Waste Today Services operates a garbage hauling company in a South Jersey city. Each year, the company must apply for a new contract with the city. The contract is in part based on the pounds of garbage hauled. Part of the analysis that goes into the contract development is an estimate of the mean pound of garbage put out by each customer in the city. The city has asked for both 99% and 90% confidence interval estimates for the mean. A sample of 100 customers was taken. It is known that the population standard deviation is 12.6 pounds and the sample mean is 40.78. What is the impact in changing the confidence level?

(please provide complete answer)

To determine the impact of changing the confidence level on the estimate for the mean pounds of garbage put out by each customer in the city, we need to calculate the confidence intervals for both the 99% and 90% confidence levels.

First, let's define the terms:

- Confidence level: It represents the level of certainty we have in the estimate. It is expressed as a percentage. In this case, we have two confidence levels, 99% and 90%.
- Confidence interval: It is the range of values within which the true population mean is expected to fall. It is calculated using the sample mean, sample size, and the confidence level.

Now, let's calculate the confidence intervals:

1. For the 99% confidence interval:
The formula to calculate the confidence interval for a known population standard deviation is:

Confidence Interval = Sample Mean ± (Z * (Population Standard Deviation / √Sample Size))

For a 99% confidence level, the corresponding Z-value is obtained from the standard normal distribution table. The Z-value for a 99% confidence level is approximately 2.576.

Plugging in the values:
Sample Mean = 40.78
Population Standard Deviation = 12.6
Sample Size = 100
Z = 2.576

Confidence Interval = 40.78 ± (2.576 * (12.6 / √100))

Simplifying the equation:
Confidence Interval = 40.78 ± (2.576 * 1.26)

Now, we can calculate the upper and lower bounds of the 99% confidence interval:
Lower bound = 40.78 - (2.576 * 1.26)
Upper bound = 40.78 + (2.576 * 1.26)

2. For the 90% confidence interval:
The process is similar to the 99% confidence interval, but this time we need to find the Z-value for a 90% confidence level. The Z-value for a 90% confidence level is approximately 1.645.

Using the same formulas as before, with a Z-value of 1.645:
Confidence Interval = 40.78 ± (1.645 * (12.6 / √100))

Calculating the upper and lower bounds of the 90% confidence interval will give us the range within which we expect the true population mean to fall.

Now, let's analyze the impact of changing the confidence level:

The main difference between the 99% and 90% confidence intervals lies in the Z-values used to calculate them. A higher confidence level requires a larger Z-value, resulting in a wider confidence interval. In other words, increasing the confidence level increases the margin of error around the estimated mean.

So, changing the confidence level from 99% to 90% would result in a narrower confidence interval, meaning a smaller range of potential values for the true population mean. However, this also means the estimate is less precise and has a higher probability of missing the true population mean.

Therefore, when comparing the two confidence levels, the 99% confidence interval will have a wider range but a higher level of certainty, while the 90% confidence interval will have a narrower range but a lower level of certainty.