Question

Set up a 95% confidence interval estimate for the population mean, based on each of the following sets of data, assuming that the population is normally distributed:
Set 1: 1, 1, 1, 1, 8, 8, 8, 8
Set 2: 1, 2, 3, 4, 5, 6, 7, 8
Explain why these data sets have different confidence intervals even though they have the same mean and range.

Answer 1

Confidence interval is 95% thus $\alpha = 0.05$

$n=8$ (small sample, less than 30)

$\bar x_1=\frac{1+1+1+1+8+8+8+8}{8}=4.5$

$s_1=\sqrt{\frac{(1-4.5)^2+(1-4.5)^2+(1-4.5)^2+(1-4.5)^2+(8-4.5)^2+(8-4.5)^2+(8-4.5)^2+(8-4.5)^2}{8-1}}=3.74$

df = n–1 = 7

$\mu=\bar x\pm t\frac{s}{\sqrt{n}}$

$\mu_1=4.5\pm2.365\cdot\frac{3.74}{\sqrt{8}}=4.5\pm3.13$

$1.37 \le\mu_1\le7.63$

$\bar x_2=\frac{1+2+3+4+5+6+7+8}{8}=4.5$

$s_2=\sqrt{\frac{(1-4.5)^2+(2-4.5)^2+(3-4.5)^2+(4-4.5)^2+(5-4.5)^2+(6-4.5)^2+(7-4.5)^2+(8-4.5)^2}{8-1}}=2.45$

$\mu_2=4.5\pm2.365\cdot\frac{2.45}{\sqrt{8}}=4.5\pm2.05$

$2.45 \le\mu_2\le6.55$

Even though these data sets have the same mean and range, they have different estimations for the population mean because they have different standard deviations which impact the calculations of intervals.