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Chapter 10: Problem 6

To test \(H_{0}: \mu=40\) versus \(H_{1}: \mu>40,\) a simple random sample of size \(n=25\) is obtained from a population that is known to be normally distributed. (a) If \(\bar{x}=42.3\) and \(s=4.3,\) compute the test statistic. (b) If the researcher decides to test this hypothesis at the \(\alpha=0.1\) level of significance, determine the critical value. (c) Draw a \(t\) -distribution that depicts the critical region. (d) Will the researcher reject the null hypothesis? Why?

Short Answer

Expert verified
Reject the null hypothesis because the test statistic (2.674) is greater than the critical value (1.318).

Step by step solution

01

- Calculate the test statistic

The test statistic for this hypothesis test is calculated using the formula: \[ t = \frac{\bar{x} - \mu_0}{s / \sqrt{n}} \] Substituting in the given values: \[ t = \frac{42.3 - 40}{4.3 / \sqrt{25}} \] First compute the denominator: \[ \frac{4.3}{\sqrt{25}} = \frac{4.3}{5} = 0.86 \] Then calculate the test statistic: \[ t = \frac{42.3 - 40}{0.86} \approx 2.674 \]
02

- Determine the critical value

For a one-tailed test with \( \alpha = 0.1 \) and degrees of freedom \( df = n - 1 = 25 - 1 = 24 \), we can determine the critical t-value using a t-distribution table. The critical t-value for \( df = 24 \) at the 0.1 significance level is approximately 1.318.
03

- Draw the t-distribution

Draw a t-distribution curve. Mark the critical t-value (1.318) on the curve. Shade the region to the right of the t-value to represent the critical region where the null hypothesis will be rejected.
04

- Make a decision

Compare the calculated test statistic (2.674) to the critical value (1.318). Since 2.674 > 1.318, the test statistic falls in the critical region. Therefore, the null hypothesis is rejected. This means there is enough evidence to support the claim that the population mean is greater than 40.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Test Statistic
The test statistic is a value calculated from sample data that is used in hypothesis testing. It helps determine whether to reject the null hypothesis.

The formula to calculate the test statistic in a t-test is: \[ t = \frac{\bar{x} - \mu_0}{\frac{s}{\sqrt{n}}} \]

This formula uses the sample mean (\( \bar{x} \)), the hypothesized population mean (\( \mu_0 \)), the sample standard deviation (\( s \)), and the sample size (\( n \)). By comparing the sample data to what is expected under the null hypothesis, you can see how far or close your sample mean is from the hypothesized mean in terms of standard deviation units, which is given by the t-statistic.

Some key points about the test statistic:
  • The higher the absolute value of the test statistic, the more likely it is that the null hypothesis is incorrect.
  • A positive test statistic suggests the sample mean is larger than the hypothesized mean, while a negative suggests it's smaller.

In our example, using the sample mean (\

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Most popular questions from this chapter

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To test \(H_{0}: \mu=40\) versus \(H_{1}: \mu>40,\) a random sample of size \(n=25\) is obtained from a population that is known to be normally distributed with \(\sigma=6\) (a) If the sample mean is determined to be \(\bar{x}=42.3\) compute the test statistic. (b) If the researcher decides to test this hypothesis at the \(\alpha=0.1\) level of significance, determine the critical value. (c) Draw a normal curve that depicts the critical region. (d) Will the researcher reject the null hypothesis? Why?

Suppose that we are testing the hypotheses \(H_{0}: \mu=\mu_{0}\) versus \(H_{1}: \mu \neq \mu_{0}\) and we find the \(P\) -value to be 0.02 Explain what this means. Would you reject \(H_{0} ?\) Why?
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