Final answer:
To create a 90% confidence interval for the population standard deviation of apple tree heights, calculate the sample standard deviation, use Chi-square distribution values.
Step-by-step explanation:
To construct a 90% confidence interval for the population standard deviation σ of the heights of the twelve two-year-old apple trees, we first need to calculate the sample standard deviation from the given data. The calculation involves finding the mean, then computing the sum of squared deviations from the mean, divided by the number of observations minus one (n-1). After finding the sample standard deviation, we use the Chi-square (χ2) distribution, since the population standard deviation is unknown and the population is assumed to be normally distributed.
For n=12 trees, with degrees of freedom df=n-1=11, and a desired confidence level of 90%, we will look up the critical Chi-square values associated with the lower and upper tails of the distribution. Specifically, we find χ2(α/2, df) and χ2(1-α/2, df), where α is the significance level (1 - confidence level). These values are then used in the formula:
({(n-1)s2}) / χ2(1-α/2, df) ≤ σ2 ≤ ({(n-1)s2}) / χ2(α/2, df)
By plugging in the calculated sample variance (s2) and the appropriate χ2 values, we can find the range for σ2, and by taking the square root, we obtain the confidence interval for σ.