Question

A random sample of n = 81 customers at a fast-food restaurant had a mean bill of x̅ = $10.71. If the population standard deviation is σ = $3.42, find the 97% confidence interval (including units) of the mean bill for all customers. Assume the variable is normally distributed and state the distribution used to find the interval.
Distribution used:
Confidence interval:

The average age of a sample of n = 73 mayors was x̅ = 46.9 years with a standard deviation of s = 3.1 years. Assuming the ages are normally distributed, find the 95% confidence interval (including units) of the mean age of all mayors. State the distribution used.

In a certain convenient care center, a random sample of n = 40 weeks was selected. It was found that an average of x̅ = 353 patients were treated per week with a standard deviation of s = 25.8. Assume the variable is normally distributed and find the 97% confidence interval (including units) of the population mean. State the distribution used.

The recorded high temperatures (in degrees Fahrenheit) in June for a sample of 17 US cities are listed below. Assuming the temperatures are normally distributed, find the 92% confidence interval (including units) of the population mean. Give the distribution used and the mean and standard deviation of the sample rounded to the nearest hundredth (two decimal places). Give all digits provided by the calculator for the confidence interval.

91        77        77        84        81        93        94        84        84

84        87        73        93        90        67        96        99

Distribution used:
x̅ ≈
s ≈
Confidence interval:

The GPAs for a sample of n = 6 freshmen at a certain college are listed below. Assuming the GPAs are normally distributed and that σ = 0.43, find the 98% confidence interval of the true mean of the freshman class (no units are needed). Give the distribution used and the mean of the sample rounded to the nearest hundredth (two decimal places). Give all digits provided by the calculator for the confidence interval.

            2.96      2.71      3.20      3.84      4.00      3.42

Distribution used:
x̅ ≈
Confidence interval:

Answer 1

Given the sample size and sample mean and population standard deviation .

Since sample size is large enough we use normal distribution.

The   two sided confidence interval for mean based on the sample data is

The hypotheses are

Here .  So the 97% CI is

Given the sample size   and sample mean and sample standard deviation .

Since sample size is large enough we use normal distribution.

The two sided confidence interval for mean based on the sample data is

The hypotheses are

Here . So the 95% CI is

Given the sample size   and sample mean   and sample standard deviation .

Since sample size not large enough we use T distribution.

The   two sided confidence interval for mean based on the sample data is

The hypotheses are

Here .  So the 97% CI is

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