Homework 2

Exploration 3.2 How Mobile Are We? 81 Note: To ensure full functionality, including saving text in input fields and adding images in image fields, please download and use Adobe Acrobat Reader (free) or any Adobe Acrobat DC product. Student Name: Exploration 3.2 How Mobile Are We? Did you move around a lot when you were younger? Suppose you wanted to estimate the proportion of students at your school that currently live in a different state from where they were born. How would you come up with an estimate? Do you think the students in your class are representative of students at your school on this issue? How would you estimate the proportion of adult Americans who currently live in a different state from where they were born? The General Social Survey (GSS) is a survey administered every two years to a random sample of adult Americans aged 18 and older. The survey asks a battery of questions focused on different social aspects of the population. In 2018, one of the questions was whether or not the adult lived in a different state from where they were born. Of the 2,348 surveyed that year, 36.2% replied that they do currently live in a different state from where they were born. 1. Identify the population and sample in this survey. Population: Sample: 2. Is it reasonable to believe that the sample of 2,348 adult Americans is representative of the larger population? Explain why or why not. 3. Explain why 36.2% is a statistic and not a parameter. What symbol would you use to represent it? 4. Identify (in words) the parameter that the Gallup organization was interested in estimating. Monkey Business Images/ Shutterstock.com

82 CHAPTER 3 Estimation: How Large Is the Effect? 5. Is it reasonable to conclude that exactly 36.2% of all adult Americans currently live in a different state from where they were born? Explain why or why not. 6. Although we expect π to be close to 0.362, we realize there may be other plausible values for the population proportion as well. First consider the value of 0.382. Is this a plausible value for π ? Use the One Proportion applet to simulate random samples of 2,348 people from such a population. ( Hint: Keep in mind that 0.382 is what we are assuming for the population proportion and 0.362 is the observed sample proportion.) What do you estimate for the two-sided p-value? Would you reject or fail to reject the null hypothesis at the 5% level of significance? 7. Also check the Summary Stats box and report the mean and standard deviation of this null distribution. 8. Now consider 0.50. Is this a plausible value for π ? Repeat #6 and record the mean and standard deviation for this null distribution as well. Clearly 0.50 is going to be “too far” from ˆ p = 0.362 to be plausible. But how far is too far? We could use the plausible values method from Section 3.1 to produce a confidence interval for the proportion of all adult Americans who currently live in a different state from where they were born. But that approach is somewhat cumbersome and time-intensive, so we’ll now learn some shortcut approaches. 9. Reconsider our first guess of π = 0.382. How many standard deviations is 0.362 from 0.382? ( Hint: Standardize the value by looking at the difference between 0.382 and 0.362 and divide by the standard deviation you found in #7.) You should notice that 0.382 and 0.362 are about 2 standard deviations apart and that the two-sided p-value is around 0.05, so this value (0.382) is close to the edge of values that can be considered plausible for π . Values between 0.362 and 0.382 are considered plausible and values larger than 0.382, or more than 2 standard deviations above 0.362, will not be plausible values for the population proportion.

84 CHAPTER 3 Estimation: How Large Is the Effect? b. Use this SD to produce a 95% confidence interval for π . ( Hint: Subtract the margin of error from p ˆ to determine the lower endpoint of the interval and then add the margin of error to ˆ p to determine the upper endpoint of the interval.) c. Interpret the confidence interval: You are 95% confident that what is between what two values? One limitation to this method is that it only applies for 95% confidence. What if we want- ed to be 90% or 99% confident instead? We can extend this 2SD method to a more general theory-based approach. Theory-Based Approach As we saw in Chapter 1, we don’t always need to simulate a sampling distribution—not if we can accurately predict what would happen if we were to sim- ulate. Instead, we can predict the standard deviation by the formula √ π ( 1 − π ) / n . But when constructing a confidence interval, we don’t have a hypothesized value of _____________ π , so to estimate this standard deviation, we will substitute the observed sample proportion. Definition An estimate of the standard deviation of a statistic, based on sample data, is called the _____________ standard error (SE) of the statistic. In this case √ p ( 1 − p ) / n is the standard error of a sample proportion ˆ ˆ p . ˆ 12. Calculate the standard error for this study. How does it compare to the standard deviations you found in #7 and #8? So a more general formula for using the 2SD method to estimate a population proportion would be ˆ _____________ p ± 2 √ ˆ p ( 1 − p ˆ ) / n But then how do we change the confidence level? The 2SD method was justified by saying 95% of samples yield a sample proportion within 2 standard deviations of the population proportion. If we want to be more confident that the parameter is within our margin of error, we can create a larger margin of error by increasing the multiplier. In fact a multiplier of 2.576 gives us a 99% confidence level, whereas a multipli- er of 1.645 gives us only 90% confidence.

Exploration 3.2 How Mobile Are We? 85 13. We will rely on technology to find the multiplier appropriate for our confidence level. a. In the Theory-Based Inference applet, specify the sample size ( n ) of 2,348 and the sample proportion of 0.362 and press Calculate . (The applet will fill in the count or specify the count of 850 and the applet will fill in the sample proportion.) b. Check the box for Confidence interval , confirm the confidence level is 95% and press Calculate CI to generate a theory-based confidence interval. Report the 95% theory-based confidence interval. 14. Is this theory-based confidence interval similar to the one you obtained using the 2SD method? Validity Condition The theory-based approach for finding a confidence interval for π (called a one- sample z-interval ) is considered valid if there are at least 10 observational units in each category of the categorical variable (i.e., at least 10 successes and at least 10 failures). Because we have a large sample size here, the theory-based approach should produce very similar results to the plausible values method and the 2SD method. In such a case, the theory- based approach is often the most convenient, especially if our confidence level is not 95%. 15. Change the confidence level in the applet from 95% to 99% and press the Calculate CI button again. Report the 99% confidence interval given by the applet. How does it compare to the 95% interval? ( Compare both the midpoint of the interval = ( lower endpoint  + upper endpoint )/2 and the margin of error = ( upper endpoint − lower endpoint )/2.) n: 2348 count: 850 sample p ˆ: 0.362 95 % Calculate CI