A pharmaceutical company needs to know if its new cholesterol drug, Praxor, is effective at lowering cholesterol levels. It believes that people who take Praxor will average a greater decrease in cholesterol level than people taking a placebo. After the experiment is complete, the researchers find that the 32
participants in the treatment group lowered their cholesterol levels by a mean of 20.4
points with a standard deviation of 4.5
points. The 42
participants in the control group lowered their cholesterol levels by a mean of 18.6
points with a standard deviation of 2.3
points. Assume that the population variances are not equal and test the company’s claim at the 0.01
level. Let the treatment group be Population 1 and let the control group be Population 2.
Step 2 of 3 :  Compute the value of the test statistic. Round your answer to three decimal places.

This scenario involves conducting a **two-sample t-test** for independent samples, specifically to test if the treatment group (Population 1) has a greater mean decrease in cholesterol than the control group (Population 2). The test assumes unequal population variances.

### Given Data:
- Treatment group (Population 1):
  - Sample size (\(n_1\)) = 32
  - Mean (\(\bar{x}_1\)) = 20.4
  - Standard deviation (\(s_1\)) = 4.5

- Control group (Population 2):
  - Sample size (\(n_2\)) = 42
  - Mean (\(\bar{x}_2\)) = 18.6
  - Standard deviation (\(s_2\)) = 2.3

- Significance level (\(\alpha\)) = 0.01

### Step 1: State the hypotheses
- Null Hypothesis (\(H_0\)): \(\mu_1 \leq \mu_2\) (no greater decrease in cholesterol for the treatment group)
- Alternative Hypothesis (\(H_A\)): \(\mu_1 > \mu_2\) (treatment group has a greater decrease)

### Step 2: Compute the test statistic

The formula for the test statistic \(t\) in a two-sample t-test with unequal variances (Welch's t-test) is:

\[
t = \frac{(\bar{x}_1 - \bar{x}_2)}{\sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}}
\]

Where:
- \(\bar{x}_1 = 20.4\), \(\bar{x}_2 = 18.6\) are the sample means
- \(s_1 = 4.5\), \(s_2 = 2.3\) are the sample standard deviations
- \(n_1 = 32\), \(n_2 = 42\) are the sample sizes

Let's calculate the value of the test statistic.The value of the test statistic is \( t = 2.066 \), rounded to three decimal places.

Do you have any questions or would you like to continue with further steps in this hypothesis test?

Here are 5 related questions:
1. How do you find the degrees of freedom for Welch’s t-test?
2. How is the critical value for a one-tailed test determined?
3. How would the test statistic change if the sample sizes were different?
4. What are the implications if the test statistic had been negative?
5. What would happen if the population variances were assumed to be equal?

**Tip:** When using Welch’s t-test, the degrees of freedom are often non-integer and depend on sample variances. Always calculate or refer to an approximation.

A pharmaceutical company needs to know if its new cholesterol drug, Praxor, is effective at lowering cholesterol levels. It believes that people who take Praxor will average a greater decrease in cholesterol level than people taking a placebo. After the experiment is complete, the researchers find that the 32 participants in the treatment group lowered their cholesterol levels by a mean of 20.4 points with a standard deviation of 4.5 points. The 42 participants in the control group lowered their cholesterol levels by a mean of 18.6 points with a standard deviation of 2.3 points. Assume that the population variances are not equal and test the company’s claim at the 0.01 level. Let the treatment group be Population 1 and let the control group be Population 2. Step 2 of 3: Compute the value of the test statistic. Round your answer to three decimal places.

Explore how to compute the test statistic for a two-sample t-test to determine if the cholesterol drug Praxor is more effective than a placebo. The problem involves hypothesis testing using Welch's t-test with unequal population variances. Includes sample size, means, standard deviations, and significance level at 0.01.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation