interval for a Mean When you are computing a confidence interval for a population mean by using raw data, round off to one more decimal place than the number of decimal places in the original data.

10 Example #1: Days It Takes to Sell an Aveo A researcher wishes to estimate the number of days it takes an automobile dealer to sell a Chevrolet Aveo. A sample of 50 cars had a mean time on the dealer s lot of 54 days. Assume the population standard deviation to be 6.0 days. Find the best point estimate of the population mean and the 95% confidence interval of the population mean.

11 Example #1: Days It Takes to Sell an Aveo

12 Example #2: Ages of Automobiles A survey of 30 adults found that the mean age of a person s primary vehicle is 5.6 years. Assuming the standard deviation of the population is 0.8 year, find the best point estimate of the population mean and the 99% confidence interval of the population mean.

13 Example #2: Ages of Automobiles

14 You Do! Data on a sample of the assets (in millions of dollars) of 30 credit unions in southwestern Pennsylvania has been used. The mean is Assume the standard deviation of the population is Find the 90% confidence interval of the mean.

15 You Do!

16 Do Now / Warm-up A recent study showed that high school students experience an average of 4.8 hours per day of distractions (phone calls, e- mails, side conversations, etc.). A random sample of 100 high school students in P.K. Yonge for the Statistics instructor found that these students were distracted an average of 3.6 hours per day and the population standard deviation was 45 minutes. Estimate the true mean population distraction time with 95% confidence, and compare your answer to the results of the study.

17 Confidence Intervals for the Mean Sample size

18 Objective Determine the minimum sample size for finding a confidence interval for the mean.

19 Sample Size Sample size determination is closely related to statistical estimation. In stats and research, quite often you ask: How large a sample is necessary to make an accurate estimate? The answer is not simple, since it depends on three things: #1: #2: #3: How close to the true mean do you want to be (2 units, 5 units, etc.)? How confident do you wish to be (90, 95, 99%, etc.)? Assumption in this topic: population standard deviation of the variable is known or has been estimated from a previous study.

20 Sample Size Formula The formula for sample size is derived from the maximum error of the estimate formula and solved for n.

21 Depth of a River A scientist wishes to estimate the average depth of a river. He wants to be 99% confident that the estimate is accurate within 2 feet. From a previous study, the standard deviation of the depths measured was 4.38 feet.

22 Depth of a River A scientist wishes to estimate the average depth of a river. He wants to be 99% confident that the estimate is accurate within 2 feet. From a previous study, the standard deviation of the depths measured was 4.38 feet. Round the value up to 32. Therefore, to be 99% confident that the estimate is within 2 feet of the true mean depth, the scientist needs at least a sample of 32 measurements.

23 Health Insurance Coverage for Children A federal report stated that 88% of children under age 18 were covered by health insurance in How large a sample is needed to estimate the true proportion of covered children with 90% confidence with a confidence interval 0.05 wide?

24 Health Insurance Coverage for Children A federal report stated that 88% of children under age 18 were covered by health insurance in How large a sample is needed to estimate the true proportion of covered children with 90% confidence with a confidence interval 0.05 wide? Solution: It does not give you a standard deviation, so for now, you can t determined! Answer is 460, because there is a way to estimate when standard deviation is not known and information is given in proportions (for example, 88% of the children) n = pq z E 2 = (0. 88)(0. 12) =

25 Health Insurance Coverage for Children A federal report stated that 88% of children under age 18 were covered by health insurance in How large a sample is needed to estimate the true proportion of covered children with 90% confidence with a confidence interval 0.05 wide?

26 Health Insurance Coverage for Children A federal report stated that 88% of children under age 18 were covered by health insurance in How large a sample is needed to estimate the true proportion of covered children with 90% confidence with a confidence interval 0.05 wide? Solution: It does not give you a standard deviation, so on Wednesday, you weren t able to determine it! Answer is 460, because there is a way to estimate when standard deviation is not known and information is given in proportions (for example, 88% of the children) n = pq z E 2 = (0. 88)(0. 12) =

27 Confidence Intervals for the Mean When σ is unknown

28 Objective Find the confidence interval for the mean when σ is unknown.

29 What we know so far When σ is known and the sample size is 30 or more, or the population is normally distributed if sample size is less than 30, the confidence interval for the mean can be found by using the z distribution

30 However, Most of the time, the value of σ is not known so it must be estimated by using s, namely, the standard deviation of the sample. When s is used, especially when the sample size is small, critical values greater than the values for the z distribution are used in confidence intervals in order to keep the interval at a given level, such as the 95%. These values are taken from the Student t distribution, most often called the.

31 Characteristics of the t distribution The t distribution shares some characteristics of the normal distribution: It is bell-shaped. It is symmetric about the mean. The mean, median, and mode are equal to 0 and are located at the center of the distribution. The curve never touches the x axis.

32 Characteristics of the t distribution The t distribution differs from the standard normal distribution in: The variance is greater than 1. The t distribution is actually a family of curves based on the concept of degrees of freedom, which is related to sample size. As the sample size increases, the t distribution approaches the standard normal distribution.

