Consider a large-sample level 0.01 test for testing h0: p = 0.2 against ha: p > 0.2.

8.2 Let the test statistic Z have a standard normal distribution when H0 is true. Give the P-value for each of the following situations. (Round your answers to four decimal places.)

(a) Ha: μ > μ0,

z = 1.78

P-value = 1 (b) Ha: μ < μ0,

z = −2.74

P-value = 2 (c) Ha: μ ≠ μ0,

z = 2.83 or z = −2.83

P-value = 3

Let the test statistic T have a t distribution when H0 is true. Give the P-value for each of the following situations. (Round your answers to three decimal places.)

(a)

Ha: μ > μ0, df = 18, t = 2.964

P-value = 1 (b)

Ha: μ < μ0, n = 24, t = 2.441

P-value = 2 (c)

Ha: μ ≠ μ0, n = 36, t = −1.791 or t = 1.791

P-value = 3

You may need to use the appropriate table in the Appendix of Tables to answer this question.

8.5 The drying time of a certain type of paint under specified test conditions is known to be normally distributed with mean value 75 min and standard deviation 9 min. Chemists have proposed a new additive designed to decrease average drying time. It is believed that drying times with this additive will remain normally distributed with σ = 9. Because of the expense associated with the additive, evidence should strongly suggest an improvement in average drying time before such a conclusion is adopted. Let μ denote the true average drying time when the additive is used. The appropriate hypotheses are H0: μ = 75 versus Ha: μ < 75. Consider the alternative value μ = 74, which in the context of the problem would presumably not be a practically significant departure from H0.

(a) For a level 0.01 test, compute β at this alternative for sample sizes n = 81, 900, and 2500. (Round your answers to four decimal places.)

n

β

81

1

900

2

2500

3

(b) If the observed value of

X

is

x = 74,

what can you say about the resulting P-value when n = 2500? Is the data statistically significant at any of the standard values of α? (Round your z to two decimal places. Round your P-value to four decimal places.)

z

=

4

P-value

=

5

Consider a large-sample level 0.01 test for testing

H0: p = 0.2

against Ha: p > 0.2.

(a) For the alternative value p = 0.21, compute β(0.21) for sample sizes n = 100, 900, 10,000, 40,000, and 90,000. (Round your answers to four decimal places.)

n

β

100

1

900

2

10,000

3

40,000

4

90,000

5

(b) For p̂ = x/n = 0.21, compute the P-value when n = 100, 900, 10,000, and 40,000. (Round your answers to four decimal places.)

n

P-value

100

6

900

7

10,000

8

40,000

9

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