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?
Use confidence interval formulas:
CI99 = mean ± (2.58)(sd/√n)
...where mean = 40.78; 2.58 represents 99% interval; sd = 12.6; n = 100.
Calculate the interval.
Use the formula again for a 90% confidence interval using the appropriate value from a z-table.
Once you calculate both intervals, you should be able to see the difference.
I hope this will help get you started.
The confidence level refers to the level of certainty or reliability in estimating the population parameter, in this case, the mean pound of garbage put out by each customer in the city. Increasing the confidence level generally results in wider confidence intervals.
To calculate the confidence interval, you can use the formula:
CI = X̄ ± (Z * σ/√n)
Where:
CI = Confidence Interval
X̄ = Sample mean
Z = Z-score (corresponding to the desired confidence level)
σ = Population standard deviation
n = Sample size
To calculate the 99% confidence interval, we need to find the Z-score associated with the 99% confidence level. The Z-score can be obtained from a Z-table or using statistical software. The Z-score for a 99% confidence level is approximately 2.58.
Using the given values:
X̄ = 40.78
Z (99%) = 2.58
σ = 12.6
n = 100
Plugging the values into the formula:
CI (99%) = 40.78 ± (2.58 * 12.6/√100)
= 40.78 ± (2.58 * 12.6/10)
= 40.78 ± 3.3252
So, the 99% confidence interval estimate for the mean pound of garbage put out by each customer in the city is (37.4548, 44.1052).
To calculate the 90% confidence interval, you need to find the Z-score associated with the 90% confidence level. The Z-score for a 90% confidence level is approximately 1.645.
Using the given values:
X̄ = 40.78
Z (90%) = 1.645
σ = 12.6
n = 100
Plugging the values into the formula:
CI (90%) = 40.78 ± (1.645 * 12.6/√100)
= 40.78 ± (1.645 * 12.6/10)
= 40.78 ± 2.1441
So, the 90% confidence interval estimate for the mean pound of garbage put out by each customer in the city is (38.6359, 42.9241).
In summary, by increasing the confidence level from 90% to 99%, the width of the confidence interval increases. This wider interval accounts for greater uncertainty and provides a higher level of confidence in capturing the true population mean within the interval. However, it also leads to a less precise estimate.