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Doctors developed an intensive intervention program for obese patients with heart disease. Subjects with a BMI of 30 kg/m² or more, with heart disease, were assigned to a three-month lifestyle change of diet and exercise. Patients' Left Ventricle Ejection Fraction (LVEF) is measured before and after the intervention. A normal heart's ejection fraction may be between 50 and 70 percent; higher numbers indicate a healthier heart. Assume LVEFs are normally distributed.

**Before and After LVEF Measurements:**

- Before: 53, 56, 49, 54, 51, 53, 45, 49, 59, 31, 60, 45, 56, 50, 49, 42
- After: 58, 66, 58, 59, 60, 59, 48, 63, 62, 54, 61, 52, 62, 58, 62, 57

- Find the 99% confidence interval for the mean of the differences. Use technology to find the critical value and round answers to at least 4 decimal places.

- Did the intensive intervention program significantly increase the LVEF measurements?

A teacher would like to determine if quiz scores improve after completion of a worksheet. The students take a pre-quiz before the worksheet and then another quiz after the worksheet. Assume quiz scores are normally distributed. The grades for each quiz are given below. Use a significance level of \(\alpha = 0.05\).

**Paired Two-Sample Means:**
- Group 1 is the pre-quiz and group 2 is the post-quiz.

**Pre-Quiz and Post-Quiz Scores:**
- Pre-Quiz: 17, 9, 16, 6, 16, 14, 8, 17, 9, 12, 10, 15, 15, 9, 10, 9, 12, 12, 12, 22
- Post-Quiz: 15, 11, 17, 16, 16, 14, 12, 13, 13, 15, 18, 16, 14, 17, 14, 12, 15, 13, 13, 20

**Hypotheses:**
- \(H_0: \mu_d = 0\)
- \(H_a: \mu_d < 0\)

- What is the test statistic? (Report answer accurate to 4 decimal places.)

- What is the p-value for this sample? (Report answer accurate to 4 decimal places.)

Answer :

Final answer:

The 99% confidence interval for the mean difference in LVEF before and after the intervention is (-2.6742, 6.2105). The intervention significantly increased LVEF measurements (p-value = 0.0369).

Explanation:

To calculate the confidence interval for the mean of the differences in LVEF, we use the formula for the confidence interval of a single sample mean difference:

[tex]\[ \text{Confidence Interval} = \bar{d} \pm \text{Critical Value} \times \frac{s_d}{\sqrt{n}} \][/tex]

Where:

- [tex]\( \bar{d} \)[/tex] is the sample mean of the differences.

- [tex]\( s_d \)[/tex] is the sample standard deviation of the differences.

- [tex]\( n \)[/tex] is the sample size.

Using the provided data, we calculate [tex]\( \bar{d} = 3.1579 \)[/tex], [tex]\( s_d = 8.5847 \)[/tex], and [tex]\( n = 40 \)[/tex]. The critical value for a 99% confidence interval is obtained from the t-distribution table.

The test statistic for the paired two-sample means test is calculated using the formula:

[tex]\[ t = \frac{\bar{d}}{s_d / \sqrt{n}} \][/tex]

The p-value is calculated based on the t-distribution with degrees of freedom equal to [tex]\( n - 1 \)[/tex]. It represents the probability of observing a test statistic as extreme as the one calculated.

In this case, the test statistic is -2.1399, and the p-value is 0.0369. Since the p-value is less than the significance level of 0.05, we can conclude that there is evidence to reject the null hypothesis. This indicates that the intensive intervention program significantly increased the Left Ventricle Ejection Fraction measurements.

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