Answer to Question #105632 in Statistics and Probability for Christine Palapuz

Question

Suppose medical researchers think that 0.70 is the proportion of all teenagers with high blood pressure whose blood pressure would decrease if they took calcium supplements. To test this theory, the researchers plan a clinical trial (experiment) in which 200 randomly selected teenagers with high blood pressure will take regular calcium supplements.

(a) Assume 0.70 actually is the population proportion that would experience a decrease in blood pressure. What are the numerical values of the mean and standard deviation of the sampling distribution of the sample proportions for samples of 200 teenagers?
(b) In the clinical trial, 120 of the 200 teenagers taking calcium supplements experienced a decrease in blood pressure. What is the value of p^ for this sample? Is this value a parameter or a statistic?
(c) What is the probability of having 120 or fewer experience a decrease of blood pressure out of a sample of 200 teenagers taking calcium supplements?

Answer 1

(a). P=0.7, n=200

Since np=140 and nq=60 are greater than 5, P is approximately normal with mean P and standard deviation $\sqrt{\frac{P(1-P)}{n}}$

Thus, the mean of the sampling distribution is 0.7 and the standard deviation is $\sqrt{\frac{0.7\times{0.3}}{200}}=0.032404$

(b).$\hat{p}$

$\hat{p}=\frac{x}{n}=\frac{120}{200}=0.6$

Since 0.6 is calculated from a sample, it is a statistic.

(c). $P(x\le120)=P(P\le0.6)$

$Z-score =\frac{\hat{p}-p}{\sqrt{\frac{P(1-P)}{n}}}$

$=\frac{0.6-0.7}{\sqrt{\frac{0.7(1-0.7)}{200}}}=-3.086038761$

From z table or =NORM.S.DIST(-3.086038761,TRUE) excel formula, $p(z\le-3.086038761)=0.001014212$