Answer: [tex]0.3012<\sigma< 0.4475[/tex]
Step-by-step explanation:
The confidence interval for population standard deviation is given by :-
[tex]s\sqrt{\dfrac{n-1}{\chi^2_{\alpha/2, n-1}}}<\sigma<s\sqrt{\dfrac{n-1}{\chi^2_{1-\alpha/2, n-1}}}[/tex]
Given : Significance level : [tex]\alpha: 1-0.95=0.05[/tex]
Sample size : n= 51
Sample standard deviation : s=0.36
Then by using the chi-square distribution table, we have
[tex]\chi^2_{1-\alpha/2, n-1}}=\chi^2_{0.975, 50}=32.36[/tex]
[tex]\chi^2_{\alpha/2, n-1}}=\chi^2_{0.025, 50}=71.42[/tex]
Confidence interval for population standard deviation will be :-
[tex](0.36)\sqrt{\dfrac{50}{32.36}}<\sigma<(0.36)\sqrt{\dfrac{50}{71.42}}\\\\0.301215683035<\sigma<0.447489909147\\\\\approx0.3012<\sigma< 0.4475[/tex]
The population is approximately normally distributed is given using a chi-square distribution is 0.3012 < σ < 0.4475.
It is also called the Gaussian Distribution. It is the most important continuous probability distribution. The curve looks like a bell, so it is also called a bell curve.
The percentage of titanium in an alloy used in aerospace castings is measured in 51 randomly selected parts.
The sample standard deviation is s = 0.36. Construct a 95% two-sided confidence interval for σ.
Assume the population is approximately normally distributed.
The confidence interval for the population standard deviation is given as
[tex]s \dfrac{n-1}{X^2_{\alpha /2, n-1}} < \sigma < s \dfrac{n-1}{X^2_{1- \alpha /2, n-1}}[/tex]
We have
Significane level: α = 1 - 0.95 = 0.05
Sample size: n = 51
Sample standard deviation: s = 0.36
Using the chi-square distribution table, we have
[tex]\rm X^2_{1- \alpha /2, n-1} = X^2_{0.975, 50} = 32 \\\\X^2_{\alpha /2, n-1} = X^2_{0.025, 50} = 71.42[/tex]
Confidence interval for population standard deviation will be
[tex]\begin{aligned} (0.36) \dfrac{50}{32.36} & < \sigma < (0.36) \dfrac{50}{71.42}\\\\0.3012 & < \sigma < 0.4475 \end{aligned}[/tex]
More about the normal distribution link is given below.
https://brainly.com/question/12421652
