Answer
Upper Limit of the 90% confidence interval for the population standard deviation = 69.3
90% Confidence interval = 13.7 < σ < 69.3
Step-by-step explanation
Confidence Interval for the population standard deviation is basically an interval of range of values where the true population standard deviation can be found with a certain level of confidence.
Mathematically,
Confidence Interval = (Sample standard deviation) ± (Margin of error)
Sample standard deviation = 41.5
Margin of Error is the width of the confidence interval about the standard deviation.
It is given mathematically as,
Margin of Error = (Critical value) × (standard Error of the distribution)
Critical value at 90% confidence interval for sample size of 8 is obtained from the t-tables because the population standard deviation is unknown.
Degree of freedom = Sample size - 1 = n - 1 = 8 - 1 = 7
Significance level = 100 - (Confidence level) = 100 - 90 = 10% = 0.1
Critical value for degree of freedom 7, 0.10 significance level = 1.8946 (From the t-value calculator or the t-value tables)
Standard error of the distribution = σₓ = (σ/√n)
σ = standard deviation of the sample = 41.5
n = sample size = 8
σₓ = (41.5/√8) = 14.67
Confidence Interval = (Sample standard deviation) ± (Margin of error)
90% Confidence Interval = (Sample standard deviation) ± [(Critical value) × (standard Error)]
CI = 41.5 ± (1.8946 × 14.67)
CI = 41.5 ± 27.798
90% CI = (13.70, 69.30)
90% Confidence interval = 13.7 < σ < 69.3
Hope this Helps!!!