Final answer:
To calculate the 90% confidence interval for the variance and standard deviation of ski lift ticket prices, one must derive the sample mean and variance, then use the chi-square distribution with n-1 degrees of freedom to find the confidence interval for variance and standard deviation.
Step-by-step explanation:
Finding a 90% Confidence Interval for Variance and Standard Deviation
To find the 90% confidence interval for the variance and standard deviation of ski lift ticket prices from the provided data, we will first calculate the sample mean (μ) and the sample variance (s2). The given prices are 59, 54, 53, 52, 51, 39, 49, 46, 49, 48. Let's compute the mean:
μ = (59 + 54 + 53 + 52 + 51 + 39 + 49 + 46 + 49 + 48) / 10 = 500 / 10 = 50
Next, we calculate the sample variance s2 using the formula:
s2 = Σ (x - μ)2 / (n - 1)
Where x is the sample data and n is the sample size. After calculating the variance, we use the fact that to estimate a population variance from sample data, the sample variance follows a chi-square distribution. We apply the chi-square distribution with degrees of freedom df = n-1 to find the critical values χ1 and χ2 corresponding to the 90% confidence interval.
The confidence interval for the variance is calculated as:
( (n-1) * s2 / χ1, (n-1) * s2 / χ2 )
For the standard deviation, take the square root of the variance interval.
Keep in mind that confidence intervals are not determined by the percentage of data values that fall within them, but rather by how confident we are that the interval estimate contains the true population parameter. When constructing a confidence interval, the area in the tails of the distribution is equal to 1 - confidence level. In this case, it would be 5% in each tail for a 90% confidence level.