Suppose medical researchers think that p = .70 is the pro-portion of all teenage

Question

Suppose medical researchers think that p = .70 is the pro-portion 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 n = 200 teenagers with high blood pressure will take regular calcium supplements. a. Assume p = .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 p, the sample proportion, for samples of n = 200 teenagers? b. Use the results of part (a) to calculate an interval that will contain the sample proportion p for about 99.7% of all samples of n = 200 teenagers. c. 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? d. Given the answers for parts (b) and (c), explain why

Explanation / Answer

a) The below things are given:

Sample size = n = 200

Population proportion = p = 0.70

The central limit theorem states that sampling distribution of a statistic will be nearly normal if sample size is large enough. Sample size of 30 is large enough.

Hence Mean of sample can be approximated as 0.70

We can calculate the standard deviation of sample proportion with the below formula:

Standard deviation = sqrt( p*(1 - p) /n ) = sqrt( 0.70 * 0.30 / 200) = 0.0324037

b) Here we need to look at the z table to find the critical value corresponding to cumulative probability of 0.997

Critical value from Z-table is 2.75

Now we need to calculate the Margin of error. The formula for calculating Margin of error is:

Margin of Error = Critical Value * Standard deviation of the statistic

Margin of Error = 2.75 * 0.0324037 = 0.08911017

Confidence interval can be calculated as below:

Confidence interval = sample statistic +- Margin of Error

Confidence interval = (0.7 - 0.08911017) to (0.7 + 0.08911017)

Confidence interval = 0.6108898 to 0.7891102

So the interval that will contain 99.7% the sample proportion p(hat) for about 99.7% of all samples of n=200 teenagers is 0.6108898 to 0.7891102

c) Here p = 120/200 = 0.6

It is a statistic because it is based on just one clinical trial of 200 teenagers. It doesnt include all the teenagers in the population.

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