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LSAT The LSAT (a test taken for law school admission) has a mean score of 151 with a standard deviation of 9 and a unimodal, symmetric distribution of scores. A test preparation organization teaches small classes of 9 students at a time. A larger organization teaches classes of 25 students at a time. Both organizations publish the mean scores of all their classes. a) What would you expect the distribution of mean class scores to be for each organization? b) If cither organization has a graduating class with a mean score of \(160,\) they'll take out a full-page ad in the local school paper to advertise. Which organization is more likely to have that success? Explain. c) Both organizations advertise that if any class has an average score below \(145,\) they'll pay for everyone to retake the LSAT. Which organization is at greater risk to have to pay?

\(t\) -models, part I Using the \(t\) tables, software, or a calculator, estimate a) the critical value of \(t\) for a \(90 \%\) confidence interval with df \(=17\) b) the critical value of \(t\) for a \(98 \%\) confidence interval with df \(=88\) c) the P-value for \(t \geq 2.09\) with 4 degrees of freedom. d) the P-value for \(|t|>1.78\) with 22 degrees of freedom.

\(t\) -models, part II Using the \(t\) tables, software, or a calculator, estimate a) the critical value of \(t\) for a \(95 \%\) confidence interval with df \(=7\) b) the critical value of \(t\) for a \(99 \%\) confidence interval with \(\mathrm{df}=102\) c) the P-value for \(t \leq 2.19\) with 41 degrees of freedom. d) the P-value for \(|t|>2.33\) with 12 degrees of freedom.

Hot Dogs A nutrition lab tested 40 hot dogs to see if their mean sodium content was less than the 325 mg upper limit set by regulations for "reduced sodium" franks. The lab failed to reject the hypothesis that the hot dogs did not meet this requirement, with a P-value of \(0.142 . \mathrm{A} 90 \%\) confidence interval estimated the mean sodium content for this kind of hot dog at 317.2 to 326.8 mg. Explain how these two results are consistent. Your explanation should discuss the confidence level, the P-value, and the decision.

Catheters During an angiogram, heart problems can be examined via a small tube (a catheter) threaded into the heart from a vein in the patient's leg. It's important that the company that manufactures the catheter maintain a diameter of \(2.00 \mathrm{mm}\). (The standard deviation is quite small.) Each day, quality control personnel make several measurements to test \(\mathrm{H}_{0}: \mu=2.00\) against \(\mathrm{H}_{\mathrm{A}}: \mu \neq 2.00\) at a significance level of \(\alpha=0.05 .\) If they discover a problem, they will stop the manufacturing process until it is corrected. a) Is this a one-sided or two-sided test? In the context of the problem, why do you think this is important? b) Explain in this context what happens if the quality control people commit a Type I error. c) Explain in this context what happens if the quality control people commit a Type II error.

Ruffles Students investigating the packaging of potato chips purchased 6 bags of Lay's Ruffles marked with a net weight of 28.3 grams. They carefully weighed the contents of each bag, recording the following weights (in grams): 29.3,28.2,29.1,28.7,28.9,28.5 a) Do these data satisfy the assumptions for inference? Explain. b) Find the mean and standard deviation of the weights. c) Create a \(95 \%\) confidence interval for the mean weight of such bags of chips. d) Explain in context what your interval means. e) Comment on the company's stated net weight of 28.3 grams.

Popcorn Yvon Hopps ran an experiment to test optimum power and time settings for microwave popcorn. His goal was to find a combination of power and time that would deliver high-quality popcorn with less than \(10 \%\) of the kernels left unpopped, on average. After experimenting with several bags, he determined that power 9 at 4 minutes was the best combination. a) He concluded that this popping method achieved the \(10 \%\) goal. If it really does not work that well, what kind of error did Hopps make? b) To be sure that the method was successful, he popped 8 more bags of popcorn (selected at random) at this setting. All were of high quality, with the following percentages of uncooked popcorn: 7,13.2,10,6,7.8 \(2.8,2.2,5.2 .\) Does this provide evidence that he met his goal of an average of no more than \(10 \%\) uncooked kernels? Explain.

Chips Ahoy In \(1998,\) as an advertising campaign, the Nabisco Company announced a " 1000 Chips Challenge," claiming that every 18 -ounce bag of their Chips Ahoy cookies contained at least 1000 chocolate chips. Dedicated Statistics students at the Air Force Academy (no kidding) purchased some randomly selected bags of cookies, and counted the chocolate chips. Some of their data are given below. (Chance, \(12,\) no. 1[1999] ) $$\begin{array}{llllllll} 1219 & 1214 & 1087 & 1200 & 1419 & 1121 & 1325 & 1345 \\ 1244 & 1258 & 1356 & 1132 & 1191 & 1270 & 1295 & 1135 \end{array}$$ a) Check the assumptions and conditions for inference. Comment on any concerns you have. b) Create a \(95 \%\) confidence interval for the average number of chips in bags of Chips Ahoy cookies. c) What does this evidence say about Nabisco's claim? Use your confidence interval to test an appropriate hypothesis and state your conclusion.