Question

The mayor is interested in finding a 95% confidence interval for the mean number of pounds of

trash per person per week that is generated in the city. The study included 186 residents whose

mean number of pounds of trash generated per person per week was 33. 6 pounds and the

standard deviation was 6. 7 pounds. Round answers to 3 decimal places where possible.

a. To compute the confidence interval use a ? distribution.

b. With 95% confidence the population mean number of pounds per person per week is between

and

pounds.

c. If many groups of 186 randomly selected members are studied, then a different confidence

percent of these confidence

interval would be produced from each group. About

intervals will contain the true population mean number of pounds of trash generated per person

percent will not contain the true population mean number of

per week and about

pounds of trash generated per person per week

Answer 1

see the explanation

Step-by-step explanation:

a. To compute the confidence interval, we will use a t-distribution because the sample size (186) is less than 30 and the population standard deviation is unknown.

b. The formula for a 95% confidence interval for the population mean is:

xbar ± t(a/2) * (s/sqrt(n))

where xbar is the sample mean, s is the sample standard deviation, n is the sample size, and ta/2 is the critical t-value from the t-distribution with n-1 degrees of freedom, where a/2 = 0.025.

Substituting the given values, we get:

Lower limit = 33.6 - (ta/2 * (6.7/sqrt(186)))

Upper limit = 33.6 + (ta/2 * (6.7/sqrt(186)))

To find the critical t-value, we can use a t-table or a calculator. For a 95% confidence interval with 185 degrees of freedom, the critical t-value is approximately 1.972.

Substituting this value, we get:

Lower limit = 33.6 - (1.972 * (6.7/sqrt(186))) ≈ 32.882

Upper limit = 33.6 + (1.972 * (6.7/sqrt(186))) ≈ 34.318

Therefore, with 95% confidence, the population mean number of pounds per person per week is between 32.882 pounds and 34.318 pounds.

c. If many groups of 186 randomly selected members are studied, then about 95% of these confidence intervals will contain the true population mean number of pounds of trash generated per person per week, and about 5% will not contain the true population mean number of pounds of trash generated per person per week. This is because the 95% confidence level means that in the long run, 95% of the intervals constructed in this way will contain the true population mean. However, it is important to note that any one particular interval may or may not contain the true population mean.