Many of the survey questions focus on the used to pay for college, so the undergraduates in the sur ing for their parents. For example, each participant is asked to report how much of the parent's current income is used to pay for college. Do you think it is wise to combine responses across the parents and undergraduates? Are you fully aware of how much money your parents are spending and borrowing for college? The authors report overall averages and percents in their repon spond. Chapter 6 Introduction to Inference 350 We will also consider this a sample from one population but this is debatable certainly Average college savings fund contribution. One survey question asked ho much money from a college savings fund, such as a 529 plan, is used to pay an answer Nonresponse should ahways be considered as a source of bias. In this case, the nonresponse is very low, so we'll proceed by treating the n1593 sample as EXAMPLE 6.4 for college. Of the 1600 who were surveyed, n= 1593 provided a The average amount is $1768. It's very likely that this distribution is highly skewed to the right with many small amounts and a few very large amounts. Nevertheless, because the sample size is quite large, we can rely if it were an unbiased sample. on the central limit theorem to assure us that the confidence interval based on the Normal distribution will be a good approximation. Let's compute an approximate 95% confidence interval for the true mean amount contributed from a college savings fund among all undergraduates We'll assume that the standard deviation for the population of college sav- ings fund contributions is $1483. For 95 % confidence, we see from Table D that 1.960. The margin of error for the 95% confidence interval for is, therefore, 1483 1.960 V1593 37.16 We have computed the margin of error with more digits than we really need. Our mean is rounded to the nearest $1, so we will do the same for the margin of error Keeping additional digits would provide no additional useful information. Therefore, we will use m = 37. The approximate 95% confidence interval is tm= 1768 37 (1731, 1805) We are 95% confident that the mean amount contributed from a college savings fund among all undergraduates is between $1731 and $1805. Suppose that the researchers who designed this study had used a different sample size. How would this affect the confidence interval? We can answer this question by changing the sample size in our calculations and assuming that the sample mean is the same. a AGURE 6.5 Confidence ervals for n 1593 andn= Examples 6.4 and 6.5, A sample size nine times as large n 177 confidence interval n 1593 Suts in a rat is one-third as wide. ollema) som llema n ben nedonu 1600 1650 1700 1750 180 TIITT 50 1900 1950 USE YOUR KNOWLEDGE bonsbAaoo noado E921beundergraduate. 6.6 Average amount paid for college. Refer to Example 6.4. The average annual amount the n = 1593 families paid for college was $24,164." If the population standard deviation is $8500, give the 95% confidence interval for , the average annual amount a family pays for a college 6.7 Changing the sample size. In the setting of the previous exercise, would the margin of error for 95% confidence be roughly doubled or haived if the sample size were raised to n 6375? Verify your answer by performing the calculations. 6.8 Changing the confidence level. In the setting of Exercise 6.7, would the margin of error for 99% confidence be larger or smaller? Verify your answer by performing the calculations. The argument leading to the form of confidence intervals for the popula- tion mean u rested on the fact that the statistic x used to estimate u has a Nor- plereini ovn mal distribution. Because many sample estimates have Normal distributions.