Answer to Question #150176 in Statistics and Probability for caarino

Question

a. From the information given here determine the 95% confidence interval estimate of the population mean.

x = 100  = 20 n = 25

b. Repeat part (a) with x = 200.

c. Repeat part (a) with x = 500.

d. Describe what happens to the width of the confidence interval estimate when the sample mean increases.

Answer 1

$\bar x=100$

$\sigma =20$

$n =25$

a) Confidence interval is:

$$\bar x \pm Z_{\frac{\alpha}{2}}\frac{\sigma}{\sqrt n}$$

$100\pm Z_{\frac{1-0.95}{2}}\frac{20}{\sqrt{25}}=100\pm Z_{0.025}\cdot 4=100\pm1.96 \cdot4=100\pm 7.84$

So given the 95% confidence level the interval is (92.16, 107.84)

b) $\bar x=200$: $200\pm 7.84$

The 95% confidence interval is (192.16, 207.84)

c) $\bar x=500$: $500\pm 7.84$

The 95% confidence interval is (492.16, 507.84)

d) As we can see, increasing the sample mean does not change the width of the confidence interval; it only changes where the interval is centered.