Introduction to Probability and Statistics for Engineers and Scientists

Problems 347

56.According to the U.S. Bureau of the Census, 25.5 percent of the population of

those age 18 or over smoked in 1990. A scientist has recently claimed that this

percentage has since increased, and to prove her claim she randomly sampled 500

individuals from this population. If 138 of them were smokers, is her claim proved?

Use the 5 percent level of significance.

57.An ambulance service claims that at least 45 percent of its calls involve life-

threatening emergencies. To check this claim, a random sample of 200 calls

was selected from the service’s files. If 70 of these calls involved life-threatening

emergencies, is the service’s claim believable at the

(a) 5 percent level of significance;

(b) 1 percent level of significance?

58.A standard drug is known to be effective in 75 percent of the cases in which it

is used to treat a certain infection. A new drug has been developed and has been

found to be effective in 42 cases out of 50. Based on this, would you accept, at

the 5 percent level of significance, the hypothesis that the two drugs are of equal

effectiveness? What is thep-value?

59.Do Problem 58 by using a test based on the normal approximation to the

binomial.

60.In a recently conducted poll, 54 out of 200 people surveyed claimed to have a

firearm in their homes. In a similar survey done earlier, 30 out of 150 people

made that claim. Is it possible that the proportion of the population having

firearms has not changed and the foregoing is due to the inherent randomness in

sampling?

61.LetX 1 denote a binomial random variable with parameters (n 1 ,p 1 ) andX 2 an

independent binomial random variable with parameters (n 2 ,p 2 ). Develop a test,

using the same approach as in the Fisher-Irwin test, of

H 0 :p 1 ≤p 2

versus the alternative

H 1 :p 1 >p 2

62.Verify that Equation 8.6.5 follows from Equation 8.6.4.

63.LetX 1 andX 2 be binomial random variables with respective parametersn 1 ,p 1

andn 2 ,p 2. Show that whenn 1 andn 2 are large, an approximate levelαtest of

H 0 :p 1 =p 2 versusH 1 :p 1 =p 2 is as follows:

reject H 0 if

|X 1 /n 1 −X 2 /n 2 |

√

X 1 +X 2

n 1 +n 2

(

1 −

X 1 +X 2

n 1 +n 2

)(

1

n 1

+

1

n 2

)>zα/2