Question

According to a Gallup poll, 11.55% of American adults have diabetes. A researcher wonders if the rate in her area is higher than the national rate. She surveys 150 adults in her area and finds that 21 of them have diabetes.

1) If the region had the same rate of diabetes as the rest of the country, how many would we expect to have diabetes?

2) Suppose you are testing the hypothesis that the diabetes rate in this area differs from the national rate. State the null and alternative hypotheses for such a test.

3) State the value of the test statistic and the p-value.

4) Use a 0.05 significance level to make a decision about the null hypothesis and state your conclusion in context.

Answer 1

Solution:

Given:

p = proportion of American adults have diabetes = 0.1155

n = Sample size = 150

x = number American adults have diabetes = 21

thus

Part 1) If the region had the same rate of diabetes as the rest of the country, how many would we expect to have diabetes?

We would expect 17 adults to have diabetes.

Part 2) Suppose you are testing the hypothesis that the diabetes rate in this area differs from the national rate. State the null and alternative hypotheses for such a test.

Since we have to test the hypothesis that the diabetes rate in this area differs from the national rate, this is non-directional statement, thus this is two tailed test.

Thus H0 and H1:

Vs

Part 3) State the value of the test statistic and the p-value.

Test statistic:

P-value:

P-value = 2 X P( Z> z test statistic)

P-value = 2 X P( Z> 0.94)

P-value = 2 X [ 1 - P( Z < 0.94) ]

Look in z table for z = 0.9 and 0.04 and find corresponding area.

P( Z< 0.94) = 0.8264

Thus

P-value = 2 X [ 1 - P( Z < 0.94) ]

P-value = 2 X [ 1 - 0.8264 ]

P-value = 2 X 0.1736

P-value = 0.3472

Part 4) Use a 0.05 significance level to make a decision about the null hypothesis and state your conclusion in context.

Decision Rule:
Reject H0, if P-value < 0.05 level of significance, otherwise we fail to reject H0

Since P-value = 0.3472 > 0.05 level of significance, we fail to reject H0.

Conclusion:

At 0.05 level of significance, we do not have sufficient evidence to conclude that: the diabetes rate in this area differs from the national rate.