Question

Suppose a random sample of 50 bottles of "CoughGone" cough syrup was collected and the alcohol content (in wt. percent) of each bottle measured. A 95 % confidence interval for bottle true mean alcohol content, μ, was then determined from these data to be (6.7, 9.3). It was assumed that the variance of the alcohol distribution was known.

a.) What is the value of the sample mean, x̄ ?(Give answer to two places past decimal.)

b.) What is the value of the population standard deviation, σ ? (Give decimal answer to two places past decimal.)  

c.) Compared to the 95 % confidence interval above, a 85% confidence interval computed from the same data would be:   
a - wider  
b - narrower  
c - same width  
d - can't tell

d.) Which of the following statements best describes the meaning of a 95 % confidence interval for μ?  
a - There is a 95 % chance that the value of μ is between 6.7 and 9.3.  
b - We can be highly confident that 95 % of all bottles of "CoughGone" cough syrup have an alcohol content between 6.7 and 9.3 wt. percent.  
c - If the process of selecting a random sample of size 50 and computing a 95 % confidence interval is repeated 100 times, we expect that 95 of the resulting intervals will include μ.  

e.) If the random sample size is increased from 50 to 80, what is the new width of the 95 % confidence interval? Assume the value of the population standard deviation, σ, remains unchanged. (Give decimal answer to two places past decimal.)  

Answer 1

a) The sample mean is computed as the middle point of the given confidence interval as:

b) From the standard normal tables, we get here:

P( -1.96 < Z < 1.96 ) = 0.95

The standard error is computed as: Upper Bound - Sample mean = 9.3 - 8.0 = 1.3

Therefore, we get the population standard deviation now as:

Therefore 4.69 is the required population standard deviation .

c) For a 85% confidence interval as we need to be less confident so the confidence interval width would be lower and therefore the confidence interval would be narrower

d) The interpretation of the confidence interval is that there is a 95% probability that the true population mean lies in the given confidence interval. Therefore c) is the correct answer here that is If the process of selecting a random sample of size 50 and computing a 95 % confidence interval is repeated 100 times, we expect that 95 of the resulting intervals will include μ.  

e) The width of the confidence interval here is computed as:

Therefore 2.0555 is the required width of the new confidence interval here.