Question

the mean of a random sample of 25 observations from a normal population whose standard deviation is 50, with a sample mean of 200. Estimate the population mean with 95% confidence.
b. repeat part a changing the population standard deviation to 25
c. repeat part a changing the population standard to 10
d. describe what happens to the confidence interval estimate when the standard deviation decreased.

Answer 1

a.

$95\%CI=(\bar{x}-1.96\frac{\sigma}{\sqrt{n}},\;\bar{x}+1.96\frac{\sigma}{\sqrt{n}})=\\=(200-1.96\frac{50}{\sqrt{25}},\;200+1.96\frac{50}{\sqrt{25}})=(180.4,\;219.6).$

b.

$95\%CI=(\bar{x}-1.96\frac{\sigma}{\sqrt{n}},\;\bar{x}+1.96\frac{\sigma}{\sqrt{n}})=\\=(200-1.96\frac{25}{\sqrt{25}},\;200+1.96\frac{25}{\sqrt{25}})=(190.2,\;209.8).$

c.

$95\%CI=(\bar{x}-1.96\frac{\sigma}{\sqrt{n}},\;\bar{x}+1.96\frac{\sigma}{\sqrt{n}})=\\=(200-1.96\frac{10}{\sqrt{25}},\;200+1.96\frac{10}{\sqrt{25}})=(196.08,\;203.92).$

d. When the standard deviation decreases, the confidence interval becomes narrower.