Question

Your city has recently purchased new, more efficient garbage trucks. Over a sample of 36 days, workers collected a daily average of 145 tons of trash, with a sample standard deviation of 21 tons.

(a) What is the point estimate for the mean daily trash collection?

(b) What is the standard error of the sampling distribution?

(c) Using Table C, determine the 90% and 95% confidence intervals for the mean daily trash collection. Show your work for each margin of error, including how you found the margin of error's critical t-value in Table C.

(d) Using Excel, calculate the 99% and 99.9% confidence intervals for the mean daily trash collection. Show your work for each margin of error, including the formula you typed into Excel to determine each margin of error's critical t-value.

Answer 1

The point estimate for the mean daily trash collection is 145 tons. The standard error of the sampling distribution is 3.5 tons. Using Table C, the 90% confidence interval is 139.0535 to 150.9465 tons, and the 95% confidence interval is 137.895 to 152.105 tons. Using Excel, the 99% confidence interval is 135.5035 to 154.4965 tons, and the 99.9% confidence interval is 132.022 to 157.978 tons.

Step-by-step explanation:

(a) The point estimate for the mean daily trash collection is the sample mean, which is 145 tons.

(b) The standard error of the sampling distribution is calculated using the formula:

Standard Error (SE) = Sample Standard Deviation / Square Root of Sample Size

SE = 21 tons / Square Root of 36 = 21 / 6 = 3.5 tons

(c) To determine the 90% and 95% confidence intervals for the mean daily trash collection, we need to calculate the margin of error and find the critical t-value from Table C. The formula to calculate the margin of error is:

Margin of Error = Critical t-value * Standard Error

For a 90% confidence interval:

Margin of Error = 1.699 * 3.5 = 5.9465 tons

90% Confidence Interval = Sample Mean - Margin of Error to Sample Mean + Margin of Error

90% Confidence Interval = 145 - 5.9465 to 145 + 5.9465

90% Confidence Interval = 139.0535 to 150.9465 tons

For a 95% confidence interval:

Margin of Error = 2.030 * 3.5 = 7.105 tons

95% Confidence Interval = Sample Mean - Margin of Error to Sample Mean + Margin of Error

95% Confidence Interval = 145 - 7.105 to 145 + 7.105

95% Confidence Interval = 137.895 to 152.105 tons

(d) To calculate the 99% and 99.9% confidence intervals for the mean daily trash collection using Excel, you can use the T.INV function. The formula to calculate the margin of error is:

Margin of Error = Critical t-value * Standard Error

For a 99% confidence interval:

Margin of Error = T.INV(0.995, Sample Size - 1) * Standard Error

Margin of Error = T.INV(0.995, 36 - 1) * 3.5 = 2.713 * 3.5 = 9.4965 tons

99% Confidence Interval = Sample Mean - Margin of Error to Sample Mean + Margin of Error

99% Confidence Interval = 145 - 9.4965 to 145 + 9.4965

99% Confidence Interval = 135.5035 to 154.4965 tons

For a 99.9% confidence interval:

Margin of Error = T.INV(0.9995, Sample Size - 1) * Standard Error

Margin of Error = T.INV(0.9995, 36 - 1) * 3.5 = 3.708 * 3.5 = 12.978 tons

99.9% Confidence Interval = Sample Mean - Margin of Error to Sample Mean + Margin of Error

99.9% Confidence Interval = 145 - 12.978 to 145 + 12.978

99.9% Confidence Interval = 132.022 to 157.978 tons