2) Studies suggest that 34% ofwomendevelop cancer in their lifetime. An Australian study of 1282tall womenfound that 386 of them developed cancer in their lifetime. Let p denote the proportion oftall womenwho develop cancer in their lifetime. Suppose we want to test, at the 5% level of significance, whether the proportion oftall womenwho develop cancer in their lifetime is less than the proportion ofwomenwho develop cancer in their lifetime. Assume the assumptions to this hypothesis test are satisfied. Answer the following questions.

(a)What are the null and alternative hypotheses, and level of significance, for this hypothesis test?

H0: p1 p2= 0 vs Ha: p1 p2> 0 ( = 0.05)

H0: p1 p20 vs Ha: p1 p2< 0 ( = 0.05)

H0: p= 0.34 vs Ha: p< 0.34 ( = 0.05)

H0: p = 0.34 vs Ha: p> 0.34 ( = 0.05)

(b)What is the distribution of the test statistic for this hypothesis test?

t1282

t1281

N(0, 1)

N(376, 0.34)

Chi-square distribution with 3 degrees of freedom

(c)What is the decision rule for this hypothesis test?

We rejectH0if z 1.6449

We rejectH0if z2.3263

We rejectH0if z1.6449

We rejectH0if t 2.5219

We rejectH0if z 2.3263

(d)What is the observed test statistic for this hypothesis test?

3.1523

2.9408

3.1523

0.8356

2.9408

(e)What is the p-value for this hypothesis test?

0.0008

0.7995

0.0712

0.0016

0.9234

(f)What is the correct conclusion for this hypothesis test?

Since the p-value is less than 0.05 we do not rejectH0at the 5% level of significance.

Since the observed test statistic is less than the critical value we do not rejectH0at the 5% level of significance.

Since the observed test statistic is greater than the critical value we do not rejectH0at the 5% level of significance.

Since the p-value is greater than 0.05 we do not rejectH0at the 5% level of significance.

Since the observed test statistic is greater than the critical value we rejectH0at the 5% level of significance.

Since the p-value is greater than 0.05 we rejectH0at the 5% level of significance.

Since the observed test statistic is less than the critical value we rejectH0at the 5% level of significance.

(g)Suppose other researchers want to estimate p using a confidence interval. How manytallwomendo they need to sample, to estimate p to within 0.01 and with 98% confidence?

13529

12893

11389

114

13530