Question

A survey of 2280 adults in a certain large country aged 18 and older conducted by reputable polling organization founded that 424 have donated blood in the past two years. Complete Parts (a) through (c).

a. Obtain a point estimate for the population proportion of adults in the country age 18 and older who have donated blood in the past two years.
b. Verify that the requirements for constructing a confidence interval.
c. Construct and interpret a 90% confidence interval for the population proportion of adults in the country who have donated blood in the past two years.

Answer 1

a. Point estimate p=0.186

b. The conditions (randomization, independence, 10% condition, sample size) are met.

c. The 90% confidence interval for the population proportion of adults in the country who have donated blood in the past two years is (0.173, 0.199).

Explanation:

The point estimate for the population proportion of adults in the country age 18 and older who have donated blood in the past two years can be calculated with the sample data as:

`p=X/n=424/2280=0.186`

The conditions are:

Randomization Condition: The data must be sampled randomly. It is considered satisfied, as it is conducted by a "reputable polling organization".

Independence Assumption: The sample values must be independent of each other. As the subjects were selected randomly, we can assume that the independence assumption is met.

10% Condition: The sample size should be no more than 10% of the population. It is satisfied, as 2280 adults in a large country must be less than 10% of the population.

Sample Size Condition: The sample size must be sufficiently large. The sample size is n=2280 and it can be considered sufficiently large.

We have to calculate a 90% confidence interval for the proportion.

The sample proportion is p=0.186.

The standard error of the proportion is:

`\sigma_p=\sqrt{(p(1-p))/(n)}=\sqrt{(0.186*0.814)/(2280)}\\\\\\ \sigma_p=√(0.000066)=0.008`

The critical z-value for a 90% confidence interval is z=1.645.

The margin of error (MOE) can be calculated as:

`MOE=z\cdot \sigma_p=1.645 \cdot 0.008=0.013`

Then, the lower and upper bounds of the confidence interval are:

`LL=p-z \cdot \sigma_p = 0.186-0.013=0.173\\\\UL=p+z \cdot \sigma_p = 0.186+0.013=0.199`

The 90% confidence interval for the population proportion of adults in the country who have donated blood in the past two years is (0.173, 0.199).