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Chapter 9: Problem 18

Consider the following null and alternative hypotheses: $$ H_{0}: \mu=60 \text { versus } \quad H_{1}: \mu>60 $$ Suppose you perform this test at \(\alpha=.01\) and fail to reject the null hypothesis. Would you state that the difference between the hypothesized value of the population mean and the observed value of the sample mean is "statistically significant" or would you state that this difference is "statistically not significant?" Explain.

Short Answer

Expert verified
The difference between the hypothesized value of the population mean and the observed value of the sample mean is 'statistically not significant'. This is because the null hypothesis was not rejected given the test at significance level \(\alpha=.01\).

Step by step solution

01

Understanding the Hypotheses

The null hypothesis \(H_0: \mu = 60\) states that the population mean is 60. The alternative hypothesis \(H_1: \mu > 60\) implies that the population mean is greater than 60. The goal is to test these hypotheses.
02

About Significance Level

The significance level \(\alpha = 0.01\) is the probability of rejecting the null hypothesis when it is true. Essentially, it's the risk tolerated of making a Type I error (rejecting a true null hypothesis).
03

Interpretation of the Results

In this exercise, the null hypothesis is not rejected. This means, based on the sample data and the chosen alpha level, there isn't enough evidence to conclude that the population mean is greater than 60. Thus, the difference between the observed sample mean and the hypothesized population mean of 60 would be 鈥榮tatistically not significant鈥�.

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Key Concepts

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

Null Hypothesis
In hypothesis testing, the null hypothesis, denoted as \(H_{0}\), serves as the statement that you initially assume to be true. It represents no effect or no difference. In our exercise, the null hypothesis is \(H_{0}: \mu = 60\). This means we assume that the average of the entire population is 60 unless we have strong evidence to prove otherwise.
  • This approach prevents us from claiming changes or differences when there aren鈥檛 any.
  • The null hypothesis is usually what you try to find evidence against.
By having a defined null hypothesis, we can conduct statistical tests to determine if the data provides enough evidence to reject it.
Alternative Hypothesis
The alternative hypothesis, denoted as \(H_{1}\), is a statement that contradicts the null hypothesis. It represents the presence of an effect or a difference. In the given exercise, \(H_{1}: \mu > 60\) suggests that the true population mean is greater than 60.
  • The alternative hypothesis is what you wish to support or prove with your test.
  • It is often formulated based on the research question or the effect you suspect exists.
By comparing data against the null hypothesis, we can see if there's significant evidence to support the alternative hypothesis. If our test concludes favorably for \(H_{1}\), we have found the difference we were looking for.
Significance Level
The significance level, denoted by \(\alpha\), is the probability of making a Type I error in hypothesis testing. It sets the threshold for the statistical significance of our test. In this problem, \(\alpha = 0.01\).
  • This means there's a 1% risk of rejecting a true null hypothesis.
  • Lower significance levels represent stricter criteria for claiming a statistically significant result.
A lower \(\alpha\) level like 0.01 indicates that you require stronger evidence to reject \(H_{0}\). Typically, common choices for \(\alpha\) levels are 0.01, 0.05, and 0.10, depending on how confident you want to be in your results.
Type I Error
A Type I error occurs when the null hypothesis is true, but we incorrectly reject it based on our sample data. With our predetermined significance level, \(\alpha\), we handle the likelihood of making such errors.
  • In the context of the exercise, this would mean claiming \(\mu > 60\) when, in fact, \(\mu = 60\) is true.
  • The chance of committing a Type I error is equal to the significance level \(\alpha\): 1% in this case.
Understanding Type I errors helps us appreciate why choosing an appropriate \(\alpha\) is crucial. It balances the probability of falsely identifying a non-existent effect with the desire to detect true effects.

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Most popular questions from this chapter

Consider \(H_{0}: p=.45\) versus \(H_{1}: p<.45\). a. A random sample of 400 observations produced a sample proportion equal to .42. Using \(\alpha=.025\), would you reject the null hypothesis? b. Another random sample of 400 observations taken from the same population produced a sample proportion of .39. Using \(a=.025\), would you reject the null hypothesis? Comment on the results of parts a and \(\mathrm{b}\).

Make the following hypothesis tests about \(p\). a. \(H_{0}: p=.45, \quad H_{1}: p \neq .45, n=100, \quad \hat{p}=.49, \quad \alpha=.10\) b. \(H_{0}: p=.72, \quad H_{1}: p<.72, n=700, \quad \hat{p}=.64, \quad a=.05\) c. \(H_{0}=p=.30, \quad H_{1}: p>.30, n=200, \quad \hat{p}=.33, \quad \alpha=.01\)

Briefly explain the procedure used to calculate the \(p\) -value for a two- tailed and for a one-tailed test, respectively.

PolicyInteractive of Eugene, Oregon conducted a study of American adults in April 2014 for the Center for a New American Dream. Seventy-five percent of the adults included in this study said that having basic needs met is very or extremely important in their vision of the American dream (www.newdream.org). A recent sample of 1500 American adults were asked the same question and \(72 \%\) of them said that having basic needs met is very or extremely important in their vision of the American dream. a. Using the critical-value approach and \(\alpha=.01\), test if the current percentage of American adults who hold the abovementioned opinion is less than \(75 \%\). b. How do you explain the Type I error in part a? What is the probability of making this error in part a? c. Calculate the \(p\) -value for the test of part a. What is your conclusion if \(\alpha=.01 ?\)

According to a 2014 CIRP Your First College Year Survey, \(88 \%\) of the first- year college students said that their college experience exposed them to diverse opinions, cultures, and values (www. heri.ucla.edu). Suppose in a recent poll of 1800 first-year college students, \(91 \%\) said that their college experience exposed them to diverse opinions, cultures, and values. Perform a hypothesis test to determine if it is reasonable to conclude that the current percentage of all firstyear college students who will say that their college experience exposed them to diverse opinions, cultures, and values is higher than \(88 \% .\) Use a \(2 \%\) significance level, and use both the \(p\) -value and the critical-value approaches.
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