The U.S. Dairy Industry wants to estimate the mean yearly milk
consumption. A sample of 16 people reveals the mean yearly consumption
to be 60 gallons with a standard deviation of 20 gallons.
a. What is the value of the population mean? What is the best
estimate of this value?
b. Explain why we need to use thetdistribution. What assumption do
you need to make?
c. For a 90 percent confidence interval, what is the value oft?
d. Develop the 90 percent confidence interval for the population
mean.
e. Would it be reasonable to conclude that the population mean is 63
gallons?
a. Best estimate of the population value = sample mean.
b-e. http://www.google.com/search?client=safari&rls=en&q=t+test+statistics&ie=UTF-8&oe=UTF-8
a sample of 16 people reveals the mean consumption to be 60 gallons with the standard of deviation of 20 gallons
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a. The value of the population mean is unknown, as it represents the average yearly milk consumption for all individuals in the U.S. To estimate this value, we can use the sample mean, which is 60 gallons.
b. We need to use the t-distribution because the population standard deviation is unknown, and we are working with a small sample size (16 individuals). The assumption we need to make is that the data approximately follows a normal distribution.
c. To determine the value of t for a 90 percent confidence interval, we need to know the degrees of freedom. For this case, the degrees of freedom (df) is calculated as df = sample size - 1 = 16 - 1 = 15. Using a t-distribution table or a statistical software, we can find the value of t that corresponds to a 90 percent confidence level with 15 degrees of freedom.
d. To develop the 90 percent confidence interval for the population mean, we can use the formula:
Confidence interval = sample mean ± (t-value * standard error)
The standard error is calculated as the sample standard deviation divided by the square root of the sample size. In this case, the standard error = 20 / √16 = 5.
Substituting in the values, the confidence interval = 60 ± (t-value * 5), where the t-value corresponds to a 90 percent confidence level with 15 degrees of freedom.
e. To determine if it is reasonable to conclude that the population mean is 63 gallons, we need to see if 63 falls within the 90 percent confidence interval calculated in part d. If 63 falls within the interval, it would be reasonable to conclude that the population mean could be 63 gallons. If 63 falls outside the interval, it would not be reasonable to conclude that the population mean is 63 gallons.