Chapter 9 Chapter Friday, June 4 th

Transcription

1 Chapter 9 Chapter 10 Sections and Friday, June 4 th

2 Parameter and Statisticti ti Parameter is a number that is a summary characteristic of a population Statistic, is a number that is computed from a sample of values taken from a larger population. Sample estimate or estimate is used when a statistic is used to estimate the unknown value of the population parameter

3 This Chapter In this Chapter we will learn the five main parameters that are of great interests In later Chapters (10-13) 13) we will learn the statistical procedures that are used to make conclusions about population parameters based on the sample statistics. Those procedures are called statistical inference procedures, and the main ideas are Confidence intervals and Hypothesis Testing.

4 Five Parameters Categorical variables (2): Parameter Statistic Population proportion: p pˆ Difference in two proportions: p p pˆ pˆ

5 Five Parameters Quantitative variables(3): Parameter Statistic Population mean: Difference in means for dependent populations: x d d d Difference in means for independent populations: x x

6 Standard d deviation Sometimes we will need the standard deviation of the population p and of the sample. Population parameter is denoted with Sample statistic is denoted with s or ˆ

7 Sampling Distributions ib ti What is the main difference between population p parameters and sample statistics? Population parameter: A fixed value that is impossible to know (why?) Sample statistic: A value that estimates the parameter but depends on the sample. Different samples give different values.

8 Sampling distribution ib ti Now since the sample statistics depend on the sample that we choose then this means there is some kind of uncertainty in them. So, we can say that they follow a distribution. This distribution ib ti is called sampling distribution ib ti Sampling distribution for a statistic is the probability bilit of possible values of the statistic ti ti for repeated samples of the same size taken from the same distribution

9 Let s play Let s say I want to see the average number of credits students in stat 200 section 103 have this semester. Let s say there are 1000 students in stat 200 section 103 this semester so that I am not able to sample all the populations. Instead I decide to take 8 samples of 5 people.

10 Common features All sampling distributions we will see today can be approximated by normal distribution, if certain conditions are met. Two features describe normal distribution. What are they? Mean Standard deviation

11 Common features The mean of sampling distributions is the mean value of a sample statistic over all possible random samples The standard deviation of a sampling distribution measures how the values of the sample statistic vary across different samples from the same population. The estimated value of the standard deviation of a statistic is called standard error

12 Sampling distribution ib ti 1 Sampling Distribution for one sample proportion p It is approximately normal with Mean = p Standard deviation = Standard deviation = p 1 p Example 1: Tossing a coin. Example 2: Find the proportion of people to vote for candidate A in the general elections n

13 Sampling distribution ib ti 1 What is the problem in previous slide? Estimated Sampling Distribution for one sample proportion p It is approximately normal with ˆp Mean = Standard d d error = pˆ 1 n pˆ

14 Example I am interviewing 100 students in PSU to tell me if they are right handed or left handed. 80 of them answer that they are right handed. What is the population proportion of right handed students? What is the sample proportion of right handed students? What is the sampling distribution of the proportion of right handed students?

15 Conditions The conditions that they have to be met in order to have a normal curve for the standard distribution are the following: np 10 and n 1 p 10 Of course if p is unknown and we are working with the estimated sampling distributions the conditions are as follows: npˆ 10 and n 1 pˆ 10

16 Forget for a while Sampling distributions Let s go back in Chapter 3 where we learn the margin of error and the Confidence Interval. Example: I asked 100 PSU students if they are right handed or left handed. 80 of them answered that t they are right handed. d What is the estimated proportion of right handed student in PSU? What is the 95% Confidence Interval?

17 Estimating The previous example shows the two ways of estimating: Point estimate: You estimate that a proportion is a fixed number. Confidence Interval: You estimate that the proportion with a probability falls into an interval.

