A researcher compares two compounds (1 and 2) used in the manufacture of car tires that are designed to reduce braking distances for SUVs equipped with the tires.  The mean braking distance for SUVs equipped with tires made with compound 1 is 61
feet, with a population standard deviation of 8.5
.  The mean braking distance for SUVs equipped with tires made with compound 2 is 66
feet, with a population standard deviation of 14.7
.  Suppose that a sample of 65
braking tests are performed for each compound.  Using these results, test the claim that the braking distance for SUVs equipped with tires using compound 1 is shorter than the braking distance when compound 2 is used.  Let μ1
be the true mean braking distance corresponding to compound 1 and μ2
be the true mean braking distance corresponding to compound 2.  Use the 0.1
level of significance.

Step 2 of 5 :  Compute the value of the test statistic. Round your answer to two decimal places.

In order to test the claim that the braking distance for SUVs equipped with tires using compound 1 is shorter than for compound 2, we need to perform a **two-sample z-test** for the difference of means.

### Step-by-step process:

1. **State the Hypotheses:**

   - Null Hypothesis (\(H_0\)): \( \mu_1 \geq \mu_2 \)
   - Alternative Hypothesis (\(H_A\)): \( \mu_1 < \mu_2 \) (This is a left-tailed test, since we are testing if the braking distance for compound 1 is shorter than for compound 2).

2. **Identify the given information:**

   - Mean braking distance for compound 1: \( \bar{x}_1 = 61 \)
   - Standard deviation for compound 1: \( \sigma_1 = 8.5 \)
   - Sample size for compound 1: \( n_1 = 65 \)
   
   - Mean braking distance for compound 2: \( \bar{x}_2 = 66 \)
   - Standard deviation for compound 2: \( \sigma_2 = 14.7 \)
   - Sample size for compound 2: \( n_2 = 65 \)

   - Significance level: \( \alpha = 0.1 \)

3. **Compute the test statistic:**

   The formula for the **z-test** statistic for two independent samples is:

   \[
   z = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}
   \]
   
   Since we are testing the difference in means and \(H_0\) assumes \( \mu_1 = \mu_2 \), we set \( (\mu_1 - \mu_2) = 0 \). The formula simplifies to:

   \[
   z = \frac{\bar{x}_1 - \bar{x}_2}{\sqrt{\frac{\sigma_1^2}{n_1} + \frac{\sigma_2^2}{n_2}}}
   \]

4. **Substitute the values:**

   \[
   z = \frac{61 - 66}{\sqrt{\frac{8.5^2}{65} + \frac{14.7^2}{65}}}
   \]
   
   Let's calculate the value.

The value of the test statistic \( z \) is \( -2.37 \).

Would you like to proceed with the next steps in the hypothesis test or need any clarification? 

Here are five related questions to consider:
1. What is the critical value for a left-tailed test at the 0.1 significance level?
2. How do we calculate the p-value for this test statistic?
3. What does the test statistic tell us about the braking distances?
4. How does the sample size affect the calculation of the test statistic?
5. What assumptions are made in this z-test for comparing two means?

**Tip:** In hypothesis testing, the test statistic helps determine how far the sample data is from the null hypothesis.

A researcher compares two compounds (1 and 2) used in the manufacture of car tires that are designed to reduce braking distances for SUVs equipped with the tires. The mean braking distance for SUVs equipped with tires made with compound 1 is 61 feet, with a population standard deviation of 8.5. The mean braking distance for SUVs equipped with tires made with compound 2 is 66 feet, with a population standard deviation of 14.7. Suppose that a sample of 65 braking tests are performed for each compound. Using these results, test the claim that the braking distance for SUVs equipped with tires using compound 1 is shorter than the braking distance when compound 2 is used. Let μ1 be the true mean braking distance corresponding to compound 1 and μ2 be the true mean braking distance corresponding to compound 2. Use the 0.1 level of significance. Step 2 of 5 : Compute the value of the test statistic. Round your answer to two decimal places.

This problem involves comparing the braking distances of SUVs using two different tire compounds with a two-sample z-test. Given means, standard deviations, and a significance level of 0.1, learn how to compute the z-test statistic for determining whether compound 1 reduces braking distances more than compound 2.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation