In the past decades there have been intensive antismoking campaigns sponsored by

Question

In the past decades there have been intensive antismoking campaigns sponsored by both federal

and private agencies. In one study of national smoking trends, two random samples of U.S. adults were selected

in different years: The first sample, taken in 1995, involved 4000 adults, of which 990 were smokers. The

second sample, taken in 2010, involved 3000 adults, of which 700 were smokers. The samples are to be

compared to determine whether the proportion of U.S. adults that smoke declined during the 15-year period

between the samples. Let p1 be the proportion of all U.S. adults that smoked in 1995. Let p2 denote the

proportion of all U.S. adults that smoked in 2010.

State: Is there significant evidence the proportion of U.S. Adults who are smokers in 1995 is greater than the

proportion in 2010? Use a significance level of 0.01.

a. Plan: State the null and alternative hypotheses.

b. Solve: Check the conditions for inference.

c. Solve: Calculate the test statistic. Show work!

d. Solve: Give the appropriate p-value and state whether it is one-sided or two-sided.

e. Solve: Calculate the appropriate 99% confidence interval. Show work!

f. Conclude: Finally, using the results from parts d and e, summarize your conclusions. Interpret the

CI and results of your test. Use the four part conclusion method.

Explanation / Answer

a) H0: p1 = p2

H1: p1 > p2

b) p1 = 990/4000 = 0.25

p2 = 700/3000 = 0.23

The pooled sample proportion p = (p1 * n1 + p2 * n2)/(n1 + n2)

                                                    = (0.25 * 4000 + 0.23 * 3000)/(4000 + 3000) = 0.2414

SE = sqrt(p * (1 - p) * (1/n1 + 1/n2))

      = sqrt(0.2414 * 0.7586 * (1/4000 + 1/3000))

      = 0.01

c) The test statistic z = (p1 -p2)/SE

                              = (0.25 -0.23)/0.01

                             = 2

d) p-value = P(Z > 2)

             = 1 - P(Z < 2)

             = 1 - 0.9772 = 0.0228

It is one sided.

e) At 99% confidence interval the critical value is 2.58

The confidence interval is

(p1 - p2) +/- z0.005 * SE

= (0.25 - 0.23) +/- 2.58 * 0.01

= 0.02 +/- 0.0258

= -0.0058, 0.0458

f) AS the p- value is greater is greater than the significance level (0.0228 > 0.01), so the null hypothesis is not rejected.

So at 0.01 significance level there is not sufficient evidence the proportion of US adults who are smokers in 1995 is greater than the proportion in 2010.

As the hypothised value 0 lies between the confidence interval , so the null hypothesis is not rejected.

So at 0.01 significance level there is not sufficient evidence the proportion of US adults who are smokers in 1995 is greater than the proportion in 2010.

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