Question

In this problem, assume that the distribution of differences is
approximately normal. *Note*: For degrees of freedom
*d*.*f*. not in the Student's *t* table, use
the closest *d*.*f*. that is *smaller*. In
some situations, this choice of *d*.*f*. may increase
the *P*-value by a small amount and therefore produce a
slightly more "conservative" answer.

Are America's top chief executive officers (CEOs) really worth all
that money? One way to answer this question is to look at row
*B*, the annual company percentage increase in revenue,
versus row *A*, the CEO's annual percentage salary increase
in that same company. Suppose a random sample of companies yielded
the following data:

B: Percent increasefor company |
24 | 25 | 25 | 18 | 6 | 4 | 21 | 37 |

A: Percent
increasefor CEO |
21 | 23 | 22 | 14 | −4 | 19 | 15 | 30 |

Do these data indicate that the population mean percentage
increase in corporate revenue (row *B*) is different from
the population mean percentage increase in CEO salary? Use a 5%
level of significance. (Let *d* = *B* −
*A*.)

(A) What is the level of significance?

_______________

State the null and alternate hypotheses.

*a) H*_{0}: *μ*_{d} = 0;
*H*_{1}: *μ*_{d} > 0

*b) H*_{0}: *μ*_{d} ≠ 0;
*H*_{1}: *μ*_{d} =
0

*c) H*_{0}: *μ*_{d} >
0; *H*_{1}: *μ*_{d} = 0

*d) H*_{0}: *μ*_{d} = 0;
*H*_{1}: *μ*_{d} ≠ 0

*e) H*_{0}: *μ*_{d} = 0;
*H*_{1}: *μ*_{d} < 0

(B) What sampling distribution will you use? What assumptions are
you making?

a) The standard normal. We assume that *d* has an
approximately uniform distribution.

b) The standard normal. We assume that *d* has an
approximately normal distribution.

c) The Student's *t*. We assume that *d* has an
approximately uniform distribution.

d) The Student's *t*. We assume that *d* has an
approximately normal distribution.

What is the value of the sample test statistic? (Round your answer
to three decimal places.)

________________

(c) Find (or estimate) the *P*-value.

*a) P*-value > 0.500

b) 0.250 < *P*-value <
0.500

c) 0.100 < *P*-value < 0.250

d) 0.050 < *P*-value < 0.100

e) 0.010 < *P*-value < 0.050

*f) P*-value < 0.010

Sketch the sampling distribution and show the area corresponding to
the *P*-value.

**Top right, top left, bottom right** or
**bottom left**?

(D) Based on your answers in parts (a) to (c), will you reject or
fail to reject the null hypothesis? Are the data statistically
significant at level *α*?

a) Since the *P*-value ≤ *α*, we reject
*H*_{0}. The data are statistically significant.

b) Since the *P*-value > *α*, we fail to reject
*H*_{0}. The data are not statistically
significant.

c) Since the *P*-value ≤ *α*, we fail to reject
*H*_{0}. The data are statistically significant.

d) Since the *P*-value > *α*, we reject
*H*_{0}. The data are not statistically
significant.

(E) Interpret your conclusion in the context of the
application.

a) Fail to reject *H*_{0}. At the 5% level of
significance, the evidence is insufficient to claim a difference in
population mean percentage increases for corporate revenue and CEO
salary.

b) Reject *H*_{0}. At the 5% level of
significance, the evidence is insufficient to claim a difference in
population mean percentage increases for corporate revenue and CEO
salary.

c) Fail to reject *H*_{0}. At the 5% level of
significance, the evidence is sufficient to claim a difference in
population mean percentage increases for corporate revenue and CEO
salary.

d) Reject *H*_{0}. At the 5% level of
significance, the evidence is sufficient to claim a difference in
population mean percentage increases for corporate revenue and CEO
salary.

Answer #1

a)

0.05

d) H0: μd = 0; H1: μd ≠ 0

b)

d) The Student's t. We assume that d has an approximately normal
distribution.

Test statistic,

t = (dbar - 0)/(s(d)/sqrt(n))

t = (2.5 - 0)/(7.5404/sqrt(8))

t = 0.938

c)

P-value Approach

P-value = 0.3795

b) 0.250 < P-value < 0.500

d)

b) Since the P-value > α, we fail to reject H0. The data are not statistically significant.

e)

a) Fail to reject H0. At the 5% level of significance, the evidence is insufficient to claim a difference in population mean percentage increases for corporate revenue and CEO salary.

In this problem, assume that the distribution of differences is
approximately normal. Note: For degrees of freedom
d.f. not in the Student's t table, use
the closest d.f. that is smaller. In
some situations, this choice of d.f. may increase
the P-value by a small amount and therefore produce a
slightly more "conservative" answer.
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that money? One way to answer this question is to look at row
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the P-value by a small amount and therefore produce a
slightly more "conservative" answer.
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more "conservative" answer.
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approximately normal. Note: For degrees of freedom d.f. not in the
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approximately normal. Note: For degrees of freedom
d.f. not in the Student's t table, use
the closest d.f. that is smaller. In
some situations, this choice of d.f. may increase
the P-value by a small amount and therefore produce a
slightly more "conservative" answer.
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approximately normal. Note: For degrees of freedom
d.f. not in the Student's t table, use
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some situations, this choice of d.f. may increase
the P-value by a small amount and therefore produce a
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