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Chapter 9: Problem 10

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 \(\$ 39,000\) 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 hours spent per week on house chores by all housewives is less than 30

Short Answer

Expert verified
a. Two-tailed test, \(H_0: \mu = 9.5\) hours, \(H_a: \mu \neq 9.5\) hours. b. Left-tailed test, \(H_0: \mu = \$105\), \(H_a: \mu < \$105\). c. Right-tailed test, \(H_0 : \mu = \$39,000\), \(H_a : \mu > \$39,000\). d. Two-tailed test, \(H_0: \mu = 10\) minutes, \(H_a: \mu \neq 10\) minutes. e. Left-tailed test, \(H_0: \mu = 30\) hours, \(H_a: \mu < 30\) hours.

Step by step solution

01

a. Null and Alternative Hypotheses

Null Hypothesis, \(H_0: \mu = 9.5\) hours. This is saying that the mean amount of time spent per week watching sports on television by all adult men equals 9.5 hours. Alternative Hypothesis, \(H_a: \mu \neq 9.5\) hours. This implies a two-tailed test as it seeks evidence of a difference, either higher or lower, from 9.5 hours.
02

b. Null and Alternative Hypotheses

Null Hypothesis, \(H_0: \mu = \$105\). This suggests that the mean amount spent by all customers is $105. Alternative Hypothesis, \(H_a: \mu < \$105\). This implies a left-tailed test as it seeks evidence that the mean spending is less than $105.
03

c. Null and Alternative Hypotheses

Null Hypothesis, \(H_0 : \mu = \$39,000\). This implies that the mean starting salary of college graduates is $39,000. Alternative Hypothesis, \(H_a : \mu > \$39,000\). This indicates a right-tailed test as it seeks evidence that the mean salary is more than $39,000.
04

d. Null and Alternative Hypotheses

Null Hypothesis, \(H_0: \mu = 10\) minutes. This is saying that the mean waiting time at the drive-through window at a fast food restaurant during rush hour is 10 minutes. Alternative Hypothesis, \(H_a: \mu \neq 10\) minutes. This implies a two-tailed test as it seeks evidence of a difference, either higher or lower, from 10 minutes.
05

e. Null and Alternative Hypotheses

Null Hypothesis, \(H_0: \mu = 30\) hours. This suggests that the mean hours spent on house chores per week by all housewives is 30 hours. Alternative Hypothesis, \(H_a: \mu < 30\) hours. This implies a left-tailed test as it seeks evidence that the mean hours is less than 30 hours.

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Key Concepts

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

Null Hypothesis
The null hypothesis, often symbolized as \(H_0\), represents a statement or assumption that there is no effect or no difference. It serves as the default or starting position that researchers aim to test against. In any given study, the null hypothesis is generally formulated to express neutrality or status quo.
For example, if we wanted to test if the average time spent watching TV is 9.5 hours, our null hypothesis would state that the mean \(\mu = 9.5\) hours. Here, we are assuming that the average remains constant at 9.5 hours without any deviation. In hypothesis testing, the null hypothesis is usually the statement that researchers hope to disprove or nullify.
  • Symbol: \(H_0\)
  • Represents: No effect or no change
  • Examples: \(\mu = 9.5\), \(\mu = 105\), \(\mu = 10\)
Alternative Hypothesis
The alternative hypothesis, symbolized as \(H_a\) or \(H_1\), stands in opposition to the null hypothesis. It reflects the claim or effect the researcher is testing for. This hypothesis is what researchers aim to provide evidence for through their study.
For example, if we suspect that the mean time watching TV might not be exactly 9.5 hours, our alternative hypothesis could state \(\mu eq 9.5\). This indicates a difference, suggesting that the time might be either more or less than 9.5 hours. Alternative hypotheses can vary: it might suggest that a mean is greater than, less than, or simply not equal to a specified value.
  • Symbol: \(H_a\) or \(H_1\)
  • Represents: A challenge or change
  • Examples: \(\mu eq 9.5\), \(\mu < 105\), \(\mu > 39,000\)
Two-Tailed Test
A two-tailed test is used when the alternative hypothesis does not specify a direction of the effect. This form of testing is appropriate when we are interested in determining if there is a difference, but we do not know beforehand if the difference will be greater or smaller.
For instance, when testing if the average waiting time at a drive-through differs from 10 minutes, our alternative hypothesis \(H_a: \mu eq 10\) suggests a two-tailed approach. This means we are checking for any significant deviation from 10 minutes, whether longer or shorter.
  • Looks for: A difference in either direction
  • Example Alternative Hypothesis: \(\mu eq 10\)
  • Benefits: Reveals any type of difference
Left-Tailed Test
A left-tailed test is designed when the objective is to show that a parameter is less than a certain value. The alternative hypothesis in this scenario usually indicates a decrease or diminishment.
Consider the case aiming to test if the mean amount spent by customers is less than $105; here, our alternative hypothesis \(H_a: \mu < 105\) supports a left-tailed test. By doing so, we specifically target evidence showing the mean is less than the benchmark value.
  • Focuses on: Values being lower
  • Example Alternative Hypothesis: \(\mu < 105\)
  • Analysis: Look only for decrease
Right-Tailed Test
In a right-tailed test, the goal is to demonstrate that a parameter exceeds a certain value. The alternative hypothesis in this context seeks an increase or augmentation.
Suppose you're interested in testing if the mean starting salary is greater than $39,000, the appropriate alternative hypothesis would be \(H_a: \mu > 39,000\), indicating a right-tailed test. This test aims to detect evidence of an increase over the specified threshold.
  • Focuses on: Values being higher
  • Example Alternative Hypothesis: \(\mu > 39,000\)
  • Assessment: Look only for increase

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Most popular questions from this chapter

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 \(\alpha=.05\) ?

A random sample of 500 observations produced a sample proportion equal to \(.38 .\) Find the critical and observed values of \(z\) for each of the following tests of hypotheses using \(\alpha=.05\). a. \(H_{0}: p=.30\) versus \(H_{1}: p>.30\) b. \(H_{0}: p=.30\) versus \(H_{1}: p \neq .30\)

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Professor Hansen believes that some people have the ability to predict in advance the outcome of a spin of a roulette wheel. He takes 100 student volunteers to a casino. The roulette wheel has 38 numbers, each of which is equally likely to occur. Of these 38 numbers, 18 are red, 18 are black, and 2 are green. Each student is to place a serics of five bets, choosing either a red or a black number before each spin of the wheel. Thus, a student who bets on red has an \(18 / 38\) chance of winning that bet. The same is true of betting on black. a. Assuming random guessing, what is the probability that a particular student will win all five of his or her bets? b. Suppose for each student we formulate the hypothesis test \(H_{0}:\) The student is guessing \(H_{1}:\) The student has some predictive ability Suppose we reject \(H_{0}\) only if the student wins all five bets. What is the significance level? c. Suppose that 2 of the 100 students win all five of their bets. Professor Hansen says, "For these two students we can reject \(H_{0}\) and conclude that we have found two students with some ability to predict." What do you make of Professor Hansen's conclusion?

The standard therapy used to treat a disorder cures \(60 \%\) of all patients in an average of 140 visits. A health care provider considers supporting a new therapy regime for the disorder if it is effective in reducing the number of visits while retaining the cure rate of the standard therapy. A study of 200 patients with the disorder who were treated by the new therapy regime reveals that 108 of them were cured in an average of 132 visits with a standard deviation of 38 visits. What decision should be made using a \(.01\) level of significance?
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