Respuesta :
Answer: [tex]0.070<\sigma< 0.128[/tex]
Step-by-step explanation:
Confidence interval for population standard deviation :-
[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: 1-0.90=0.10[/tex]
Sample size : n= 17
Sample standard deviation: [tex]s= 0.09[/tex]
Then by using the chi-square distribution table, we have
[tex]\chi^2_{1-\alpha/2, n-1}}=\chi^2_{0.95, 16}=7.96[/tex]
[tex]\chi^2_{\alpha/2, n-1}}=\chi^2_{0.05, 16}=26.30[/tex]
Confidence interval for population standard deviation will be :-
[tex]( 0.09)\sqrt{\dfrac{16}{26.30}}<\sigma<( 0.09)\sqrt{\dfrac{16}{7.96}}\\\\0.070197981837<\sigma<0.12759861690\\\\\approx0.070<\sigma< 0.128[/tex]
Hence, 90% confidence interval of the standard deviation of the skating performance test.: [tex]0.070<\sigma< 0.128[/tex]
90% confidence interval of the standard deviation of the skating performance test is [tex]0.070 < \sigma < 0.127[/tex]
What is a confidence interval for population standard deviation?
It is defined as the sampling distribution following an approximately normal distribution for known standard deviation.
The formula for finding the confidence interval for population standard deviation as follow:
[tex]\rm 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]
Where [tex]\rm s[/tex] is the standard deviation.
[tex]\rm n[/tex] is the sample size.
[tex]\rm{\chi ^2_{\alpha /2,n-1[/tex] and [tex]\rm\chi ^2_{1-\alpha /2,n-1[/tex] are the constant based on the Chi-Square distribution table.
[tex]\rm\alpha[/tex] is the significance level.
[tex]\rm\sigma[/tex] is the confidence interval for population standard deviation.
Calculating the confidence interval for population standard deviation:
We know significance level = 1 - confidence level
[tex]\rm \alpha = 1 - 0.90 = 0.10[/tex]
[tex]\rm n = 17[/tex]
[tex]\rm s = 0.09[/tex]
By the chi-square distribution table, the values of constants are below:
[tex]\rm \chi ^2_{1-\alpha/2,n-1} = \chi ^2_{0.95,16} = 7.96\\\rm \chi ^2_{ \alpha /2,n-1} = \chi ^2_{0.05,16} = 26.30[/tex]
putting all values in the above formula we will get the confidence interval for population standard deviation:
[tex]\begin{aligned} \rm (0.09) \sqrt{\dfrac{17-1}{26.30}}} & < \sigma < (0.09) \sqrt{\dfrac{17-1}{7.96}}}\\\\\rm s \sqrt{\dfrac{16}{26.30}}} & < \sigma < s \sqrt{\dfrac{16}{7.96}}\\\\\rm 0.0701197 & < \sigma < 0.12759 \\\\\end{aligned}\\[/tex]
or [tex]\approx 0.070 < \sigma < 0.127[/tex]
Thus, 90% confidence interval of the standard deviation of the skating performance test is [tex]0.070 < \sigma < 0.127[/tex]
Learn more about the confidence interval for population standard deviation here:
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