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Chapter 9: Problem 32

Make the following tests of hypotheses. a. \(H_{0}: \mu=25, \quad H_{1}: \mu \neq 25, \quad n=81, \quad \bar{x}=28.5, \quad \sigma=3, \quad \alpha=.01\) b. \(H_{0}=\mu=12, \quad H_{1}: \mu<12, \quad n=45, \quad \bar{x}=11.25, \quad \sigma=4.5, \quad \alpha=.05\) c. \(H_{0}=\mu=40, \quad H_{1}: \mu>40, \quad n=100, \quad \bar{x}=47, \quad \sigma=7, \quad \alpha=.10\)

Short Answer

Expert verified
For part a, we are using a two-tailed hypothesis test and for parts b and c, we are using left-tailed and right-tailed tests, respectively. The test statistic Z is calculated and compared with the critical value(s). The null hypothesis \(H_0\) is rejected if the absolute value of Z is greater than the critical value (two-tailed) or if Z is less than the negative critical value (left-tailed) or greater than the critical value (right-tailed).

Step by step solution

01

Identify the Type of Hypothesis

For part a, the alternative hypothesis \(H_1: \mu \neq 25\) is a two-tailed hypothesis and because the population standard deviation \(\sigma\) is given, we can use the z-test. For part b, \(H_1: \mu<12\) is a left-tailed hypothesis and we'll again use the z-test. For part c, \(H_1: \mu>40\) is a right-tailed hypothesis and we'll use the z-test.
02

Calculate the Test Statistic

We can calculate the test statistic using the formula \[Z = (\bar{x} - \mu) / (\sigma / \sqrt{n})\].For part a, it will be \[Z = (28.5 - 25) / (3 / \sqrt{81})\].For part b, \[Z = (11.25 - 12) / (4.5 / \sqrt{45})\].For part c, \[Z = (47 - 40) / (7 / \sqrt{100})\].
03

Identify the Critical Value(s)

For part a, the critical values for a two-tailed test with \(\alpha = .01\) is 卤2.576. For part b, the critical value for a left-tailed test with \(\alpha = .05\) is -1.64485.For part c, the critical value for a right-tailed test with \(\alpha = .10\) is 1.28155.
04

Make a Conclusion

For each part, compare the calculated test statistic with the critical value(s). For part a, if \(|Z|\) > 2.576, reject \(H_0\); otherwise, do not reject \(H_0\).For part b, if \(Z\) < -1.64485, reject \(H_0\); otherwise, do not reject \(H_0\).For part c, if \(Z\) > 1.28155, reject \(H_0\); otherwise, do not reject \(H_0\).

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Key Concepts

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

Z-Test
A z-test is a type of hypothesis test used to determine if there is a significant difference between sample and population means. It's applicable when the population standard deviation is known and the sample size is large (usually >30).
This test compares the sample mean to the population mean using a z-score, which measures how many standard deviations the sample mean is from the population mean. The z-test formula is:
\[Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}\]Where:
  • \(\bar{x}\) is the sample mean.
  • \(\mu\) is the population mean.
  • \(\sigma\) is the population standard deviation.
  • \(n\) is the sample size.
If the calculated z-score is beyond the critical value, the null hypothesis is rejected.
Critical Value
The critical value is a cut-off point described by the significance level (\(\alpha\)) of a hypothesis test. It determines the boundaries for the rejection region of the null hypothesis. The type of hypothesis test (two-tailed, left-tailed, or right-tailed) affects the critical value used.
In a two-tailed test, like in part a of the exercise, the critical values would be symmetric around zero:
  • For \(\alpha = 0.01\), the critical values are \(卤2.576\).
In a left-tailed test, the critical value lies to the left:
  • For \(\alpha = 0.05\), it is \(-1.64485\).
Right-tailed test critical values lie to the right:
  • For \(\alpha = 0.10\), it is \(1.28155\).
The comparison between test statistic and critical value leads to a decision about rejecting or not rejecting the null hypothesis.
Test Statistic
The test statistic in a z-test is a particular z-score calculated from the sample data. It helps in determining the likelihood of observing the sample results under the assumption that the null hypothesis is true.
The formula for calculating the z-score, or the test statistic, is:\[Z = \frac{\bar{x} - \mu}{\sigma / \sqrt{n}}\]Using this formula, we develop a z-score which indicates whether the difference between the sample and population means is significant.
  • For part a: \(Z = 3.5\)
  • For part b: \(Z = -0.8348\)
  • For part c: \(Z = 10\)
These z-scores are then compared against the critical values to determine if a null hypothesis can be rejected.
Null Hypothesis
The null hypothesis (\(H_0\)) is a statement that assumes no effect or no difference exists. In hypothesis testing, it serves as the default position that one aims to test against. For each of the parts in the exercise, the null hypothesis specifies a particular population mean:
  • Part a: \(H_0: \mu = 25\)
  • Part b: \(H_0: \mu = 12\)
  • Part c: \(H_0: \mu = 40\)
The process of hypothesis testing involves examining the evidence provided by sample data. If the data suggests the null hypothesis is unlikely, it is rejected in favor of the alternative hypothesis. However, if the data does not provide sufficient evidence against \(H_0\), it is not rejected. Instead, researchers continue to operate under the assumption that the null hypothesis is true.

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Most popular questions from this chapter

The mean balance of all checking accounts at a bank on December 31,2011, was \(\$ 850 .\) A random sample of 55 checking accounts taken recently from this bank gave a mean balance of \(\$ 780\) with a standard deviation of \(\$ 230 .\) Using a \(1 \%\) significance level, can you conclude that the mean balance of such accounts has decreased during this period? Explain your conclusion in words. What if \(\alpha=.025\) ?

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 hours spent per week on house chores by all housewives is less than 30

Briefly explain the conditions that must hold true to use the \(t\) distribution to make a test of hypothesis about the population mean.

A random sample of 80 observations produced a sample mean of \(86.50 .\) Find the critical and observed values of \(z\) for each of the following tests of hypothesis using \(\alpha=.10 .\) The population standard deviation is known to be \(7.20\). a. \(H_{0}: \mu=91 \quad\) versus \(\quad H_{1}: \mu \neq 91\) b. \(H_{0}=\mu=91\) versus \(\quad H_{1}: \mu<91\)

What is the difference between the critical value of \(z\) and the observed value of \(z\) ?
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