Final answer:
To find the 90% confidence interval for the standard deviation of peanut weights, calculate the sample variance, then determine the chi-square critical values for your desired confidence level. Plug these into the confidence interval formula to find the lower and upper limits of the interval.
Step-by-step explanation:
Finding a 90% Confidence Interval for the Standard Deviation of Peanut Weights
To answer the student's question regarding the confidence interval for the standard deviation of the peanut weights, we must understand the distribution and know how to utilize the appropriate statistical methods. Since the student mentioned that the weights are normally distributed, we can use the chi-square distribution for estimating the confidence interval for the standard deviation.
First, we calculate the sample variance s^2 using the sample data provided. The formula for sample variance is:
s^2 = Σ(xi - μ)^2 / (n-1)
Where Σ(xi - μ)^2 is the sum of the squared differences between the sample observations (xi) and the sample mean (μ), and (n-1) is the sample size minus one. After calculating the sample variance, we find the chi-square value (χ^2) corresponding to the desired confidence level. In this case, it is a 90% confidence interval.
The boundaries of the confidence interval for the standard deviation σ are then calculated using the formula:
Lower Limit = √[(n - 1) * s^2 / χ^2_{α/2}]
Upper Limit = √[(n - 1) * s^2 / χ^2_{1-α/2}]
Where α is 1 minus the confidence level (for 90%, α = 0.10), χ^2_{α/2} and χ^2_{1-α/2} are the chi-square values for α/2 and 1-α/2 degrees of freedom respectively. These values act as critical values for the chi-square distribution.
By substituting the calculated sample variance and the appropriate chi-square values, the manager will obtain the 90% confidence interval for the population standard deviation.