STAT 200 Homework 8 Due Date: Oct 31, 2016 1. Assume a sample size of n=16. Draw a picture of the distribution of the t statistic under the null hypothesis. a) Use Table D and your picture to illustrate the values of the test statistic that would lead to rejection of the significance level 5% for a two-sided alternative. b) Find the critical value that should be used to calculate the margin of error for a 95% confidence interval for the mean of the population. c) Let the sample size n=10. Draw a picture of the distribution of the t statistic under the null hypothesis. Then repeat part a) and b) using n=10 instead of n=16. 2. Computers in some vehicles calculate various quantities related to performance. One of these is the fuel efficiency, or gas mileage, usually expressed as miles per gallon (mpg). For one vehicle equipped in this way, the miles per gallon were recorded each time the gas was filled, and the computer was then reset. Here are the mpg values for a random sample of 20 of these records: 41.5 50.7 36.6 37.3 34.2 45.0 48.0 43.2 47.7 42.2 43.2 44.6 48.4 46.4 46.8 39.2 37.3 43.5 44.3 43.3 a) Describe the distribution using graphical methods. Is it appropriate to analyze these data using methods based on Normal distribution? Explain why or why not. b) Find the mean, standard deviation, standard error, and margin of error for 95% confidence. c) Report the 95% confidence interval for , the mean miles per gallon for this vehicle based on these data. d) Do these data provide evidence that is less than 45? Using a significance level of 5%, state your hypotheses, the P-value, and your conclusion. 3. Corporate advertising tries to enhance the image of the corporation. A study compared two ads from two sources, the Wall Street Journal and the National Enquirer. Subjects were asked to pretend that their company was considering a major investment in Performax, the fictitious, sportswear firm in the ads. Each subject was asked to respond to the question \"How trustworthy was the source in the sportswear company ad for Performax\" on a 7-point scale. Higher values indicated more trustworthiness. Here is a summary for the results: Ad source n Wall Street Journal 66 4.77 1.50 National Enquirer 61 2.43 1.64 a) Compare the two sources of ads using a t test. Be sure to state your null and alternative hypotheses, the test statistic with degrees of freedom, the P-value, and your conclusion (significance level 10%). b) Give a 95% confidence interval for the difference. c) Interpret the confidence interval in part b). 4. A study of young children was designed to increase their intake of whole- grain, rather than regular-grain, snacks, At the end of the study the 76 children who participated in the study were presented with a choice between a regular-grain snack and a whole-grain alternative. The whole- grain alternative was chosen by 52 children. You want to examine the possibility that the children are equally likely to choose each type of snack. a) Formulate the null and alternative hypotheses for this setting. b) Are the guidelines for using the large-sample significance test satisfied for testing this null hypothesis? Explain your answer. c) Perform the 5% significance test and state your conclusion. d) Find a 90% confidence interval for the proportion of children who chose the regular-grain snack. 5. Young children need calcium in their diet to support the growth of their bones. The Institute of Medicine provides guidelines for how much calcium should be consumed by people of different ages. One study examined whether or not a sample of children consumed an adequate amount of calcium based on these guidelines. Since there are different guidelines for children aged 5 to 10 years and those ages 11 to 13 years, the children were classified into these two age groups. Each student's calcium intake was classified as meeting or not meeting the guideline. There were 2029 children in the study. Here are the data: Age (years) Met requirement 5 to 10 11 to 13 No 194 557 Yes 861 417 a) Find the proportion of children who met requirement in 5 to 10 group and its standard error. Do the same for children aged 11 to 13 years. b) Give a 90% confidence interval for the difference and interpret the confidence interval. c) Use a 10% significance test to examine whether the two proportions are equal, and state your conclusion