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Chapter 9: Problem 12

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

Short Answer

Expert verified
The difference between the critical value of \(z\) and the observed value of \(z\) indicates how significantly your result departs from the null hypothesis, with larger differences indicating more significance. It is calculated by simply subtracting the observed value from the critical value.

Step by step solution

01

Understand the Concepts

In statistics, the Z-score is a measure of how many standard deviations an element is from the mean. The critical value of \(z\) (also known as the Z critical value) is the threshold that the test statistic must exceed to reject the null hypothesis. It's typically found in a Z table. The observed value of \(z\), on the other hand, is the actual calculated z-score based on your sample data.
02

Calculating the Difference

To calculate the difference between the critical value of \(z\) and the observed value of \(z\), you just need to subtract the observed value from the critical value.

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Most popular questions from this chapter

Two years ago, \(75 \%\) of the customers of a bank said that they were satisfied with the services provided by the bank. The manager of the bank wants to know if this percentage of satisfied customers has changed since then. She assigns this responsibility to you. Briefly explain how you would conduct such a test.

Briefly explain the meaning of each of the following terms. a. Null hypothesis b. Alternative hypothesis c. Critical point(s) d. Significance level e. Nonrejection region f. Rejection region \(\mathrm{g}\). Tails of a test h. Two types of errors

According to the National Association of Colleges and Employers, the average starting salary of 2014 college graduates with a bachelor's degree was \(\$ 45,473\) (www.naceweb.org). A random sample of 1000 recent college graduates from a large city showed that their average starting salary was \(\$ 44,930\). Suppose that the population standard deviation for the starting salaries of all recent college graduates from this city is \(\$ 7820\). a. Find the \(p\) -value for the test of hypothesis with the alternative hypothesis that the average starting salary of recent college graduates from this city is less than \(\$ 45,473 .\) Will you reject the null hypothesis at \(\alpha=.01 ?\) Explain. What if \(\alpha=.025 ?\) b. Test the hypothesis of part a using the critical-value approach. Will you reject the null hypothesis at \(\alpha=.01 ?\) What if \(\alpha=.025 ?\)

In a Gallup poll conducted July \(7-10,2014,45 \%\) of Americans said that they actively try to include organic foods into their diets (www.gallup.com). In a recent sample of 2100 Americans, 1071 said that they actively try to include organic foods into their diets. Is there significant evidence at a \(1 \%\) significance level to conclude that the current percentage of all Americans who will say that they actively try to include organic foods into their diets is different from \(45 \%\) ? Use both the \(p\) -value and the critical-value approaches.

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 \(a=0\) ?
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