33 Degrees of Freedom Many statistical distributions use the concept of degrees of freedom, and the formulas for finding the degrees of freedom vary for different statistical tests. The degrees of freedom are the number of values that are free to vary after a sample statistic has been computed, and they tell the researcher which specific curve to use when a distribution consists of a family of curves. The symbol d.f. will be used for degrees of freedom. The degrees of freedom for a confidence interval for the mean are found by subtracting 1 from the sample size. That is, d.f. = n 1.

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35 Find the t α/2 value for a 95% confidence interval when the sample size is 22.

36 Sleeping Time Ten randomly selected people were asked how long they slept at night. The mean time was 7.1 hours, and the standard deviation was 0.78 hour. Find the 95% confidence interval of the mean time. Assume the variable is normally distributed.

37 Sleeping Time

38 Home Fires Started by Candles Data from the National Fire Protection Association on a sample of the number of home fires started by candles for the past several years have a mean of 7,041.4 and a standard deviation of 1, Find the 99% confidence interval for the mean number of home fires started by candles each year.

39 Home Fires Started by Candles

40 DO NOW Find your number in the lists posted in the bulletin boards (back wall and main entrance). Find the seat with your number and that will be your seat from now on. SILENCE YOUR PHONE and put it in the pocket that has your number in the bulletin board (back wall). NO EXCEPTIONS and NO PHONES in backpack. If I see your phone, I will take it!!! No food allowed in my room. Finish your food outside before you enter my classroom.

41 Confidence Intervals and Sample Size for Proportions

42 Objectives Find the confidence interval for a proportion. Determine the minimum sample size for finding a confidence interval for a proportion.

43 Proportions In USA TODAY it was revealed that 12% of the pleasure boats in the United States were named Serenity. The parameter 12% is called a. It means that of all the pleasure boats in the United States, 12 out of every 100 are named Serenity. A proportion represents a of a whole. It can be expressed as a fraction, decimal, or percentage. Proportions can also represent probabilities. In this case, if a pleasure boat is selected at random, the probability that it is called Serenity is.

44 For example, In a study, 200 people were asked if they were satisfied with their job or profession; 162 said that they were. If p represents the proportion of those surveyed who were satisfied with their job or profession, determine the value of q*.

45 For example, In a study, 200 people were asked if they were satisfied with their job or profession; 162 said that they were. If p represents the proportion of those surveyed who were satisfied with their job or profession, determine the value of q*. p, = = The proportion of people who did not respond favorably when asked if they were satisfied with their job or profession constituted q,. q, = 1 p, = = 0. 19

46 Air Conditioned Households In a recent survey of 150 households, 54 had central air conditioning. Find p and q*, where p is the proportion of households that have central air conditioning.

47 Air Conditioned Households In a recent survey of 150 households, 54 had central air conditioning. Find p and q*, where p is the proportion of households that have central air conditioning. p, = = The proportion of households that do not have central air conditioned constituted q,. q, = 1 p, = = 0. 64

48 Confidence Intervals To construct a confidence interval about a proportion, you must use the maximum error of the estimate, which is E = z α/2 p,q, Remember that p n 5 and q*n 5. When you find E you add it to and subtract it from the proportion. Rounding Rule for a Confidence Interval for a Proportion Round off to three decimal places. n

49 Male Nurses A sample of 500 nursing applications included 60 from men. Find the 90% confidence interval of the true proportion of men who applied to the nursing program.

50 Male Nurses A sample of 500 nursing applications included 60 from men. Find the 90% confidence interval of the true proportion of men who applied to the nursing program. E = z α/2 p,q, n = (0. 88) 500 = true proportion of men who applied to the nursing program. This is 0.12 ± error < p < Hence, you can be 90% confident that the percentage of applicants who are men is between 9.6 and 14.4%.

51 Religious Books A survey of 1721 people found that 15.9% of individuals purchase religious books at a Christian bookstore. Find the 95% confidence interval of the true proportion of people who purchase their religious books at a Christian bookstore.

52 Religious Books A survey of 1721 people found that 15.9% of individuals purchase religious books at a Christian bookstore. Find the 95% confidence interval of the true proportion of people who purchase their religious books at a Christian bookstore. You can say with 95% confidence that the true percentage is between and 17. 6%.

53 Sample Size for Proportions Just a reminder of what we have done so far with this topic To find the sample size needed to determine a confidence interval about a proportion, use this formula: n = p,q, z α/2 E 2

54 Home Computers A researcher wishes to estimate, with 95% confidence, the proportion of people who own a home computer. A previous study shows that 40% of those interviewed had a computer at home. The researcher wishes to be accurate within 2% of the true proportion. Find the minimum sample size necessary.

55 Home Computers A researcher wishes to estimate, with 95% confidence, the proportion of people who own a home computer. A previous study shows that 40% of those interviewed had a computer at home. The researcher wishes to be accurate within 2% of the true proportion. Find the minimum sample size necessary. 2, 305 executives to ask!

56 That s it for Unit 4!!! Time to work!!! UNIT 4 EXAM is on Thursday, November 30, The exam will have 10 questions. Right after you finish the exam, I want you to work on your research proposal and share with me an update of your progress. For today: Exercises 1-8 are for you to work with your partner in your table. Exercise 9-10 are for you to work independently to prove that YOU CAN find the confidence interval for a proportion and that YOU CAN determine the minimum sample size for finding a confidence interval for a proportion.