Final answer:
To construct a 0.99 confidence interval for the population mean, calculate the critical value, margin of error, and construct the interval by adding and subtracting the margin of error from the sample mean. To construct a 0.99 confidence interval for the population variance, calculate the critical values, and use the formula to calculate the upper and lower limits of the interval. The interpretation of the 0.99 confidence interval for the population mean is that we are 99% confident that the true population mean falls within the interval.
Step-by-step explanation:
To construct a 0.99 confidence interval for the population mean, we first need the critical value which corresponds to a 0.01 significance level on each tail of the distribution. Since the sample size is 6, we can use the t-distribution. The critical value for a 0.01 significance level and 6 degrees of freedom is approximately 3.707. Next, we calculate the margin of error by multiplying the standard deviation by the critical value and dividing it by the square root of the sample size: (0.021 * 3.707) / sqrt(6) = 0.061. Finally, we construct the confidence interval by subtracting the margin of error from the sample mean and adding it to the sample mean: 84.327 - 0.061 and 84.327 + 0.061. The 0.99 confidence interval for the population mean is (84.266, 84.388) meters.
To construct a 0.99 confidence interval for the population variance, we first need the critical values which correspond to a 0.005 significance level on each tail of the chi-square distribution. Since the sample size is 6, the degrees of freedom are 6 - 1 = 5. The critical values are 0.0637 and 19.6758. Next, we calculate the upper and lower limits of the confidence interval using the formula: [(n-1)s^2] / X^2(0.005/2) and [(n-1)s^2] / X^2(1 - (0.005/2)). Finally, we substitute the sample size, sample variance, and critical values into the formula and calculate the limits: [(6-1)(0.021^2)] / 19.6758 and [(6-1)(0.021^2)] / 0.0637. The 0.99 confidence interval for the population variance is (0.00006002, 0.00143934) meters squared.
An interpretation of the 0.99 confidence interval for the population mean is that we are 99% confident that the true population mean distance is between 84.266 and 84.388 meters. This means that if we were to repeat the experiment many times and construct a confidence interval using each sample, 99% of these intervals would contain the true population mean.