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The formula used to calculate a confidence interval for the mean of a normal population is $$ \bar{x} \pm(t \text { critical value }) \frac{s}{\sqrt{n}} $$ What is the appropriate \(t\) critical value for each of the following confidence levels and sample sizes? a. \(95 \%\) confidence, \(n=17\) b. \(99 \%\) confidence, \(n=24\) c. \(90 \%\) confidence, \(n=13\)

The paper "The Effects of Adolescent Volunteer Activities on the Perception of Local Society and Community Spirit Mediated by Self-Conception" (Advanced Science and Technology Letters [2016]: 19-23) describes a survey of a large representative sample of middle school children in South Korea. One question in the survey asked how much time per year the children spent in volunteer activities. The sample mean was 14.76 hours and the sample standard deviation was 16.54 hours. a. Based on the reported sample mean and sample standard deviation, explain why it is not reasonable to think that the distribution of volunteer times for the population of South Korean middle school students is approximately normal. b. The sample size was not given in the paper, but the sample size was described as "large." Suppose that the sample size was 500 . Explain why it is reasonable to use a one-sample \(t\) confidence interval to estimate the population mean even though the population distribution is not approximately normal. c. Calculate and interpret a confidence interval for the mean number of hours spent in volunteer activities per year for South Korean middle school children. (Hint: See Example 12.7.)

Students in a representative sample of 65 first-year students selected from a large university in England participated in a study of academic procrastination ("Study Goals and Procrastination Tendencies at Different Stages of the Undergraduate Degree," Studies in Higher Education [2016]: 2028-2043). Each student in the sample completed the Tuckman Procrastination Scale, which measures procrastination tendencies. Scores on this scale can range from 16 to \(64,\) with scores over 40 indicating higher levels of procrastination. For the 65 first-year students in this study, the mean score on the procrastination scale was 37.02 and the standard deviation was 6.44 . a. Construct a \(95 \%\) confidence interval estimate of \(\mu,\) the mean procrastination scale for first-year students at this college. (Hint: See Example 12.7.) b. Based on your interval, is 40 a plausible value for the population mean score? What does this imply about the population of first-year students?

The formula used to calculate a confidence interval for the mean of a normal population is $$ \bar{x} \pm(t \text { critical value }) \frac{s}{\sqrt{n}} $$ What is the appropriate \(t\) critical value for each of the following confidence levels and sample sizes? a. \(90 \%\) confidence, \(n=12\) b. \(90 \%\) confidence, \(n=25\) c. \(95 \%\) confidence, \(n=10\)

Acrylic bone cement is sometimes used in hip and knee replacements to secure an artificial joint in place. The force required to break an acrylic bone cement bond was measured for six specimens, and the resulting mean and standard deviation were 306.09 Newtons and 41.97 Newtons, respectively. Assuming that it is reasonable to believe that breaking force has a distribution that is approximately normal, use a confidence interval to estimate the mean breaking force for acrylic bone cement.

What percentage of the time will a variable that has a distribution with the specified degrees of freedom fall in the indicated region? a. 5 df, between -2.02 and 2.02 b. 14 df, between -2.98 and 2.98 c. \(22 \mathrm{df}\), outside the interval from -1.72 to 1.72 d. \(26 \mathrm{df},\) to the left of -2.48

The formula used to calculate a confidence interval for the mean of a normal population when \(n\) is small is $$ \bar{x} \pm(t \text { critical value }) \frac{s}{\sqrt{n}} $$ What is the appropriate \(t\) critical value for each of the following confidence levels and sample sizes? a. \(95 \%\) confidence, \(n=15\) b. \(99 \%\) confidence, \(n=20\) c. \(90 \%\) confidence, \(n=26\)

Give as much information as you can about the \(P\) -value of a \(t\) test in each of the following situations. (Hint: See discussion on page \(594 .\) ) a. Upper-tailed test, \(\mathrm{df}=8, t=2.0\) b. Lower-tailed test, \(\mathrm{df}=10, t=-2.4\) c. Lower-tailed test, \(n=22, t=-4.2\) d. Two-tailed test, \(\mathrm{df}=15, t=-1.6\)

The Economist collects data each year on the price of a Big Mac in various countries around the world. A sample of McDonald's restaurants in Europe in July 2016 resulted in the following Big Mac prices (after conversion to U.S. dollars): \(\begin{array}{llll}4.44 & 3.15 & 2.42 & 3.96\end{array}\) \(\begin{array}{llll}4.51 & 4.17 & 3.69 & 4.62\end{array}\) \(\begin{array}{lll}3.80 & 3.36 & 3.85\end{array}\) The mean price of a Big Mac in the U.S. in July 2016 was \$5.04. For purposes of this exercise, you can assume it is reasonable to regard the sample as representative of European McDonald's restaurants. Does the sample provide convincing evidence that the mean July 2016 price of a Big Mac in Europe is less than the reported U.S. price? Test the relevant hypotheses using \(\alpha=0.05 .\) (Hint: See Example 12.12.)

Give as much information as you can about the \(P\) -value of a \(t\) test in each of the following situations: a. Two-tailed test, \(\mathrm{df}=9, t=0.73\) b. Upper-tailed test, \(\mathrm{df}=10, t=0.5\) c. Lower-tailed test, \(n=20, t=-2.1\) d. Two-tailed test, \(n=40, t=1.7\)