Question

a simple random sample of 800 individuals provides 300 yes responses.
a. what is the point estimate of the proportion of the population that would provide yes responses (to 4 decimals)?
b. what is your estimate of the standard error of the proportion (to 4 decimals)?
c. compute the 95% confidence interval for the population proportion (to 4 decimals).

Explanation:

Step1: Calculate the point estimate

The point estimate of the population proportion \( \hat{p} \) is given by \( \hat{p}=\frac{x}{n} \), where \( x = 300 \) (number of Yes responses) and \( n=800 \) (sample size).
\( \hat{p}=\frac{300}{800}=0.3750 \)

Step2: Calculate the standard error

The formula for the standard error of the proportion \( SE=\sqrt{\frac{\hat{p}(1 - \hat{p})}{n}} \).
Substitute \( \hat{p}=0.3750 \) and \( n = 800 \) into the formula:

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Step3: Calculate the confidence interval

For a 95% confidence interval, the critical value \( z = 1.96 \).
The confidence interval is \( \hat{p}\pm z\times SE \)
Lower limit: \( 0.3750-1.96\times0.0171=0.3750 - 0.0335 = 0.3415 \)
Upper limit: \( 0.3750+1.96\times0.0171=0.3750+ 0.0335=0.4085 \)

Answer:

a. \( 0.3750 \)
b. \( 0.0171 \)
c. \( (0.3415,0.4085) \)