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Chapter 8: Problem 106

Find the test statistic \(z \star\) and the \(p\) -value for each of the following situations. a. \(\quad H_{o}: \mu=22.5, H_{a}: \mu>22.5 ; \bar{x}=24.5, \sigma=6, n=36\) b. \(\quad H_{o}: \mu=200, H_{a}: \mu<200 ; \bar{x}=192.5, \sigma=40, n=50\) c. \(\quad H_{o}: \mu=12.4, H_{a}: \mu \neq 2.4 ; \bar{x}=11.52, \sigma=2.2, n=16\)

Short Answer

Expert verified
The calculation of the z test statistic and the p-value depends on the sample data given in the problem. The z score measures how many standard deviations an element is from the mean. The p-value determines the probability that we would obtain the observed sample, assuming the null hypothesis is true.

Step by step solution

01

Calculate the Sample Error

The sample error(SE) can be calculated using the formula SE = \(\sigma/\sqrt{n}\). The symbol \(\sigma\) represents the population standard deviation and \(n\) is the sample size. The values for \(\sigma\) and \(n\) are given in the problem.
02

Calculate the Z-Score

Then we can calculate the Z-score or test statistic (\(z^*\)) which can be found using the formula \(z^* = (\bar{x}-\mu_0)/SE\). Here, \(\bar{x}\) is the sample mean and \(\mu_0\) is the population mean (from null hypothesis \(H_0\)).
03

Calculate the P-value

Next, we can calculate the p-value. This is done by referring to Z-tables or using a calculator or a software that can calculate the probability associated with the Z-Score. The decision rule is to reject \(H_0\) if the p-value is less than the level of significance (usually 0.05).

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Key Concepts

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

Test Statistic
The test statistic is a crucial component in hypothesis testing. It serves as a way to quantify how much the sample data deviates from the null hypothesis. This value aids us in determining whether or not there is enough evidence to reject the null hypothesis.To calculate the test statistic, we use the formula for the Z-score:
  • First, determine the sample mean ( \( \bar{x} \)) and the hypothetical population mean from the null hypothesis ( \( \mu_0 \)).
  • Next, compute the standard error (SE), which is the standard deviation ( \( \sigma \)) divided by the square root of the sample size ( \( n \)):
\[ SE = \frac{\sigma}{\sqrt{n}} \]
  • Then, find the Z-score (or test statistic) using this formula:
\[ z^* = \frac{\bar{x} - \mu_0}{SE} \]This computed Z-score informs us how many standard deviations the sample mean is from the population mean hypothesized under the null hypothesis.
P-Value
The p-value is an essential concept in hypothesis testing. It provides a method to gauge the statistical significance of the test results. A p-value essentially tells us the probability of obtaining test results at least as extreme as the observed data, assuming that the null hypothesis is true. To calculate the p-value:
  • First, find the test statistic (Z-score)
  • Then, use statistical tables or software tools to find the p-value associated with the Z-score.
A low p-value (typically less than 0.05) suggests that the observed data is unlikely under the null hypothesis, leading to the rejection of the null hypothesis. Conversely, a high p-value indicates that the observed data is consistent with the null hypothesis.
Null Hypothesis
The null hypothesis, denoted as \( H_0 \), is the default assumption that there is no effect or difference. In other words, it's a statement of no change or status quo.In hypothesis testing, we start by assuming that the null hypothesis is true, and then analyze the sample data to determine the likelihood of observing the sample results under this assumption. For instance:
  • If we say \( H_0: \mu = 22.5 \), we're stating that the population mean is hypothesized to be 22.5.
  • The null hypothesis often acts as a baseline or a control against which the alternative hypothesis is tested.
The purpose of hypothesis testing is to evaluate whether there's enough statistical evidence in the sample data to "reject" the null hypothesis.
Z-Score
The Z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. It's expressed in terms of standard deviations from the mean.In the context of hypothesis testing:
  • The Z-score reveals how far away the sample mean is from the population mean.
  • A higher absolute Z-score indicates that the sample mean is further from the population mean, suggesting a less likely chance that the sample mean occurred by random chance under the null hypothesis.
To calculate the Z-score, subtract the population mean from the sample mean, then divide by the standard error:\[ z^* = \frac{\bar{x} - \mu_0}{SE} \]The Z-score helps determine the p-value, which in turn helps decide whether to reject the null hypothesis based on the level of statistical significance.

