1. A random sample of 26 Inuit was taken and their age at death was recorded.

62, 91, 24, 36, 19, 17, 45, 42, 33, 51, 60, 18, 23, 49, 37, 38, 55, 66, 64, 86, 93, 77, 54, 72, 49, 62.

1. Construct a histogram of the data.
2. Construct a box plot of the data.
3. Compute the sample mean of the data.
4. Compute the sample median of the data.
5. Compute the sample mode of the data.
6. Compute the sample standard deviation of the data.
7. Compute the sample range of the data.
8. Compute the sample coefficient of variation of the data.

1. A sample of the daily amount of protein from beef (in grams) people in New Mexico receive yielded a sample mean weight of grams and a sample standard deviation of s = 12 grams.  Assume this data is approximately normally distributed and use what you know about the normal distribution to answer the following questions.
   1. About what percentage of people receive between 46 and 70 grams of protein per day?
   2. What is the approximate median amount of protein people in New Mexico received per day?
   3. If an individual received 96 grams of protein from beef per day would you think it likely that this person was from New Mexico?  Why or why not?
   4. What is the probability a randomly selected person from New Mexico receives more than 62 grams of protein from beef per day?
   5. What is the probability a randomly selected person from New Mexico receives less than 50 grams of protein from beef per day?
2. The following data are the number of surviving children of females of age > 50 in two different villages in Northwest Papua New Guinea.  Assume that the two sets of data are independent from each other and that the populations that the data are sampled from follow normal distributions.

   Village 1	Village 2
   4	4
   3	8
   2	10
   5	5
   1	1
   0	0
   7	3
   8	6
   4	9

   1. Do females from village 1 tend to have less surviving children than females from village 2?  Conduct the appropriate hypothesis test and use alpha = 0.05.
   2. Compute a 95% confidence interval for the mean number of surviving children of females from village 1. What do you find?
   3. Conduct a test to see if the mean number of surviving children that the females from village 2 have is different than 6.  Use alpha = 0.05. What do you find?

1. A nurse walks into a bar and says to the bartender: "Here is my data, give me a 2-sample t-test analysis."  The bartender does the analysis and hands the nurse a p-value.  The nurse says: "Thanks for the p-value, but is this result significant?"  The bartender says: "I don’t know, you didn’t tell me the ______."  What does the bartender need to know in order to tell the nurse whether or not to reject the validity of the null hypothesis?  What particular numerical value would you tell the bartender to use?  When you give the bartender a specific value to use to evaluate the significance of the p-value you are also deciding upon the chance that what type of error might be made?
2. A researcher would like to find out whether a woman’s nickname affects her cholesterol reading . She records the cholesterol readings of 23 women nicknamed Sue, 24 women nicknamed Deb and 19 women nicknamed Becky; her data appears in the table below. She wants to know whether the differences in the mean readings are significant; i.e., whether there is any difference in the mean cholesterol readings of the women nicknamed Sue, women nicknamed Deb, and women nicknamed Becky.  Use the ANOVA method to conduct this analysis, and do the following:
   1. State the null and alternative hypotheses you are testing.
   2. Use an alpha level of 0.05 for the ANOVA test.
   3. Clearly state the conclusion of your test.  In other words, what did you find out?

1. The probabilities associated with the expected principal source of payment for hospital discharges in the U.S. in the year 1990 are list below:

Principal Source of Payment Probability

Private insurance 0.387

Medicare 0.345

Medicaid 0.116

Other govt. program 0.033

Self-payment 0.058

Other/No charge 0.028

Not stated 0.033

Total 1.000

1. What is the probability that the principal source of payment for a given hospital discharge is the patient’s private insurance?
2. What is the probability that the principal source of payment is Medicare, Medicaid, or some other government program?
3. Given that the principal source of payment is a government program, what is the probability that it is Medicare?

