Question

4 (a) In a random sample of 200 observations, we found the proportion of successes to be 48%. Estimate with 95% confidence the population proportion of successes.

(b) Repeat part (a) with n = 500.

(c) Repeat part (a) with n = 1000.

(d) Describe the effect on the confidence interval

estimate of increasing the sample size.

Answer 1

(a) $95\%CI=(0.48-1.96\sqrt{\frac{0.48(1-0.48)}{200}},0.48+1.96\sqrt{\frac{0.48(1-0.48)}{200}})=(0.4108,0.5492).$

(b) $95\%CI=(0.48-1.96\sqrt{\frac{0.48(1-0.48)}{500}},0.48+1.96\sqrt{\frac{0.48(1-0.48)}{500}})=(0.4352,0.5238).$

(c) $95\%CI=(0.48-1.96\sqrt{\frac{0.48(1-0.48)}{1000}},0.48+1.96\sqrt{\frac{0.48(1-0.48)}{1000}})=(0.4490,0.5110).$

(d) As the sample size increases, the confidence interval narrows.