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Chapter 9: Problem 79

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 \(H_{1}: p \neq .30\)

Short Answer

Expert verified
The observed z-score and critical z-score are calculated for each hypothesis test. The null hypothesis is rejected if the observed z-score is extreme relative to the critical z-score in the direction specified by the alternative hypothesis.

Step by step solution

01

Calculate Observed Z-Scores

The observed value of z can be calculated using the formula \[ Z = \frac{(\hat{p} - p_{0})}{\sqrt{\frac{p_{0}(1-p_{0})}{n}}} \]\nWhere:\n- \(\hat{p}\) is the sample proportion\n- \(p_{0}\) is the null hypothesis population proportion\n- \(n\) is the sample size. \n For both tests, \(\hat{p}\) is .38, \(p_{0}\) is .30 and \(n\) is 500. So, plug in these values to find the observed z-score for each test.
02

Calculate Critical Z-Scores

To find the critical value of z, use the z-score associated with the significance level (伪) of the hypothesis test.\nFor the one-tailed test (a), find the z-score associated with 伪 = .05 in the positive direction, because the alternative hypothesis is \(H_{1}: p > .30\). This value is 1.645 (from z-table or statistical software).\nFor the two-tailed test (b), the significance level is split between both tails of the distribution, so find the z-score associated with 伪 = .025 in both directions. This value is 卤1.96.
03

Evaluate Hypothesis Tests

Compare the observed z-score to the critical z-score. If the observed value exceeds the critical value in the direction specified by the alternative hypothesis, the null hypothesis is rejected.\nFor the one-tailed test (a), if the observed z-score is greater than 1.645, reject \(H_{0}: p = .30\).\nFor the two-tailed test (b), if the observed z-score is less than -1.96 or greater than 1.96, reject \(H_{0}: p = .30\).

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Most popular questions from this chapter

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. \(H_{1}: \mu \neq 625 \quad\) b. \(H_{1}: \mu>625 \quad\) c. \(H_{1}: \mu<625\)

Consider \(H_{0^{\circ}} \mu=45\) versus \(H_{1}: \mu<45\). a. A random sample of 25 observations produced a sample mean of \(41.8 .\) Using \(\alpha=.025\), would you reject the null hypothesis? The population is known to be normally distributed with \(\sigma=6 .\) b. Another random sample of 25 observations taken from the same population produced a sample mean of \(43.8\). Using \(\alpha=.025\), would you reject the null hypothesis? The population is known to be normally distributed with \(\sigma=6\). Comment on the results of parts a and \(\mathrm{b}\).

For each of the following significance levels, what is the probability of making a Type I error? \(\begin{array}{lll}\text { a. } \alpha=.10 & \text { b. } \alpha=.02 & \text { c. } \alpha=.005\end{array}\)

For each of the following examples of tests of hypothesis about the population proportion, show the rejection and nonrejection regions on the graph of the sampling distribution of the sample proportion. a. A two-tailed test with \(\alpha=.05\) b. A left-tailed test with \(\alpha=.02\) c. A right-tailed test with \(\alpha=.025\)

A business school claims that students who complete a 3-month typing course can type, on average at least 1200 words an hour. A random sample of 25 students who completed this course typed, on aver 1125 words an hour with a standard deviation of 85 words. Assume that the typing speeds for all stu lents. who complete this course have an approximate normal distribution Suppose the probability of making a Type I error is selected to be zero. Can you conclude that the claim of the business school is true? Answer without performing the five steps of a test of hypothe b. Using a \(5 \%\) significance level, can you conclude that the claim ousiness school is true? \(\mathrm{Us}\) both approaches
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