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Getting Enough Sleep? It is generally recommended that adults sleep at least 8 hours each night. One of the authors recently asked some of her students (undergraduate and graduate students at Harvard) how many hours each had slept the previous night, curious as to whether her students are getting enough sleep. The data are displayed in Figure 6.13 . The 12 students sampled averaged 6.2 hours of sleep with a standard deviation of 1.70 hours, Assuming this sample is representative of all her students, and assuming students need at least 8 hours of sleep a night, does this provide evidence that, on average, her students are not getting enough sleep?

Football Air Pressure During the National Football League's 2014 AFC championship game, officials measured the air pressure on 11 of the game footballs being used by the New England Patriots. They found that the balls had an average air pressure of 11.1 psi, with a standard deviation of 0.40 psi. (a) Assuming this is a representative sample of all footballs used by the Patriots in the 2014 season, perform the appropriate test to determine if the average air pressure in footballs used by the Patriots was significantly less than the allowable limit of 12.5 psi. There is no extreme skewness or outliers in the data, so it is appropriate to use the \(\mathrm{t}\) -distribution. (b) Is it fair to assume that this sample is representative of all footballs used by the Patriots during the 2014 season?

Number of Fouls in a Season by NBA Players The variable Fouls in the dataset NBAPlayers2015 shows the total number of fouls during the \(2014-2015\) season for all players in the \(\mathrm{NBA}\) (National Basketball Association) who played at least 24 minutes per game that season. We use this group as a sample of all NBA players in all seasons who play regularly. Use this information to test whether there is evidence that NBA players who play regularly have a mean number of fouls in a season less than 160 (or roughly 2 fouls per game).

Situations comparing two proportions are described. In each case, determine whether the situation involves comparing proportions for two groups or comparing two proportions from the same group. State whether the methods of this section apply to the difference in proportions. (a) Compare the proportion of students who use a Windows-based \(\mathrm{PC}\) to the proportion who use a Mac. (b) Compare the proportion of students who study abroad between those attending public universities and those at private universities. (c) Compare the proportion of in-state students at a university to the proportion from outside the state. (d) Compare the proportion of in-state students who get financial aid to the proportion of outof-state students who get financial aid.

In Exercises 6.139 to \(6.142,\) use the normal distribution to find a confidence interval for a difference in proportions \(p_{1}-p_{2}\) given the relevant sample results. Give the best estimate for \(p_{1}-p_{2},\) the margin of error, and the confidence interval. Assume the results come from random samples. A 95\% confidence interval for \(p_{1}-p_{2}\) given counts of 240 yes out of 500 sampled for Group 1 and 450 ves out of 1000 sampled for Group \(2 .\)

Metal Tags on Penguins and Survival Data 1.3 on page 10 discusses a study designed to test whether applying metal tags is detrimental to penguins. One variable examined is the survival rate 10 years after tagging. The scientists observed that 10 of the 50 metal tagged penguins survived, compared to 18 of the 50 electronic tagged penguins. Construct a \(90 \%\) confidence interval for the difference in proportion surviving between the metal and electronic tagged penguins \(\left(p_{M}-p_{E}\right)\). Interpret the result.

In Exercises 6.156 to 6.161: (a) Find the relevant sample proportions in each group and the pooled proportion. (b) Complete the hypothesis test using the normal distribution and show all details. Test whether there is a difference between two groups in the proportion who voted, if 45 out of a random sample of 70 in Group 1 voted and 56 out of a random sample of 100 in Group 2 voted.

(a) Find the relevant sample proportions in each group and the pooled proportion. (b) Complete the hypothesis test using the normal distribution and show all details. Test whether patients getting Treatment \(\mathrm{A}\) are more likely to survive, if 63 out of 82 getting Treatment A survive and 31 out of 67 getting Treatment B survive.

(a) Find the relevant sample proportions in each group and the pooled proportion. (b) Complete the hypothesis test using the normal distribution and show all details. Test whether people with a specific genetic marker are more likely to have suffered from clinical depression than people without the genetic marker, using the information that \(38 \%\) of the 42 people in a sample with the genetic marker have had clinical depression while \(12 \%\) of the 758 people in the sample without the genetic marker have had clinical depression.

Do Ovulating Women Affect Men's Speech? Studies suggest that when young men interact with a woman who is in the fertile period of her menstrual cycle, they pick up subconsciously on subtle changes in her skin tone, voice, and scent. A study introduced in Exercise \(\mathrm{B} .23\) suggests that men may even change their speech patterns around ovulating women. The men were randomly divided into two groups with one group paired with a woman in the fertile phase of her cycle and the other group with a woman in a different stage of her cycle. The same women were used in the two different stages. For the men paired with a less fertile woman, 38 of the 61 men copied their partner's sentence construction in a task to describe an object. For the men paired with a woman at peak fertility, 30 of the 62 men copied their partner's sentence construction. The experimenters hypothesized that men might be less likely to copy their partner during peak fertility in a (subconscious) attempt to attract more attention to themselves. Use the normal distribution to test at a \(5 \%\) level whether the proportion of men copying sentence structure is less when the woman is at peak fertility.