A random sample of 22 garbage cans yields a mean of 33.2 pounds with a standard deviation of 3.2 pounds. Is there evidence that residents overload garbage cans beyond 30 pounds on average?

A random sample of 22 garbage cans yields a mean of 33.2 pounds with a standard deviation of 3.2 pounds. Is there evidence that residents overload garbage cans beyond 30 pounds on average?

• A. One-sample z-test
• B. 1-sample t-test with 21 degrees of freedom
• C. Two-sample t-test
• D. Sign test

Answer: (B)

Testing whether the population mean exceeds a specific value using a single sample when the population standard deviation is unknown. Since sigma isn’t known and the sample is of moderate size (n = 22), the appropriate method is a one-sample t-test with 21 degrees of freedom. Set up: null hypothesis that the mean is 30 pounds, versus the alternative that the mean is greater than 30 pounds. The data give X̄ = 33.2 and s = 3.2. The t statistic is t = (X̄ − μ0) / (s/√n) = (33.2 − 30) / (3.2/√22) ≈ 3.2 / 0.682 ≈ 4.69. With 21 degrees of freedom, a t value of about 4.69 is far into the upper tail, giving a p-value well below 0.001. Conclusion: there is strong evidence that residents overload garbage cans beyond 30 pounds on average. The z-test isn’t appropriate here because the population standard deviation isn’t known; a two-sample t-test isn’t the scenario since this is a test about a single mean; a sign test isn’t necessary when you have the actual numerical data and can use the more powerful t-test.

Testing whether the population mean exceeds a specific value using a single sample when the population standard deviation is unknown. Since sigma isn’t known and the sample is of moderate size (n = 22), the appropriate method is a one-sample t-test with 21 degrees of freedom.

Set up: null hypothesis that the mean is 30 pounds, versus the alternative that the mean is greater than 30 pounds. The data give X̄ = 33.2 and s = 3.2. The t statistic is

t = (X̄ − μ0) / (s/√n) = (33.2 − 30) / (3.2/√22) ≈ 3.2 / 0.682 ≈ 4.69.

With 21 degrees of freedom, a t value of about 4.69 is far into the upper tail, giving a p-value well below 0.001.

Conclusion: there is strong evidence that residents overload garbage cans beyond 30 pounds on average. The z-test isn’t appropriate here because the population standard deviation isn’t known; a two-sample t-test isn’t the scenario since this is a test about a single mean; a sign test isn’t necessary when you have the actual numerical data and can use the more powerful t-test.