The mayor is interested in finding a 95% confidence interval for the mean number of pounds of trash per person per week generated in the city. The study included 147 residents whose mean number of pounds of trash generated per person per week was 33.6 pounds, and the standard deviation was 8.2 pounds. Round answers to 3 decimal places where possible.

a. To compute the confidence interval, use a [insert appropriate distribution] distribution.

b. With 95% confidence, the population mean number of pounds per person per week is between [insert lower limit] and [insert upper limit] pounds.

c. If many groups of 147 randomly selected members are studied, then a different confidence interval would be produced from each group. About [insert percentage]% of these confidence intervals will contain the true population mean number of pounds of trash generated per person per week, and about [insert percentage]% will not contain the true population mean number of pounds of trash generated per person per week.

The answers are: a) t-distribution will be used b) the confidence interval is 32.260 < Population Mean < 34.940 c) sampling process and assumptions are valid.

a. To compute the confidence interval, we will use the t-distribution since the sample size is relatively small (147 residents).

b. The formula for the confidence interval for the population mean is given by:

Confidence Interval = Sample Mean ± Margin of Error

Where the margin of error is:

Margin of Error = Critical Value × Standard Error

Standard Error = Standard Deviation / √Sample Size

For a 95% confidence level, the critical value for a t-distribution with 146 degrees of freedom (147 - 1) is approximately 1.9785 (you can look this up in a t-table or calculator).

Given:

Sample Mean = 33.6 pounds

Standard Deviation (σ) = 8.2 pounds

Sample Size (n) = 147

Critical Value = 1.9785 (for a 95% confidence level)

Calculate the Standard Error:

Standard Error = 8.2 / √147 ≈ 0.6772

Calculate the Margin of Error:

Margin of Error = 1.9785 × 0.6772 ≈ 1.3399

So, the confidence interval is:

33.6 - 1.3399 < Population Mean < 33.6 + 1.3399

32.2601 < Population Mean < 34.9399

Rounded to three decimal places, the interval is:

32.260 < Population Mean < 34.940

c. About 95% of these confidence intervals will contain the true population mean number of pounds of trash generated per person per week, while about 5% will not contain the true population mean. This is because of the nature of confidence intervals, which are constructed to provide a certain level of confidence (in this case, 95%) that the true population parameter lies within the interval. This assumes that the sampling process and assumptions are valid.

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