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PBHL-B300/B301/B304 Homework #5

Note: For all hypothesis testing problems, please make sure that you write the hypotheses, calculate the p-value, and conclude the problem regarding the null hypothesis.

• (8 points) The following data contains the number of friends on Facebook for an SRS of 20 Facebook users from a large university.

137 74 85 104 106 72 119 160 99 148 158 126 118 112 103 111 154 90 120 127

• Manually compute the mean, standard deviation, standard error and margin of error for the 95% confidence interval. (Use R to find the appropriate critical t* value)

• Manually compute the 95% confidence interval for the mean number of friends on Facebook for Facebook users at this large university.

• (10 points) If we increase our food intake, we generally gain weight. Nutrition scientists can calculate the amount of weight gain that would be associated with a given increase in calories. In one study, 16 nonobese adults, aged 25 to 36 years, were fed 1000 calories per day in excess the calories needed to maintain a stable body weight. The subjects maintained this diet for 8 weeks, so they consumed a total of 56,000 extra calories. According to theory, 7700 extra calories will translate into a weight gain of 1 kilogram. Therefore, we expect each of these subjects to gain 56000/7700=7.3 kilograms. The weights in kilograms before and after the 8-week period are shown below:

ID: 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

Before: 55.9 61.7 55.7 54.9 59.6 62.3 74.2 75.6 70.7 53.3 73.3 63.4 68.1 73.7 91.7 57.8

After: 63.0 68.3 61.7 58.8 66.0 66.2 79.0 82.4 74.3 59.3 79.1 67.0 73.4 76.9 93.1 60.3

• For each subject, manually calculate the weight gain subtracting the weight before from the weight after.
• Manually find the mean and the standard deviation for the weight gain in your output and report/paste it here
• Calculate the standard error and the margin of error for a 95% confidence interval for the mean weight gain. Report the 95% confidence interval in a sentence that explains the meaning of the 95%.
• Test the null hypothesis that the mean weight gain is 7.3 kilograms. Be sure to specify the null and alternative hypotheses, calculate the test statistic with degrees of freedom, and find the p-value (using R). What do you conclude? Use = 0.05.

• (5 points) A crossover study was conducted to investigate whether oat bran cereal helps to lower serum cholesterol in hypercholesterolemic males. Fourteen such individuals were randomly placed on a diet that included either oat bran or corn flakes; after two weeks, their low-density lipoprotein (LDL) cholesterol levels were recorded. Each man was then switched to the alternative diet. After a second two-week period, the LDL cholesterol level of each individual was again recorded. The data from this study are shown below:

Subject	Corn flakes	Oat bran
1	4.61	3.84
2	6.42	5.57
3	5.4	5.85
4	4.54	4.8
5	3.98	3.68
6	3.82	2.96
7	5.01	4.41
8	4.34	3.72
9	3.8	3.49
10	4.56	3.84
11	5.35	5.26
12	3.89	3.73
13	2.25	1.84
14	4.24	4.14

• Are the two samples of data paired or independent?
• What are the appropriate null and alternative hypotheses?

• (7 points) Researchers were interested in comparing the long-term psychological effects of dieters on a high-carbohydrate, low-fat (LF) diet with those on a high-fat, low-carbohydrate (LC) diet. A total of 106 overweight and obese individuals were randomly assigned to one of these two energy-restricted diets. At 52 weeks a total of 32 LC dieters and 33 LF dieters remained. Mood was assessed using a total mood disturbance score (TMDS), where a lower score is associated with a less negative mood. A summary of these results are:

Group	Sample size	Sample mean	Sample standard deviation
LC	32	47.3	28.3
LF	33	39.3	25.8

Is there a difference in the TMDS at week 52 between the two groups of dieters? Test the null hypothesis that the dieters' average mood in the two groups is the same. Be sure to clearly state they appropriate hypotheses, p-value (use R to find it), and conclusion. Use a significance level of 0.05. Notice that the test-statistic for this problem would be:

• (6 points) As shown in the R Lab, download and load the 2004 North Carolina births data set, nc.

• As shown in the R Lab, use the R inference function to do a hypothesis test to determine if there is a statistically significant difference in the means of the length of pregnancy in weeks, comparing the married versus the not married mothers, using the variables weeks and marital. Use = 0.05. (Note: make sure you write the hypotheses, p-value and conclusion.)

• From the R output, report the group means, standard deviations, and sample sizes.

• From the boxplot graph created, does it appear there is a difference between the means?

• (6 points) Repeat the hypothesis test instructions above from #5 (1), except now compare the means of the Mother's age, comparing the low birth weight babies versus the non-low birth weight babies, using the variables mage and lowbirthweight.

• (8 points) An airline wants to evaluate the depth perception of its pilots over the age of fifty. A random sample of n = 14 airline pilots over the age of fifty are asked to judge the distance between two markers placed 20 feet apart at the opposite end of the laboratory. Assume we don't know population standard deviation. The sample data listed here are the pilots' error (recorded in feet) in judging the distance. 2.9 2.6 2.9 2.6 2.4 2.1 2.3 2.2 2.5 2.3 2.8 2.5 2.7 2.6

• (1 point) Enter these data into R, as we did in Homework #2, creating a variable depth using the c() function:

> depth= c(enter the 14 raw data values from above, all separated by commas)

• (4 points) As shown in the R Lab, use the R inference function to do a hypothesis test to determine if there evidence that the mean error in depth perception for the company's pilots over the age of fifty is greater than 2.3? Use = 0.05. Since this is a t-test comparing a single mean, you delete the x= variable from the inference call. (Note: be sure you write the hypotheses, p-value and conclusion.) (You must first load the 2004 North Carolina data package from #5 above for the inference function to work.)

• (4 points) As shown in the R Lab, use the R inference function to determine the 90% confidence interval for the mean error in depth perception?