High School

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

31.9, 36.9, 40.7, 36.3, 39.4, 34.5, 36.9, 37.2, 40.3, 42.0, 41.0, 39.3

Construct a 99% confidence interval for the population standard deviation [tex]\sigma[/tex].

Answer :

Final answer:

To construct a 99% confidence interval for population standard deviation σ of two-year-old apple tree heights, calculate the sample variance, obtain the Chi-Square critical values, and apply the confidence interval formula for variance.

Explanation:

The student has asked to construct a 99% confidence interval for the population standard deviation σ. To calculate this, we need to use the Chi-Square distribution since we are dealing with standard deviations and the data is assumed to be normally distributed.

  • First, calculate the sample variance. This is done by squaring the difference of each observation from the sample mean, summing these squared differences, and dividing by the number of observations minus one.
  • With the sample variance, use the Chi-Square distribution to find the critical values for the 99% confidence interval. For a sample size of 12, we use degrees of freedom equal to n - 1, which is 11 in this case.
  • Finally, plug these values into the formula for the confidence interval for variance and take the square root to obtain the interval for standard deviation.

This approach will provide upper and lower bounds for the population standard deviation σ, giving us a range that, with 99% confidence, contains the true parameter.

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