Question

A random sample of 86 observations produced a mean x 25.4 and a standard deviation s 2.7 a. Find a 95% confidence interval for b. Find a 90% confidence interval for u c. Find a 99% confidence interval for

Answer 1

SOLUTION:

From given data,

A random sample of 86 observations produced a mean $\bar{x}$ = 25.4 and a standard deviation s = 2.7.

sample size = n= 86

mean = $\bar{x}$ = 25.4

standard deviation = s = 2.7

Formula for for the population mean µ

$\bar{x}$  $\pm$ Z_$\alpha$/2$\times$ ($\sigma$ / $\sqrt{n}$ )

(a) Find a 95% confidence interval for $\mu$

95% confidence interval

Confidence interval is 95%

95% = 95/100 = 0.95

$\alpha$ = 1 - Confidence interval = 1-0.95 = 0.05

$\alpha$/2 = 0.05 / 2

= 0.025

Z_$\alpha$/2 = Z_0.025 = 1.96

for the population mean µ

$\bar{x}$  $\pm$ Z_$\alpha$/2$\times$ ( / $\sqrt{n}$ )

$\bar{x}$  $\pm$ Z_0.025$\times$ (s / $\sqrt{n}$ )

25.4 $\pm$ 1.96$\times$ (2.7 / $\sqrt{86}$ )

25.4 $\pm$ 1.96$\times$ (2.7 / 9.2736 )

25.4 $\pm$ 0.57065

(24.82935, 25.97065)

24.829 < $\mu$ < 25.971

(b) Find a 90% confidence interval for $\mu$

90% confidence interval

Confidence interval is 90%

90% = 90/100 = 0.90

$\alpha$ = 1 - Confidence interval = 1-0.90 = 0.10

$\alpha$/2 = 0.10 / 2

= 0.05

Z_$\alpha$/2 = Z_0.05 = 1.64

for the population mean µ

$\bar{x}$  $\pm$ Z_$\alpha$/2$\times$ ( / $\sqrt{n}$ )

$\bar{x}$  $\pm$ Z_0.05$\times$ (s / $\sqrt{n}$ )

25.4 $\pm$ 1.64$\times$ (2.7 / $\sqrt{86}$ )

25.4 $\pm$ 1.64$\times$ (2.7 / 9.2736 )

25.4 $\pm$ 0.477484

(24.922516, 25.877484)

24.923 < $\mu$ < 25.877

(c) Find a 99% confidence interval for $\mu$ ​​​​​​​

99% confidence interval

Confidence interval is 99%

99% = 99/100 = 0.99

$\alpha$ = 1 - Confidence interval = 1-0.99 = 0.01

$\alpha$/2 = 0.01 / 2

= 0.005

Z_$\alpha$/2 = Z_0.005  = 2.57

for the population mean µ

$\bar{x}$  $\pm$ Z_$\alpha$/2$\times$ ( / $\sqrt{n}$ )

$\bar{x}$  $\pm$ Z_0.05$\times$ (s / $\sqrt{n}$ )

25.4 $\pm$ 2.58$\times$ (2.7 / $\sqrt{86}$ )

25.4 $\pm$ 2.58$\times$ (2.7 / 9.2736 )

25.4 $\pm$ 0.751164

(24.648836, 26.151164)

24.649 < $\mu$ < 26.151

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