18 Confidence interval A confidence interval is an interval of values computed from sample data that is likely to include the true population value

19 Confidence interval A confidence interval is always accompanied by a percentage; for example 95% confidence Interval or 90% confidence interval etc This percentage is called confidence level The percentage is interpreted like this: Example (with right handed people) Based on this sample we are 95% confident that t the proportion of right handed student in PSU is between 0.7 and 0.9.

20 How to calculate Confidence Interval There is a general rule of calculating Confidence intervals: Sampleestimate Multiplier * Standard error The difficult part is to find the multiplier We will see in each case how to find it.

21 Now let s go back to sample proportion Sampling Distribution for one sample proportion p It is approximately normal with Mean = p p 1 p Standard deviation = n if we do not know the actual p we have pˆ ˆ standard error= 1 p n How do we find confidence intervals in this case?

22 Example Let say that I am asking 200 PSU students if they believe that marijuana should be legalized. 48 of them answer Yes. What is the estimate of the proportion of students that say they believe marijuana should be legalized? What is the sampling distribution of that proportion? p What is a 95% Confidence Interval for that proportion?

23 Using the Tables If you are looking for a multiplier: Take the confidence level, let s say 95% Take the rest of it 100%-95%=5% and divide it by 2. So we have 5% / 2 =2.5% Add 95%+2.5% =97.5% (Why do we do this procedure?) Look in the probabilities reported in the Table (those are the four decimal points number that the table is full of). Try to find the one closest to 97.5% or 0.975

24 Using the Tables When you find the probability. Move horizontally and report the first number in that row; for that is 1.9 and then move vertically and report the first number in that column that is.06 This will give us =1.96 So the multiplier in this interval will be 1.96 Do the same thing for a 90% confidence interval

25 How to interpret Confidence Intervals We are 95% confident that the proportion of interest will be in the range that is given by the 95% Confidence Interval We are 90% confident that the proportion of interest will be in the range given by the 90% Confidence Interval

26 Size of Confidence intervals Is affected by: Sample size: Larger sample size produce narrower intervals (why?) Level of confidence intervals: Larger level of confidence Intervals gives wider (why?) Natural variability among the populations. More variability results in wider intervals

27 Sampling Distribution ib ti 2 We have learn the sampling distribution and how to calculate confidence intervals for one sample proportion The next step is to do the same thing for the difference of two proportions. We repeat one of the first slides of the lecture to remind ourselves what is going on

28 Two categorical Parameters Categorical variables (2): Parameter Statistic Population proportion: p pˆ Difference in two proportions: p p p ˆ p ˆ

29 Sampling distribution ib ti 2 p 1 We denote with the proportion of the first population p and with p 2 the proportion p of the second population. The estimated sample proportions are denoted with ˆp 1 and ˆp 2

30 Sampling distribution for the difference of two population proportions p The sampling Distribution for the difference of two population p proportions p is normal with Mean: p p 1 2 Standard deviation= p 1 p p 1 p n n

31 Sampling distribution for the difference of two population proportions p In case the two population proportions are unknown we have again normal distribution with: Mean= pˆ pˆ 1 2 Standard error= 1 1 pˆ pˆ pˆ pˆ n n

32 Conditions For the difference of two population proportions p to have a normal distribution we need the following conditions: n p 10, n 1 p 10, n p 10 and n 1 p Now if the population proportion is unknown and we will use the sample estimate the conditions are: n pˆ 10, n 1 pˆ 10, n pˆ 10and n 1 pˆ

33 Example I asked 200 students in Penn State if they believe that marijuana should be legalized. From 120 males I asked I got 36 Yes and from the 80 female I got 12 Yes. What is the estimate for the proportion of male students that believe marijuana should be legalized? What is the estimate t of the proportion of female students t that believe marijuana should be legalized? What is the estimate of the difference of the two proportions?

34 Example Now calculate the sampling distribution of the difference of the two proportions p

35 Example Now calculate a 98% Confidence interval for the difference of the two proportions p