Consider the following hypothesis test: *H0*: *μ* ≤ 75 , *HA*: *μ* > 75. A random sample of 100 observations yields a sample mean of 80. The population standard deviation is 30. What is the value of the test statistic?

To calculate the test statistic for the given hypothesis test, we can use the formula for the z-test statistic when the population standard deviation is known:

\[
z = \frac{\bar{x} - \mu_0}{\sigma / \sqrt{n}}
\]

Where:
- \(\bar{x}\) is the sample mean,
- \(\mu_0\) is the population mean under the null hypothesis,
- \(\sigma\) is the population standard deviation,
- \(n\) is the sample size.

Given:
- \(\bar{x} = 80\)
- \(\mu_0 = 75\)
- \(\sigma = 30\)
- \(n = 100\)

Plugging these values into the formula:

\[
z = \frac{80 - 75}{30 / \sqrt{100}} = \frac{5}{30 / 10} = \frac{5}{3} \approx 1.67
\]

Thus, the value of the test statistic is approximately **1.67**.

If you need further details or have any questions, let me know! Here are some related questions for your consideration:

1. What is the significance level typically used in hypothesis testing?
2. How do you interpret the z-test statistic in the context of this hypothesis test?
3. What would be the critical value for a one-tailed test at a 0.05 significance level?
4. How would you determine whether to reject or fail to reject the null hypothesis?
5. What are the potential consequences of a Type I and Type II error in this context?

**Tip:** Always check the assumptions of the test (e.g., normality, independence) before applying the hypothesis test.

Consider the following hypothesis test: *H0*: *μ* ≤ 75 , *HA*: *μ* > 75. A random sample of 100 observations yields a sample mean of 80. The population standard deviation is 30. What is the value of the test statistic?

Discover how to calculate the z-test statistic for hypothesis testing with a sample mean of 80 and a population standard deviation of 30. This example demonstrates key statistical concepts and provides insights into hypothesis testing for students in Grades 11-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation