Question

Following are the heights in inches of 12 two-year-old apple trees. Assume that the population is normally distributed.

37.4, 34.8, 37.8, 38.6, 36.4, 33.1, 34.8, 33.8, 38.0, 31.5, 38.0, 38.3
Construct a 95% confidence interval for the population standard deviation σ.

Answer 1

To construct a 95% confidence interval for the population standard deviation of tree heights, calculate the sample standard deviation and use the chi-square distribution with n-1 degrees of freedom to find the confidence interval bounds.

Step-by-step explanation:

To construct a 95% confidence interval for the population standard deviation σ of the height of two-year-old apple trees, we use the sample standard deviation and the chi-square (χ2) distribution since the population is normally distributed. First, calculate the sample standard deviation (s).

Then, use the chi-square values that correspond to the lower and upper tail areas in a chi-square distribution with n-1 degrees of freedom.

The steps are:

1. Calculate the sample variance (s2) and then the sample standard deviation (s).
2. Look up the chi-square values (χ2) for the 95% confidence level with 11 degrees of freedom (n-1, where n is the sample size of 12).
3. Use the chi-square values to calculate the confidence interval bounds with the formula: lower bound = √[(n-1)s2/χ2upper], upper bound = √[(n-1)s2/χ2lower].
4. Compute and report the confidence interval.

Note that 'n' is the number of trees, and s2 is the calculated sample variance. For the chi-square distribution, the area is divvied up between two tails, with half the alpha level in each tail for a two-tailed test.