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Most popular questions from this chapter

The expected mean of a continuous population is \(100,\) and its standard deviation is \(12 .\) A sample of 50 measurements gives a sample mean of 96. Using a 0.01 level of significance, a test is to be made to decide between "the population mean is 100 " and "the population mean is different from \(100 . "\) State or find each of the following: a. \(H_{o}\) b. \(H_{a}\) c. \(\alpha\) d. \(\mu\left(\text { based on } H_{o}\right)\) e. \(\bar{x}\) f . \(\boldsymbol{\sigma}\) g. \(\sigma_{x}\) h. \(z *, z\) -score for \(\bar{x}\) i. \(p\) -value j. Decision k. Sketch the standard normal curve and locate \(z \star\) and \(p\) -value.

For each of the following pairs of values, state the decision that will occur and why. a. \(\quad p\) -value \(=0.014, \alpha=0.02\) b. \(\quad p\) -value \(=0.118, \alpha=0.05\) c. \(\quad p\) -value \(=0.048, \alpha=0.05\) d. \(\quad p\) -value \(=0.064, \alpha=0.10\)

Use a computer and generate 50 random samples, each of size \(n=28,\) from a normal probability distribution with \(\mu=18\) and \(\sigma=4\) a. Calculate the \(z\) * corresponding to each sample mean. b. In regard to the \(p\) -value approach, find the proportion of \(50 z \star\) values that are "more extreme" than the \(z=-1.04\) that occurred in Exercise 8.201 \(\left(H_{a}: \mu \neq 18\right) .\) Explain what this proportion represents. c. In regard to the classical approach, find the critical values for a two- tailed test using \(\alpha=0.01 ;\) find the proportion of \(50 z \star\) values that fall in the critical region. Explain what this proportion represents.

At a very large firm, the clerk-typists were sampled to see whether salaries differed among departments for workers in similar categories. In a sample of 50 of the firm's accounting clerks, the average annual salary was \(\$ 16,010 .\) The firm's personnel office insists that the average salary paid to all clerk-typists in the firm is \(\$ 15,650\) and that the standard deviation is \(\$ 1800 .\) At the 0.05 level of significance, can we conclude that the accounting clerks receive, on average, a different salary from that of the clerk-typists? a. Solve using the \(p\) -value approach. b. Solve using the classical approach.

Jack Williams is vice president of marketing for one of the largest natural gas companies in the nation. During the past 4 years, he has watched two major factors erode the profits and sales of the company. First, the average price of crude oil has been virtually flat, and many of his industrial customers are burning heavy oil rather than natural gas to fire their furnaces, regardless of added smokestack emissions. Second, both residential and commercial customers are still pursuing energy-conservation techniques (e.g., adding extra insulation, installing clockdrive thermostats, and sealing cracks around doors and windows to eliminate cold air infiltration). In previous years, residential customers bought an average of 129.2 mcf of natural gas from Jack's company \((\sigma=18 \mathrm{mcf})\) based on internal company billing records, but environmentalists have claimed that conservation is cutting fuel consumption up to \(3 \%\) per year. Jack has commissioned you to conduct a spot check to see if any change in annual usage has transpired before his next meeting with the officers of the corporation. A sample of 300 customers selected randomly from the billing records reveals an average of \(127.1 \mathrm{mcf}\) during the past 12 months. Is there a significant decline in consumption? a. Complete the appropriate hypothesis test at the 0.01 level of significance using the \(p\) -value approach so that you can properly advise Jack before his meeting. b. Because you are Jack's assistant, why is it best for you to use the \(p\) -value approach?
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