Question

in the 1980s, it was generally believed that congenital abnormalities affected about 8% of a large nation’s children. some people believe that the increase in the number of chemicals in the environment has led to an increase in the incidence of abnormalities. a recent study examined 373 randomly selected children and found that 40 of them showed signs of an abnormality. is this strong evidence that the risk has increased? (we consider a p - value of around 5% to represent reasonable evidence.) complete parts a through f. assume the independence assumption is met.

c) perform the mechanics of the test. what is the p - value?
p - value = 0.027 (round to three decimal places as needed.)

d) explain carefully what the p - value means in this context. choose the correct answer below.
a. the p - value is the chance of observing 40 or more children with genetic abnormalities in a random sample of 373 children if 8% of children actually have genetic abnormalities.
b. the p - value is the chance of observing 8% of children with genetic abnormalities.
c. the p - value is the actual percentage of children who have genetic abnormalities.
d. the p - value is the chance of observing 40 or more children with genetic abnormalities in a random sample of 373 children.

Explanation:

Step1: Identify hypothesis test type

This is a one-sample proportion z-test. The null hypothesis is $H_0: p = 0.08$, alternative hypothesis is $H_a: p > 0.08$.

Step2: Calculate sample proportion

Sample proportion $\hat{p} = \frac{40}{373} \approx 0.1072$

Step3: Compute test statistic

$$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}} = \frac{0.1072 - 0.08}{\sqrt{\frac{0.08(1-0.08)}{373}}} \approx 1.92$$

Step4: Find P-value

For right-tailed test, P-value is $P(Z > 1.92) \approx 0.027$

Step5: Interpret P-value

The P-value is the probability of observing a sample statistic as extreme or more extreme than the observed one, assuming the null hypothesis is true.

Answer:

c) $0.027$
d) A. The P-value is the chance of observing 40 or more children with genetic abnormalities in a random sample of 373 children if 8% of children actually have genetic abnormalities.