Question

A sanitation department is interested in estimating the mean amount of garbage per bin for all bins in the city. In a random sample of 40 bins, the sample mean amount was 51.5 pounds and the sample standard deviation was 3.6 pounds. Construct 95% and 99% confidence intervals for the mean amount of garbage per bin for all bins in the city.

a) What is the lower limit of the 95% interval? Give your answer to three decimal places.  

b) What is the upper limit of the 95% interval? Give your answer to three decimal places.  

c) What is the lower limit of the 99% interval? Give your answer to three decimal places.  

d) What is the upper limit of the 99% interval? Give your answer to three decimal places.  

e) Consider the claim that the mean amount of garbage per bin is 52.791 pounds. Is the following statement true or false? The decision about the claim would depend on whether we use a 95% or 99% confidence interval.

TrueFalse    

Answer 1

We will use the one-sample t-test formula to calculate the confidence interval. The formula is:

Mean = 51.5

n = 40

s = 3.6

Now we need to find the t-critical value at 95% and 99% confidence level.

When df = n - 1 = 39

t-critical at alpha=0.05 is: 2.023

t-critical at alpha=0.01 is: 2.708

When alpha=0.05:

a) Lower limit: 51.5 - 2.023*3.6/√40 = 51.5 - 1.152 = 50.348

b) Upper limit: 51.5 + 2.023*3.6/√40 = 51.5 + 1.152 =  52.652

When alpha = 0.01:

c) Lower limit: 51.5 - 2.708*3.6/√40 = 51.5 - 1.541= 49.959

d) Upper limit: 51.5 + 2.708*3.6/√40 = 51.5 + 1.541 = 53.041

As the claim is that the mean is 52.791. The 95% CI does not contain the value 52.791 but 99% CI contains it. Hence, the decision will depend on whether we use 95% or 99% CI

Statement is TRUE