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Chapter 9: Problem 10

What does the level of significance represent in a test of hypothesis? Explain.

Short Answer

Expert verified
The level of significance in a hypothesis test, denoted \( \alpha \), is the threshold probability of rejecting the null hypothesis when it is true. It defines the degree of risk one is willing to take in accepting that the observed results are not due to chance, given that the null hypothesis is true.

Step by step solution

01

Understanding the concept of Hypothesis Testing

Hypothesis testing is a statistical method used to make inferences or draw conclusions about a population based on a sample. Typically, this involves making a statement (the null hypothesis), and then testing whether the evidence contradicts this statement (against an alternative hypothesis). It is important to understand this process before moving on to the concept of level of significance.
02

Defining Level of Significance

In the context of hypothesis testing, the level of significance is a threshold set for making a decision about the null hypothesis. It's usually denoted with the Greek letter \( \alpha \). The significance level defines how much risk you're willing to accept when you reject the null hypothesis, even though it might be true.
03

Interpreting Level of Significance

In simpler terms, the level of significance is the probability of rejecting the null hypothesis when it is true. If the p-value (the smallest significance level at which the null hypothesis would be rejected given the observed sample) is less than the chosen level of significance, we reject the null hypothesis in favor of the alternative. Therefore, a smaller level of significance requires stronger evidence before you can reject the null hypothesis.

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Most popular questions from this chapter

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 longdistance calls made by residential customers of this company 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 \(\alpha=.05\) ?

What are the four possible outcomes for a test of hypothesis? Show these outcomes by writing a table. Briefly describe the Type and Type II errors.

Make the following tests of hypotheses. 1\. \(H_{0}: \mu=80, H_{1}: \mu \neq 80, n=33, \bar{x}=76.5, \sigma=15, \alpha=.10\) b. \(H_{0}: \mu=32, H_{1}: \mu<32, n=75, \bar{x}=26.5, \sigma=7.4, \alpha=.01\) \& \(H_{0}: \mu=55, H_{1}: \mu>55, n=40, \bar{x}=60.5, \sigma=4, \quad \alpha=.05\)

Consider the null hypothesis \(H_{0}: \mu=625 .\) Suppose that a random sample of 29 observations is taken from a normally distributed population with \(\sigma=32 .\) Using a significance level of \(.01\), show the rejection and nonrejection regions on the sampling distribution curve of the sample mean and find the critical value(s) of \(z\) when the alternative hypothesis is as follows. a. \(H_{1}: \mu \neq 625\) b. \(H_{1}: \mu>625\) c. \(H_{1}: \mu<625\)

You read an article that states "50 hypothesis tests of \(H_{0}\) - \(\mu=35\) versus \(H_{1}: \mu \neq 35\) were performed using \(\alpha=.05\) on 50 different samples taken from the same population with a mean of \(35 .\) Of these, 47 tests failed to reject the null hypothesis." Explain why this type of result is not surprising.
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