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Chapter 9: Problem 11

What are the five steps of a test of hypothesis using the critical value approach? Explain briefly.

Short Answer

Expert verified
The five steps in a hypothesis test using the critical value approach are: 1) State the null and alternative hypotheses. 2) Identify a test statistic. 3) Set up a decision rule based on the chosen significance level. 4) Calculate the test statistic using your data. 5) Compare your test statistic to your critical value and make a decision to reject or not reject the null hypothesis.

Step by step solution

01

State the Hypotheses

The first step in the process is to state the null hypothesis, usually denoted as \(H_0\), and the alternative hypothesis, usually denoted as \(H_1\) or \(H_a\). The null hypothesis assumes that there is no effect or difference, while the alternative hypothesis assumes that there is an effect or difference.
02

Identify a Test Statistic

Depending on the nature of the data and the hypotheses, a suitable test statistic should be identified. The test statistic (like t-score or z-score) is used to compare the observed data with what is expected under the null hypothesis.
03

Set up the Decision Rule

Based on the significance level, typically denoted as \(\alpha\), the critical value is identified. This value is then compared to the test statistic to decide whether to reject or not reject the null hypothesis.
04

Calculate the Test Statistic

With the data collected, calculate the test statistic. The formula for this calculation will depend on the type of test statistic chosen (t or z).
05

Make a Decision

Finally, compare the calculated test statistic to the critical value. If the test statistic is more extreme than the critical value, the decision is to reject the null hypothesis. Otherwise, we fail to reject the null hypothesis. Conditional upon the decision, we say that the results were statistically significant or not significant.

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Most popular questions from this chapter

The police that patrol a heavily traveled highway claim that the average driver exceeds the 65 miles per hour speed limit by more than 10 miles per hour. Seventy-two randomly selected cars were clocked by airplane radar. The average speed was \(77.40\) miles per hour, and the standard deviation of the speeds was \(5.90\) miles per hour. Find the range for the \(p\) -value for this test. What will your conclusion be using this \(p\) -value range and \(\alpha=.02\) ?

A telephone company claims that the mean duration of all long-distance phone calls made by its residential customers is 10 minutes. A random sample of 100 long-distance calls made by its residential customers taken from the records of this company showed that the mean duration of calls for this sample is \(9.20\) minutes. The population standard deviation is known to be \(3.80\) minutes. a. Find the \(p\) -value for the test that the mean duration of all long- distance calls made by residential customers is different from 10 minutes. If \(\alpha=.02\), based on this \(p\) -value, would you reject the null hypothesis? Explain. What if \(\alpha=.05\) ? b. Test the hypothesis of part a using the critical-value approach and \(\alpha=.02\). Does your conclusion change if \(\alpha=.05 ?\)

Consider the following null and alternative hypotheses: $$ H_{0}: \mu=120 \text { versus } H_{1}: \mu>120 $$ A random sample of 81 observations taken from this population produced a sample mean of \(123.5 .\) The population standard deviation is known to be 15 . a. If this test is made at the \(2.5 \%\) significance level, would you reject the null hypothesis? Use the critical-value approach. b. What is the probability of making a Type I error in part a? c. Calculate the \(p\) -value for the test. Based on this \(p\) -value, would you reject the null hypothesis if \(\alpha=.01 ?\) What if \(\alpha=.05\) ?

For each of the following examples of tests of hypothesis about \(\mu\), show the rejection and nonrejection regions on the \(t\) distribution curve. a. A two-tailed test with \(\alpha=.02\) and \(n=20\) b. A left-tailed test with \(\alpha=.01\) and \(n=16\) c. A right-tailed test with \(\alpha=.05\) and \(n=18\)

A study claims that all adults spend an average of 14 hours or more on chores during a weekend. A researcher wanted to check if this claim is true. A random sample of 200 adults taken by this researcher showed that these adults spend an average of \(14.65\) hours on chores during a weekend. The population standard deviation is known to be \(3.0\) hours. a. Find the \(p\) -value for the hypothesis test with the alternative hypothesis that all adults spend more than 14 hours on chores during a weekend. Will you reject the null hypothesis at \(\alpha=.01\) ? b. Test the hypothesis of part a using the critical-value approach and \(\alpha=.01\).
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