Question

The report titled "Digital Democracy Survey"† stated that 31% of the people in a representative sample of adult Americans age 33 to 49 rated a landline telephone among the three most important services that they purchase for their home. In a representative sample of adult Americans age 50 to 68, 48% rated a landline telephone as one of the top three services they purchase for their home. Suppose that the samples were independently selected and that the sample size was 700 for the 33 to 49 age group sample and 750 for the 50 to 68 age group sample. Does this data provide convincing evidence that the proportion of adult Americans age 33 to 49 who rate a landline phone in the top three is less than this proportion for adult Americans age 50 to 68? Test the relevant hypotheses using α = 0.05. (Let p1 be the proportion of adult Americans ages 33 to 49 who rate a landline phone in the top three, and p2 be the proportion of adult Americans ages 50 to 68 who rate a landline phone in the top three.)

State the appropriate null and alternative hypotheses.

H0: p1 − p2 > 0 Ha: p1 − p2 < 0

H0: p1 − p2 < 0 Ha: p1 − p2 > 0

H0: p1 − p2 = 0 Ha: p1 − p2 ≠ 0

H0: p1 − p2 = 0 Ha: p1 − p2 > 0

H0: p1 − p2 = 0 Ha: p1 − p2 < 0

Find the test statistic and P-value. (Use a table or technology. Round your test statistic to two decimal places and your P-value to four decimal places.)

z =

P-value =

State your conclusion.

We reject H0. We have convincing evidence that the proportion of adult Americans age 33 to 49 who rate a landline phone in the top three is less than this proportion for adult Americans age 50 to 68.

We fail to reject H0. We do not have convincing evidence that the proportion of adult Americans age 33 to 49 who rate a landline phone in the top three is less than this proportion for adult Americans age 50 to 68.

We fail to reject H0. We have convincing evidence that the proportion of adult Americans age 33 to 49 who rate a landline phone in the top three is less than this proportion for adult Americans age 50 to 68.

We reject H0. We do not have convincing evidence that the proportion of adult Americans age 33 to 49 who rate a landline phone in the top three is less than this proportion for adult Americans age 50 to 68.

Answer 1

H0: p1 − p2 = 0 Ha: p1 − p2 < 0

tets statistics:

p1 = 0.31 , p2 = 0.48
n1 = 700 , n2 =750

p = (p1 * n1 + p2 * n2) / (n1 + n2)
= ( 0.31 *700 + 0.48 * 750)/(700+750)
=0.3979

SE = sqrt{ p * ( 1 - p ) * [ (1/n1) + (1/n2) ] }
= sqrt(0.3979 * (1-0.3979) *((1/700) + (1/750)))
= 0.0257

z = (p1 - p2)/SE
= ( 0.31 - 0.48) / 0.0257
= -6.61

p value = 0

We reject H0. We have convincing evidence that the proportion of adult Americans age 33 to 49 who rate a landline phone in the top three is less than this proportion for adult Americans age 50 to 68.