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Business / Statistics for Business and Economics 12 / Chapter 7 / Problem 142SE

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

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QUESTION:

Problem 142SE

Ages of cable TV shoppers. In a paper presented at the 2000 Conference of the International Association for Time Use Research, professor Margaret Sanik of Ohio State University reported the results of her study on American cable TV viewers who purchase items from one of the home shopping channels. She found that the average age of these cable TV shoppers was 51 years. Suppose you want to test the null hypothesis, H0: = 51, using a sample of n = 50 cable TV shoppers.

a. Find the p-value of a two-tailed test if = 52.3 and s = 7.1.

b. Find the p-value of an upper-tailed test if = 52.3 and s = 7.1.

c. Find the p-value of a two-tailed test if = 52.3 and s = 10.4.

d. For each of the tests, parts a–c, give a value of that will lead to a rejection of the null hypothesis.

e. If = 52.3, give a value of s that will yield a two-tailed p-value of .01 or less.

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QUESTION:

Problem 142SE

Ages of cable TV shoppers. In a paper presented at the 2000 Conference of the International Association for Time Use Research, professor Margaret Sanik of Ohio State University reported the results of her study on American cable TV viewers who purchase items from one of the home shopping channels. She found that the average age of these cable TV shoppers was 51 years. Suppose you want to test the null hypothesis, H0: = 51, using a sample of n = 50 cable TV shoppers.

a. Find the p-value of a two-tailed test if = 52.3 and s = 7.1.

b. Find the p-value of an upper-tailed test if = 52.3 and s = 7.1.

c. Find the p-value of a two-tailed test if = 52.3 and s = 10.4.

d. For each of the tests, parts a–c, give a value of that will lead to a rejection of the null hypothesis.

e. If = 52.3, give a value of s that will yield a two-tailed p-value of .01 or less.

ANSWER:

Answer

Step 1 of 5

(a)

Suppose you want to test the null hypothesis, ${H}_{0}:\ \mu{}\ =51,$ using a sample of $n\ =50$ cable TV shoppers.

Find the $p-value$ of a two-tailed test if $\overline{x}\ =52.3\ and\ s\ =7.1.$

In hypothesis testing, the general form for a two-tailed test about a population mean is

as follows:

${H}_{0}:\ \mu{}\ =\ {\mu{}}_{0}$

$\ \ {H}_{a}:\ \ \mu{}\ \ne{}\ {\mu{}}_{0}$

Figure 1: One Proportion $Z-test$ (two tail test)

Hence the test statistic for hypothesis tests about a population mean when $\sigma{}$ is known,

$z\ =\frac{\overline{x}\ -{\mu{}}_{0}}{{\sigma{}}_{\overline{x}}}$

We know ${\sigma{}}_{\overline{x}}\ =\frac{s}{\sqrt{n}},$

$z\ =\frac{\overline{x}\ -{\mu{}}_{0}}{\frac{s}{\sqrt{n}}}$

$=\ \frac{52.3\ -51}{7.1\ /\sqrt{50}}\ =1.2947$

Hence the $p-value$ using figure 1,

$p-value\ =P(z\geq{}1.295)+P(z\leq{}-1.295)$

$p-value\ =2\times{}P(z\leq{}-1.295)$

[using symmetry property of $z-distribution$ ]

From the $z\$ table II, Appendix D, the value of $P(z\leq{}-1.295),$

$P(z\leq{}-1.295)\ =0.0985$

$p-value\ =2\times{}0.0985\ =0.197$

Hence the $p-value$ is $0.197.$

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