Question

A random sample of 25 glass sheets is obtained and their thicknesses are measured. The sample mean is x= 3.54 mm and sample standard deviation is S = 0.20 mm. Construct a 95% two-sided confidence interval for the mean glass thickness.

Answer 1

Answer: (3.46, 3.62)

Step-by-step explanation:

The formula to find the confidence interval for population mean is given :-

$\overline{x}\ \pm\ t_{\alpha/2}\dfrac{s}{\sqrt{n}}$

, where n = sample size.

$t_{\alpha/2}$ = Two-tailed t-value for significance level of $(\alpha)$ and degree of freedom df= n-1.

$s$ = sample standard deviation.

As per given , we have

$\overline{x}= 3.54$ mm

s= 0.20 mm

n= 25

Significance level $=\alpha=1-0.95=0.05$

Since population standard deviation is not given , it means the given problem has t- distribution.

Two-tailed t-value for significance level of $(0.05)$ and degree of freedom df= 24:

$t_{\alpha/2\ ,df}=t_{0.025,\ 24}=2.0639$

95% Confidence interval for population mean:

$3.54\ \pm\ (2.0639)\dfrac{0.20}{\sqrt{25}}$

$=3.54\ \pm\ (2.0639)\dfrac{0.20}{5}$

$=3.54\ \pm\ (2.0639)(0.04)$

$=3.54\ \pm\ 0.082556$

$=(3.54- 0.082556,\ 3.54+ 0.082556 )=(3.457444,\ 3.622556)\approx(3.46,\ 3.62)$

Hence, the 95% two-sided confidence interval for the mean glass thickness = (3.46, 3.62)