Question 1 (b)

Step 1 : Identifies the appropriate confidence interval (by name or by formula) and checks appropriate conditions. The appropriate procedure is a one-sample t-interval for a population mean. Conditions: 1. The sample is randomly selected from the population. 2. The population has a normal distribution, or the sample size is large. The first condition is met because we were told that the crows were randomly selected. The sample size of 23 is not considered large, so we need to examine the sample data to assess whether it is reasonable to assume that the population distribution of lead levels for all crows in this region is normal. The stem-and-leaf plot shows no strong skewness or outliers, so we will consider the second condition to be met. Step 2: Correct mechanics \* \_é— The critical value for A 95% confidence interval for the population mean is given by: i ± t n 95% confidence, based on 23 — 1 = 22 degrees of freedom, is t\* = 2.074. The 95% confidence interval for p is therefore 4.90 ± 2.074 x 4.90 ± 0.484, uää which is the interval (4.416, 5.384) ppm. Using the raw data rather than the given summary statistics, the 95% confidence interval for p is (4.411, 5.3803). Step 3: Interpretation We can be 95% confident that the population mean lead level among all crows in this region is between 4.416 and 5.384 parts per million.

Question 2 (c)

Stratifying by campus would be more advantageous than stratifying by gender provided that opinions about appearance of university buildings and grounds between the two campuses differ more than the opinions about appearance of university buildings and grounds between the two genders.

Question 4

Step 2: Identifies a correct test procedure (by name or by formula) and checks appropriate conditions. The appropriate test is a chi-square test of independence. The conditions for this test were satisfied because: 1. The question states that the sample was randomly selected. 2. The expected counts for all six cells of the table were all at least 5, as seen in the following table that lists expected counts in parentheses beside the observed counts: 18—34 years 35—54 years 55+ years Total Five or more servings of fruit and vegetables 231 (240.2) 669 (719.4) 1291 (1231.4) 2191 Four or fewer servings of Total fruit and vegetables 741 (731.8) 2242 (2191.6) 3692 (3751.6) 6675 972 2911 4983 8866

Statistic/Parameter Condition/Assumption How do we check?

More two Sample Proportions

(Test for Homogeneity)

1. Count Condition: The data are counts.

2. Independent Condition: Data in groups are independent.

3. Large sample

1. Verify this.

2. SRS and 10n < N

3. Count > 5

Step 4: States a correct conclusion in the context of the study, using the result of the statistical test. Because the p-value is very small (for instance, much smaller than a = 0.05 ), we would reject the null h ypothesis at the 0.05 level and conclude that the sample data provide strong evidence that there is an association between age group and consumption of fruits and vegetables for adults in the United States. In particular, older (55+ years of age) people were more likely to eat five or more servings of fruits and vegetables, and middle-aged people (35—54 years of age) were less likely to eat five or more servings of fruits and vegetables.

Question 5 (a)

No, it would not be reasonable to conclude that meditation causes a reduction in blood pressure for men in the retirement community. Because this is an observational study and not an experiment, no cause-and-effect relationship between meditation and lower blood pressure can be inferred. It is quite possible that men who choose to meditate could differ from men who do not choose to meditate in other ways that were also associated with blood pressure.

Question 5 (b)

The sample sizes were too small, relative to the overall sample proportion of successes, to justify using a normal approximation. One way to check this is to note that the combined sample proportion of 0+8 0.286, so neither = 3.143 nor nab = 172 z 4.857 is a 8 successes is — - 11+17 - 28 28 28 least 1

Statistic/Parameter Condition/Assumption How do we check?

Two Sample Proportions

(Independent)

1. Randomization Condition: Samples in each group are random samples (SRS) or representatives of their populations or in experiments the treatments are randomly assigned.

2. Normality Condition: n1p1 and n2p2 ≥ 10 and n1q1 and n2q2 ≥ 10.

3. Independent Condition: The selection of each subject is independent of each other (10n < N) for each sample. In some experiments this is not necessary.

4. Independence of Groups Condition: The groups are independent of each other.

1. Based on the information provided.

2. Show that the inequalities are true.

3. Show that the inequality is true.

4. Based on the information provided.

Question 5 (c)

g — Y —0.47. The graph of simulation results The observed value of the sample statistic i m — is 11 17 reveals that a difference of —0.47 or more extreme was very rare. In fact, the value —0.47 was the smallest possible outcome and occurred in only 76 of the 10,000 repetitions in the simulation. Thus, assuming that all men in the retirement community were equally likely to have high blood pressure whether they meditate or not, there is an approximate probability of 0.0076 of getting a difference of —0.47 or smaller by chance alone. Because this approximate p-value is very small, there is convincing evidence that men in this retirement community who meditate were less likely to have high blood pressure than men in this retirement community who do not meditate. However , because this is an observational study, even though we can conclude that meditation is associated with a lower chance of having high blood pressure, we cannot conclude that meditation causes a reduction in the likeliness of having high blood pressure.

Question 6 (a)

The Westem Pacific Ocean had more typhoons than the Eastem Pacific Ocean in all but one of these years. The average seems to have been about 31 typhoons per year in the Westem Pacific Ocean, which is higher than the average of about 19 typhoons per year in the Eastern Pacific Ocean. The Western Pacific Ocean also saw more variability (in number of typhoons per year) than the Eastern Pacific Ocean; for example, the range of the frequencies for the Western Pacific is about 21 typhoons and only 10 typhoons for the Eastern Pacific.

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