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Chapter 7: Q29 E (page 408)

Suppose you are interested in conducting the statistical test of \({H_0}:\mu = 255\) against \({H_a}:\mu > 225\), and you have decided to use the following decision rule: Reject H0 if the sample mean of a random sample of 81 items is more than 270. Assume that the standard deviation of the population is 63.

a. Express the decision rule in terms of z.

b. Find \(\alpha \), the probability of making a Type I error by using this decision rule.

Short Answer

Expert verified

a. Reject the null hypothesis when\(z > 2.14\).

b. The probability of a Type I error is: \(\alpha = 0.0150\).

Step by step solution

01

Given information

The hypothesis test is:\({H_0}:\mu = 255\)versus\({H_a}:\mu > 255\).

The population standard deviation is 63.

The decision rule is: to reject the null hypothesis if the sample mean is more than 270.

02

Expressing the decision rule in terms of z

a.

Reject the null hypothesis if the sample mean is more than 270

That is,

\(\begin{aligned}\bar x > 270\\\frac{{\bar x - \mu }}{{\frac{\sigma }{{\sqrt n }}}} > \frac{{270 - \mu }}{{\frac{\sigma }{{\sqrt n }}}}\\z > \frac{{270 - 255}}{{\frac{{63}}{{\sqrt {81} }}}}\\z > \frac{{15}}{{\frac{{63}}{9}}}\\z > \frac{{15}}{7}\\z > 2.14\end{aligned}\)

Therefore, reject the null hypothesis when \(z > 2.14\).

03

Computing the probability of Type I error

b.

The probability of Type I error is obtained as follows:

\(\begin{aligned}\alpha &= P\left( {Z > 2.14} \right)\\ &= 1 - P\left( {Z \le 2.14} \right)\\ &= 1 - 0.9850\\ &= 0.0150\end{aligned}\).

The probability of a z-score less than or equal to 2.14 is obtained using the z-table.

Therefore, the probability of a Type I error is: \(\alpha = 0.0150\).

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Most popular questions from this chapter

For each of the following situations, determine the p-value and make the appropriate conclusion.

a.\({H_0}:\mu \le 25\),\({H_a}:\mu > 25\),\(\alpha = 0.01\),\(z = 2.02\)

b.\({H_0}:\mu \ge 6\),\({H_a}:\mu < 6\),\(\alpha = 0.05\),\(z = - 1.78\)

c.\({H_0}:\mu = 110\),\({H_a}:\mu \ne 110\),\(\alpha = 0.1\),\(z = - 1.93\)

d. \({H_0}:\mu = 10\), \({H_a}:\mu \ne 10\), \(\alpha = 0.05\), \(z = 1.96\)

If a hypothesis test were conducted using 伪= 0.05, for which of the following p-values would the null hypothesis be rejected?

a. .06

b. .10

c. .01

d. .001

e. .251

f. .042

Oxygen bubble velocity in a purification process. Refer to the Chemical Engineering Research and Design (March 2013) study of a method of purifying nuclear fuel waste, Exercise 6.35 (p. 349). Recall that the process involves oxidation in molten salt and tends to produce oxygen bubbles with a rising velocity. To monitor the process, the researchers collected data on bubble velocity (measured in meters per second) for a random sample of 25 photographic bubble images. These data (simulated) are reproduced in the accompanying table. When oxygen is inserted into the molten salt at a rate (called the sparging rate) of \(3.33 \times {10^{ - 6}}\) , the researchers discovered that the true mean bubble rising velocity \(\mu = .338\)

a. Conduct a test of hypothesis to determine if the true mean bubble rising velocity for the population from which the sample is selected is\(\mu = .338\)Use\(\alpha = .10\).

0.275 0.261 0.209 0.266 0.265 0.312 0.285 0.317 0.229 0.251 0.256 0.339 0.213 0.178 0.217 0.307 0.264 0.319 0.298 0.169 0.342 0.270 0.262 0.228 0.22

Refer to Exercise 7.99.

a. Find b for each of the following values of the population mean: 74, 72, 70, 68, and 66.

b. Plot each value of b you obtained in part a against its associated population mean. Show b on the vertical axis and m on the horizontal axis. Draw a curve through the five points on your graph.

c. Use your graph of part b to find the approximate probability that the hypothesis test will lead to a Type II error when m = 73.

d. Convert each of the b values you calculated in part a to the power of the test at the specified value of m. Plot the power on the vertical axis against m on the horizontal axis. Compare the graph of part b with the power curve of this part.

e. Examine the graphs of parts b and d. Explain what they reveal about the relationships among the distance between the true mean m and the null hypothesized mean m0, the value of b, and the power.

Customers who participate in a store鈥檚 free loyalty card program save money on their purchases but allow the store to keep track of the customer鈥檚 shopping habits and potentially sell these data to third parties. A Pew Internet & American Life Project Survey (January 2016) revealed that 225 of a random sample of 250 U.S. adults would agree to participate in a store loyalty card program, despite the potential for information sharing. Letp represent the true proportion of all customers who would participate in a store loyalty card program.

a. Compute a point estimate ofp

b. Consider a store owner who claims that more than 80% of all customers would participate in a loyalty card program. Set up the null and alternative hypotheses for testing whether the true proportion of all customers who would participate in a store loyalty card program exceeds .8

c. Compute the test statistic for part b.

d. Find the rejection region for the test if 伪=.01.

e. Find the p-value for the test.

f. Make the appropriate conclusion using the rejection region.

g. Make the appropriate conclusion using the p-value.
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