Question

1) A simple random sample of the running time of movies of 70 international movies resulted in a sample mean length of 112 minutes and a sample standard deviation of 7 minutes. Test the claim that international movies have a mean running time of more than 110 minutes at the 5% level of significance. Assume that the lengths of movies are normally distributed. 2) A researcher wishes to determine whether the blood pressure of vegetarians is, on average, lower than the blood pressure of nonvegetarians. Independent simple random samples of 85 vegetarians and 75 nonvegetarians yielded the following sample statistics for systolic blood pressure: Vegetarians Nonvegetarians sample size n = 85 n = 75 sample mean x = 124.1 mmHg x = 138.7 mmHg sample standard deviation s = 38.7 mmHg s = 39.2 mmHg Use a significance level of 0.01 to test the claim that the mean systolic blood pressure of vegetarians is lower than the mean systolic blood pressure of nonvegetarians.

2) A researcher wishes to determine whether the blood pressure of vegetarians is, on average,
lower than the blood pressure of nonvegetarians. Independent simple random samples of 85
vegetarians and 75 nonvegetarians yielded the following sample statistics for systolic blood
pressure:
Vegetarians Nonvegetarians
sample size n = 85 n = 75
sample mean x = 124.1 mmHg x = 138.7 mmHg
sample standard deviation s = 38.7 mmHg s = 39.2 mmHg
Use a significance level of 0.01 to test the claim that the mean systolic blood pressure of
vegetarians is lower than the mean systolic blood pressure of nonvegetarians.

Answer 1

1) Based on the given sample data, we reject the null hypothesis and conclude that there is sufficient evidence at the 5% level of significance to support the claim that international movies have a mean running time of more than 110 minutes.

2) At a 0.01 significance level, the data provides enough evidence to reject the null hypothesis, indicating that the mean systolic blood pressure of vegetarians is indeed lower than that of nonvegetarians.

Step-by-step explanation:

1) In the first scenario, we perform a one-sample t-test for the mean running time of international movies. The null hypothesis
`(\(H_0\))` assumes that the mean running time is 110 minutes or less, while the alternative hypothesis
`(\(H_a\))` suggests that it is more than 110 minutes. Using the given sample mean
`(\(\bar{x} = 112\))`, sample standard deviation (s = 7), sample size (n = 70), and a significance level of 0.05, we calculate the t-statistic and compare it to the critical value. With a t-statistic greater than the critical value, we reject the null hypothesis.

2) For the comparison of systolic blood pressure between vegetarians and nonvegetarians, a two-sample t-test is employed. The null hypothesis
`(\(H_0\))` posits that there is no difference in mean blood pressure, while the alternative hypothesis
`(\(H_a\))` suggests that the mean blood pressure of vegetarians is lower. Utilizing the provided sample means
`(\(x_1 = 124.1\)` ,
`\(x_2 = 138.7\))` , sample standard deviations
`(\(s_1 = 38.7\), \(s_2 = 39.2\))` , and sample sizes
`(\(n_1 = 85\), \(n_2 = 75\))` , along with a significance level of 0.01, we calculate the t-statistic and compare it to the critical value. The obtained t-statistic supports the rejection of the null hypothesis.

In both cases, the rejection of the null hypothesis implies that there is enough statistical evidence to support the respective claims at the specified significance levels.

Answer 2

The claim about international movie lengths was tested using a one-sample z-test, while the claim about the blood pressure of vegetarians versus nonvegetarians was tested using a two-sample t-test, both assuming normal distributions of the underlying populations.

Step-by-step explanation:

To test the claim that international movies have a mean running time of more than 110 minutes, we use a one-sample z-test because the sample size is large (n>30) and the population standard deviation is known to be 7 minutes. The null hypothesis (H0) is μ = 110 minutes, and the alternative hypothesis (H1) is μ > 110 minutes. This is a one-tailed test since we are checking for 'more than' not 'different from'.

For the second claim, that the mean systolic blood pressure of vegetarians is lower than that of nonvegetarians, we'll use a two-sample t-test since sample standard deviations are provided, not population standard deviations, making it appropriate for samples of sizes n=85 and n=75. The null hypothesis (H0) is μ1 - μ2 ≥ 0, and the alternative hypothesis (H1) is μ1 - μ2 < 0, denoting that the mean blood pressure of vegetarians is less than nonvegetarians. This is a one-tailed test because the claim is 'lower than' not 'different from'.