SOLUTION: A research firm conducted a survey to determine the mean amount Americans spend on coffee during a week. They found the distribution of weekly spending followed the normal dist

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Question 1153359: A research firm conducted a survey to determine the mean amount Americans spend on coffee during a week.
They found the distribution of weekly spending followed the normal distribution with a population standard deviation of $5.
A sample of 49 Americans revealed that Xbar=$20.

What is the point estimate of the population mean?
Using the 95% level of confidence, determine the confidence interval for μ.
Answer by jim_thompson5910(35256) About Me  (Show Source):
You can put this solution on YOUR website!

Question: What is the point estimate of the population mean?
Answer: xbar = 20

Question: What is the 95% confidence interval for mu?
Answer: approximately (18.6, 21.4), in which we can alternatively write as 18.6 < mu < 21.4 or as

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Work Shown:

n = 49 is the sample size
mu = population mean
xbar = 20 is the sample mean
sigma = 5 is the population standard deviation

xbar is the point estimate of the population mean mu. It is our best guess at the population mean based on the sample information we gathered/computed. Of course we aren't likely to have xbar and mu match up perfectly. This is where the confidence interval comes in. The confidence interval is like a net we cast out and we say we are 95% confident that the population mean is in this range. For any confidence interval involving mu, the point estimate xbar is always at the center of the confidence interval.

At 95% confidence, the critical z value is roughly z = 1.96; use a table or calculator to determine this.

L = lower bound of confidence interval
L = xbar - z*(sigma/sqrt(n))
L = 20 - 1.96*(5/sqrt(49))
L = 20 - 1.96*(5/7)
L = 20 - 1.96*(0.714285714285714)
L = 20 - 1.4
L = 18.6

U = upper bound of confidence interval
U = xbar + z*(sigma/sqrt(n))
U = 20 + 1.96*(5/sqrt(49))
U = 20 + 1.96*(5/7)
U = 20 + 1.96*(0.714285714285714)
U = 20 + 1.4
U = 21.4

(L, U) = (18.6, 21.4) is the approximate 95% confidence interval.

If you want to write the confidence interval in the form L < mu < U, then you would write 18.6 < mu < 21.4

Another format is , which has P as the point estimate and M as the margin of error. So, meaning we have a point estimate of 20 and a margin of error of 1.4; this is reflected in L = 20-1.4 and U = 20+1.4