CHAPTER 21A. What is a Confidence Interval?

Transcription

1 CHAPTER 21A What is a Confidence Interval?

2 RECALL Parameter fixed, unknown number that describes the population Statistic known value calculated from a sample a statistic is used to estimate a parameter Sampling Variability different samples from the same population may yield different values of the sample statistic estimates from samples will be closer to the true values in the population if the samples are larger 2

3 RECALL Example: The amount by which the proportion obtained from the sample ( ˆp ) will differ from the true population proportion (p) rarely exceeds the margin of error. Sampling Distribution tells what values a statistic takes and how often it takes those values in repeated sampling. Example: sample proportions ( ˆp s) from repeated sampling would have a normal distribution with a certain mean and standard deviation. 3

4 SAMPLING DISTRIBUTION OF P-HAT An opinion poll asks an SRS of 1500 adults, Do you happen to jog? Suppose that the population proportion who jog is p = In a large number of samples, the sample proportion who answer Yes will follow the normal distribution. The mean of the sampling distribution will equal. The standard deviation of the sampling distribution will equal. 4

5 SAMPLING DISTRIBUTION OF P-HAT Sketch the curve from the previous slide and use it to answer the following questions. What is the probability that ˆp will take a value between and 0.159? What is the probability that does not lie between and 0.159? ˆp 5

6 ESTIMATING Statistical inference draws a conclusion about a population from evidence provided by a sample. Since we don t know the true value of the parameter (unless we do what?), we use a statistic to estimate the parameter. Recall that the statistic is calculated using our sample data. We then make a conclusion about the value of the parameter (remember confidence statements?). 6

7 ESTIMATING WITH 95% CONFIDENCE The quick method for our margin of error allowed us to estimate the parameter with 95% confidence. Recall that a 95% confidence interval is a range of numbers calculated from the sample data such that it is guaranteed to capture the true population parameter in 95% of the samples. What did the quick method look like? 7

8 SAMPLING DISTRIBUTION OF A SAMPLE PROPORTION (CHAPTER 18) If the sample is large enough, the sampling distribution of the sample proportion ( ˆp ) is approximately normal. This normal distribution has a mean equal to p (the population proportion). It has standard deviation equal to p *(1 p) n 8

9 NEW 95% CONFIDENCE INTERVAL FORMULA Recall from our empirical rule that about 95% of observations fall within 2 standard deviations of the mean when we have a normal distribution. So a 95% confidence interval for our sample proportion is given by p (1 ) 2 p p n 9

10 NEW 95% CONFIDENCE INTERVAL FORMULA But this gives us a problem. Notice that this depends on the unknown parameter p. What do we do? We can use the sample proportion in place of p. We also change the 2 to 1.96 to get the exact middle 95% of the normal distribution. ˆ(1 ˆ) ˆ 1.96 p p p n 10

11 EXAMPLE 21.1 The student newspaper at a college asks a SRS of 250 students, Do you favor eliminating Spring Break so the semester will end a week earlier? 100 of the 250 were in favor What is our sample proportion? Construct a 95% confidence interval to estimate the population proportion of satisfied new car purchasers. Use the new method & compare to the quick method. (Round your MOE to 3 decimals.) Use the results of the new method to make a 95% confidence statement. 11

12 EXAMPLE 21.2 The state of California was trying to estimate the proportion of college graduates that had engaged in binge drinking. A sample of 2166 found that 279 had engaged in binge drinking. 279 The sample proportion was pˆ The 95% confidence interval was ( ) (0.115, 0.143)

13 EXAMPLE 21.2 What if they take another sample from the same population? A second sample of 2166 found that 301 had engaged in binge drinking. 301 The sample proportion was pˆ The 95% confidence interval was ( ) (0.124,0.154)

14 EXAMPLE 21.2 Each new sample yields a new p-hat and a new confidence interval. If we sample forever, 95% of these intervals will capture the true parameter. Remember that probability describes the long run. If we make only 20 confidence intervals, it does not mean that exactly 95% of them will capture the true parameter. 14

15 REMINDERS We will finish chapter 21 next class. Chapter 21 homework is posted online and is due Thursday. Chapter 18 homework is due today. 15