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Chapter 9: Problem 8

Write the null and alternative hypotheses for each of the following examples. Determine if each is a case of a two-tailed, a left-tailed, or a right-tailed test. a. To test if the mean amount of time spent per week watching sports on television by all adult men is different from \(9.5\) hours b. To test if the mean amount of money spent by all customers at a supermarket is less than \(\$ 105\) c. To test whether the mean starting salary of college graduates is higher than \(\$ 47,000\) per year d. To test if the mean waiting time at the drive-through window at a fast food restaurant during rush hour differs from 10 minutes e. To test if the mean time spent per week on house chores by all housewives is less than 30 hours

Short Answer

Expert verified
a. Null hypothesis: \( H_0: \mu = 9.5 \), Alternative hypothesis: \( H_1: \mu \neq 9.5 \), Test: Two-tailed; b. Null hypothesis: \( H_0: \mu \geq 105 \), Alternative hypothesis: \( H_1: \mu < 105 \), Test: Left-tailed; c. Null hypothesis: \( H_0: \mu \leq 47000 \), Alternative hypothesis: \( H_1: \mu > 47000 \), Test: Right-tailed; d. Null hypothesis: \( H_0: \mu = 10 \), Alternative hypothesis: \( H_1: \mu \neq 10 \), Test: Two-tailed; e. Null hypothesis: \( H_0: \mu \geq 30 \), Alternative hypothesis: \( H_1: \mu < 30 \), Test: Left-tailed.

Step by step solution

01

Hypothesis for the first scenario

In this situation, the null hypothesis \( H_0: \mu = 9.5 \) states that the mean time spent watching sports on television by all adult men per week equals 9.5 hours. The alternative hypothesis \( H_1: \mu \neq 9.5 \) suggests that it isn't equal to 9.5 hours. This is a case of a two-tailed test because we're testing if the mean is different from 9.5, not specifically less or greater.
02

Hypothesis for the second scenario

Here, the null hypothesis \( H_0: \mu \geq 105 \) posits that the mean amount of money spent by all customers at a supermarket is at least \(\$ 105\). The alternative hypothesis \( H_1: \mu < 105 \) suggests that it's less than \(\$ 105\). This is a left-tailed test because we're testing if the mean is less than a specified value.
03

Hypothesis for the third scenario

In this case, the null hypothesis \( H_0: \mu \leq 47000 \) asserts that the mean starting salary of college graduates is less than or equal to \(\$ 47,000\). The alternative hypothesis \( H_1: \mu > 47000 \) indicates it is more than \(\$ 47,000\). This is a right-tailed test because we're testing if the mean is more than a certain value.
04

Hypothesis for the fourth scenario

For this situation, the null hypothesis \( H_0: \mu = 10 \) states that the mean waiting time at a fast-food restaurant's drive-through window during rush hour is 10 minutes. The alternative hypothesis \( H_1: \mu \neq 10 \) suggests that it's not equal to 10 minutes. This is a two-tailed test because we're testing if the mean differs from a set value, not if it鈥檚 specifically less or more.
05

Hypothesis for the fifth scenario

In this scenario, the null hypothesis \( H_0: \mu \geq 30 \) posits that the mean time spent on house chores by all housewives per week is at least 30 hours. The alternative hypothesis \( H_1: \mu < 30 \) implies that the mean is less than 30 hours. This is a left-tailed test because we're investigating if the mean is less than a particular value.

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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, typically denoted by \( H_0 \), is a statement that suggests no effect or no difference in a certain context. The purpose of the null hypothesis is to serve as a starting point that researchers can test against. By saying there's no difference, it's like the status quo, allowing you to have a baseline to compare with your alternative. For example, if you're testing whether the mean starting salary of college graduates is higher than \\(47,000, the null hypothesis would be \( H_0: \mu \leq 47000 \). Here, you assume the mean is less than or equal to \\)47,000 until you have enough evidence to suggest otherwise.
Alternative Hypothesis
The alternative hypothesis, denoted as \( H_1 \) or \( H_a \), suggests there is an effect, a difference, or a relationship in the situation being analyzed. This is what researchers hope to prove. For instance, if you want to check if the amount of money spent by customers at a supermarket is less than \( \$105 \), the alternative hypothesis would be \( H_1: \mu < 105 \). This sets the stage for the hypothesis test, as you're looking for evidence to reject the null hypothesis in favor of the alternative.
Two-Tailed Test
A two-tailed test is used when you want to check if there is a significant difference on either side of a certain parameter. You're testing if the actual value is either higher or lower than the specified value. In statistics speak, you're not interested in direction but rather in whether it *differs* at all. For example, if you're investigating if the waiting time at a fast-food drive-through during rush hour differs from 10 minutes, the two hypotheses would be \( H_0: \mu = 10 \) and \( H_1: \mu eq 10 \). This means you're open to finding the mean waiting time being either less than or more than 10 minutes.
Left-Tailed Test
In a left-tailed test, the focus is on finding out if a particular statistic is significantly less than a given number. You'd use this when you're interested in decreases. Consider a supermarket testing if the mean amount of money spent is less than \( \\(105 \). Here, the null and alternative hypotheses are \( H_0: \mu \geq 105 \) and \( H_1: \mu < 105 \). By conducting a left-tailed test, you're checking if you have enough evidence to claim that the mean spending is indeed lower than \\)105.
Right-Tailed Test
A right-tailed test looks for evidence of a statistic being significantly greater than a predefined value. This type of test is ideal when hypothesizing about increases. For instance, if you are examining whether the mean starting salary for college graduates is more than \\(47,000, your set would be \( H_0: \mu \leq 47000 \) against \( H_1: \mu > 47000 \). The hypothesis test is done to see if you can confidently assert that the average salary exceeds \\)47,000 based on the data you gather.

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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 \(\alpha=.05\) ?

According to Moebs Services Inc., the average cost of an individual checking account to major U.S. banks was \(\$ 380\) in 2013 (www. moebs.com). A bank consultant wants to determine whether the current mean cost of such checking accounts at major U.S. banks is more than \(\$ 380\) a year, A recent random sample of 150 such checking accounts taken from major U.S. banks produced a mean annual cost to them of \(\$ 390\). Assume that the standard deviation of annual costs to major banks of all such checking accounts is \(\$ 60 .\) a. Find the \(p\) -value for this test of hypothesis. Based on this \(p\) -value, would you reject the null hypothesis if the maximum probability of Type I error is to be \(05 ?\) What if the maximum probability of Type I error is to be \(.01\) ? b. Test the hypothesis of part a using the critical-value approach and \(\alpha=.05\). Would you reject the null hypothesis? What if \(\alpha=.01 ?\) What if \(a=0\) ?

Consider the following null and alternative hypotheses: $$ H_{0}: p=.82 \text { versus } H_{1}: p \neq .82 $$ A random sample of 600 observations taken from this population produced a sample proportion of \(.86\). a. If this test is made at a \(2 \%\) 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 \(\alpha=.025\) ? What if \(\alpha=.01 ?\)

Consider \(H_{0}: \mu=72\) versus \(H_{1}: \mu>72 . \Lambda\) random sample of 16 observations taken from this population produced a sample mean of \(75.2\). The population is normally distributed with \(\sigma=6\). a. Calculate the \(p\) -value. b. Considering the \(p\) -value of part a, would you reject the null hypothesis if the test were made at a significance level of \(.01\) ? c. Considering the \(p\) -value of part a, would you reject the null hypothesis if the test were made at a significance level of \(.025 ?\)

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. \(H_{1}: \mu<625\)
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