Suppose we collect a random sample 27 rabbits and weigh them. We find that among these 27 rabbits, the average weight is 1.7 pounds with a standard deviation of 0.44 pounds.

Question

Suppose we collect a random sample 27 rabbits and weigh them. We find that among these 27 rabbits, the average weight is 1.7 pounds with a standard deviation of 0.44 pounds. (a) Construct a 90% one-sided confidence interval with no upper bound for the population mean weight. (b) Construct a 95% one-sided confidence interval with no lower bound for the population standard deviation

Answer 1

(a) To construct a 90% one-sided confidence interval with no upper bound for the population mean weight, we can use the t-distribution. Since we don't have the population standard deviation, we'll use the sample standard deviation as an estimate.

The formula for the confidence interval for the mean with no upper bound is:

CI = x̄ + t * (s/√n)

Where:

CI = Confidence interval

x̄ = Sample mean

t = Critical value from the t-distribution (corresponding to the desired confidence level and degrees of freedom)

s = Sample standard deviation

n = Sample size

In this case, the sample mean (x̄) is 1.7 pounds, the sample standard deviation (s) is 0.44 pounds, and the sample size (n) is 27.

To find the critical value (t), we need to determine the degrees of freedom. Since we have a sample size of 27, the degrees of freedom will be n - 1 = 27 - 1 = 26.

Looking up the critical value for a one-sided 90% confidence level and 26 degrees of freedom in the t-distribution table or using statistical software, we find t = 1.706.

Substituting the values into the formula, we have:

CI = 1.7 + 1.706 * (0.44/√27)

Calculating this expression, we find:

CI ≈ 1.7 + 1.706 * (0.083)

CI ≈ 1.7 + 0.141

CI ≈ 1.841

Therefore, the one-sided 90% confidence interval with no upper bound for the population mean weight is approximately (1.7, ∞).

(b) To construct a 95% one-sided confidence interval with no lower bound for the population standard deviation, we can use the chi-square distribution.

The formula for the confidence interval for the standard deviation with no lower bound is:

CI = √[(n - 1) * s^2 / χ^2]

Where:

CI = Confidence interval

n = Sample size

s = Sample standard deviation

χ^2 = Critical value from the chi-square distribution (corresponding to the desired confidence level and degrees of freedom)

In this case, the sample size (n) is 27, and the sample standard deviation (s) is 0.44 pounds.

To find the critical value (χ^2), we need to determine the degrees of freedom. Since we have a sample size of 27, the degrees of freedom will be n - 1 = 27 - 1 = 26.

Looking up the critical value for a one-sided 95% confidence level and 26 degrees of freedom in the chi-square distribution table or using statistical software, we find χ^2 ≈ 40.113.

Substituting the values into the formula, we have:

CI = √[(27 - 1) * (0.44^2) / 40.113]

Calculating this expression, we find:

CI ≈ √[(26) * (0.44^2) / 40.113]

CI ≈ √[(26) * (0.1936) / 40.113]

CI ≈ √[5.0336 / 40.113]

CI ≈ √0.1255

CI ≈ 0.354

Therefore, the one-sided 95% confidence interval with no lower bound for the population standard deviation is approximately (0.354, ∞).