1. A rehabilitation center researcher was interested in examining the relationship between physical fitness prior to surgery of persons undergoing corrective knee surgery and time required in physical therapy until successful rehabilitation.  Patient records in the rehabilitation center were examined, and 24 male subjects ranging in age from 18 to 30 years who had undergone similar corrective knee surgery during the past year were selected for the study.  The number of days required for successful completion of physical therapy and the prior physical fitness status (below average, average, above average) for each patient follows:

1. Please test the hypothesis that the mean amount of time (in days) required for successful completion of physical therapy is equal for patients with all three types of prior physical fitness status vs. the hypothesis that differences exist using the ANOVA method.  Test the hypothesis with alpha = 0.05. What do you find?
2. If differences between means exist, have SPSS conduct the Bonferroni multiple comparisons test and make sure that the significance level says .05.  Use the SPSS output from the Bonferroni test to determine which means differ from which others and in what way. What do you find?
3. Summarize your findings based on this analysis.  In other words, does prior physical fitness status matter in relation to length of time required for successful completion of physical therapy?  If so, how does it matter?

8.     This problem makes use of the attached SPSS data set lowbwt . The data set lowbwt contains information for a sample of 100 low birth weight infants born in two hospitals. Measurements of systolic blood pressure are saved under the variable name sbp and indicators of gender -- with 1 representing a male and 0 a female under the name sex.

(a) Construct a histogram of systolic blood pressure measurements for this sample. Based on the graph, do you believe that blood pressure is approximately normally distributed?

(b) Test the null hypothesis that among low birth weight infants, the mean systolic blood pressure for boys is equal to the mean for girls. Use a two-sided test at the 0.05 level of significance. What do you conclude?

9.     This problem makes use of the attached SPSS data set heart . The Bayley Scales of Infant Development yield scores on two indices – the Psychomotor Development Index (PDI) and the Mental Development Index (MDI) – which can be used to assess a child’s level of functioning in each of these areas at approximately one year of age. Among normal healthy infants, both indices have a mean value of 100. As part of a study assessing the development and neurologic status of children who have undergone reparative heart surgery during the first three months of life, the Bayley Scales were administered to a sample of one-year-old infants born with congenital heart disease. The data are contained in the attached SPSS data set heart; PDI scores are saved under the variable name pdi, while MDI scores are saved under mdi.

(a) At the 0.05 level of significance, test the null hypothesis that the mean PDI score for children born with congenital heart disease who undergo reparative heart surgery during the first three months of life is equal to 100, the mean score for healthy children. Use a two-sided test. What is the p-value? What do you conclude?

(b) Conduct the analogous test of hypothesis for the mean MDI score. What do you conclude?

(c) Construct 95% confidence intervals for the true mean PI score and the true mean MDI score for this population of children with congenital heart disease. Does either of these intervals contain the value 100? Would you have expected that they would?

10.  The population of male industrial workers in London who have never experienced a major coronary event has a mean systolic blood pressure 136 mm Hg and mean diastolic blood pressure 84 mm Hg. You might be interested in determining whether these values are the same as those for the population of industrial workers who have suffered a coronary event.

(a) A sample of 86 workers who have experienced a major coronary event has mean systolic blood pressure = 143 mm Hg and standard deviation = 24.4 mm Hg. Test the null hypothesis that the mean systolic blood pressure for the population of industrial workers who have experienced such an event is identical to the mean for the workers who have not, using a two-sided test at the 0.05 level of significance.

(b) The same sample of men has mean diastolic blood pressure = 87 mm Hg and standard deviation = 16.0 mm Hg. Test the null hypothesis that the mean diastolic blood pressure for the population of workers who have experienced a major coronary event is identical to the mean for the workers who have not.

(c) How do the two groups of workers compare?

11.   In the Nile river, the distribution of lead levels in tilapia fish is skewed to the right; it has a mean of \(\bar{X}\) = 8 mg/kg and a standard deviation of \(\sigma \) = 1.6 mg / kg.  Describe the distribution of means of samples of size 45 selected from the population of tilapia fish in the Nile river.

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