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Business / Statistics for Business and Economics 12 / Chapter 7 / Problem 100E

Statistics for Business and Economics | 12th Edition | ISBN: 9780321826237 | Authors: James T. McClave, P. George Benson, Terry T Sincich

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Refer to Exercise 7.99.a. Find for each of the following

Chapter 7, Problem 100E

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QUESTION:

Problem 100E

Refer to Exercise 7.99.

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

b. Plot each value of you obtained in part a against its associated population mean. Show on the vertical axis and 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 = 73.

d. Convert each of the values you calculated in part a to the power of the test at the specified value of . Plot the power on the vertical axis against on the horizontal axis. Compare the graph of part b to 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 and the null hypothesized mean 0, the value of , and the power.

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QUESTION:

Problem 100E

Refer to Exercise 7.99.

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

b. Plot each value of you obtained in part a against its associated population mean. Show on the vertical axis and 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 = 73.

d. Convert each of the values you calculated in part a to the power of the test at the specified value of . Plot the power on the vertical axis against on the horizontal axis. Compare the graph of part b to 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 and the null hypothesized mean 0, the value of , and the power.

ANSWER:

Solution

Step 1 of 5

From the given exercise we are testing $H{\ }_{0}:\mu{}=75\ against\ H{\ }_{1}:\mu{}<75$

And $\alpha{}=0.1$

$\sigma{}=15$

$n=49$

$\overline{x}=72.257$

a) We have to find the $\beta{}$ value for population mean :74, 72, 70, 68, and 66

If $\mu{}=74\$ $\beta{}=P(\overline{x}>72.275)$

= $P(z>\frac{72.275-74}{15/\sqrt{49}})$

$=P(z>-0.81)$

=1- $P(z<-0.81)$

=1-0.20897

=0.79103

If $\mu{}=72\$ $\beta{}=P(\overline{x}>72.275)$

= $P(z>\frac{72.275-72}{15/\sqrt{49}})$

$=P(z>0.12)$

=1- $P(z<0.12)$

=1-0.54116

=0.45884

If $\mu{}=70\$ $\beta{}=P(\overline{x}>72.275)$

= $P(z>\frac{72.275-70}{15/\sqrt{49}})$

$=P(z>1.05)$

=1- $P(z<1.05)$

=1-0.8531

=0.1469

If $\mu{}=68\$ $\beta{}=P(\overline{x}>72.275)$

= $P(z>\frac{72.275-68}{15/\sqrt{49}})$

$=P(z>1.99)$

=1- $P(z<1.99)$

=1-0.9767

=0.0233

If $\mu{}=66\$ $\beta{}=P(\overline{x}>72.275)$

= $P(z>\frac{72.275-66}{15/\sqrt{49}})$

$=P(z>2.92)$

=1- $P(z<2.92)$

=1-0.9982

=0.0018

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