Question 1170607: Fast-food restaurant located in Dubai in half-pound hamburgers, fish sandwiches, and chicken sandwiches. Soft drinks and French fries are also available. The Planning Department of the restaurant reports that the distribution of daily sales for restaurants follows the normal distribution and that the population standard deviation is $3,000. A sample of 40 showed the mean daily sales to be $20,000.
(a) What is the population mean?
(b) What is the best estimate of the population mean? What is this value called?
(c) Develop a 99 % confidence interval for the population mean.
(d) Interpret the confidence interval.
Answer by math_tutor2020(3816)   (Show Source):
You can put this solution on YOUR website!
Part (a)
The population mean is unknown. We'll let mu be the population mean.
The goal of statistics is to estimate population parameters based on sample statistics. This is due to the high cost (of money and time) dealing with taking a census, so resorting to a representative sample is the next best thing. Therefore, it is quite common to not know the population mean. The same goes for the population standard deviation; however, this was given to be sigma = 3000.
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Part (b)
The variable xbar is the best estimate of the population mean. The statistic xbar is an unbiased estimate of the parameter mu.
In this problem, xbar = 20000
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Part (c)
n = 40 is the sample size
sigma = 3000 is the population standard deviation
We'll use the z distribution because sigma is known.
At 99% confidence, the z critical value is about 2.576 which is determined through use of a calculator or table.
The margin of error is
E = z*sigma/sqrt(n)
E = 2.576*3000/sqrt(40)
E = 1221.90408788907
The confidence interval for mu is
xbar-E < mu < xbar+E
20000-1221.90408788907 < mu < 20000+1221.90408788907
18778.095912111 < mu < 21221.904087889
18778.096 < mu < 21221.904
The 99% confidence interval for mu is 18778.096 < mu < 21221.904
This is equivalent to saying (18778.096, 21221.904)
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Part (d)
We are 99% confident that mu is between 18778.096 and 21221.904
In the context of the problem, it means we're 99% confident that the population mean of daily sales is between $18,778.096 and $21,221.904
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