A sanitation department is interested in estimating the mean amount of garbage p

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 52.8 pounds and the sample standard deviation was 3.9 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 54.1985 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. False True

Explanation / Answer

a-b)

Note that              
Margin of Error E = z(alpha/2) * s / sqrt(n)              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.025          
X = sample mean =    52.8          
z(alpha/2) = critical z for the confidence interval =    1.959963985          
s = sample standard deviation =    3.9          
n = sample size =    40          
              
Thus,              
Margin of Error E =    1.208600313          

Lower bound =    51.59139969   [ANSWER, PART A]      
Upper bound =    54.00860031   [ANSWER, PART B]

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c-d)

Note that              
Margin of Error E = z(alpha/2) * s / sqrt(n)              
Lower Bound = X - z(alpha/2) * s / sqrt(n)              
Upper Bound = X + z(alpha/2) * s / sqrt(n)              
              
where              
alpha/2 = (1 - confidence level)/2 =    0.005          
X = sample mean =    52.8          
z(alpha/2) = critical z for the confidence interval =    2.575829304          
s = sample standard deviation =    3.9          
n = sample size =    40          
              
Thus,              
Margin of Error E =    1.588370055          

Lower bound =    51.21162994   [ANSWER, PART C]      

Upper bound =    54.38837006   [ANSWER, PART D]      
**********************************

e)

As 54.1985 is not is the 95% interval but inside the 99% interval, then it depends on the interval we use.

Hence,

ANSWER: TRUE [ANSWER]              

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