Calculating a Confidence Interval for a Population Mean
Calculating a confidence interval for a population mean when the population standard deviation is known and the sample size is at least 30 involves the Zdistribution. When the population standard deviation is unknown, it involves a tdistribution. Calculate confidence intervals for population means in the following problems.
Sample questions

In a random sample of 50 intramural basketball players at a large university, the average points per game was 8, with a standard deviation of 2.5 points and a 95% confidence level.
Which of the following statements is correct?
(A) With 95% confidence, the average points scored by all intramural basketball players is between 7.3 and 8.7 points.
(B) With 95% confidence, the average points scored by all intramural basketball players is between 7.7 and 8.4 points.
(C) With 95% confidence, the average points scored by all intramural basketball players is between 5.5 and 10.5 points.
(D) With 95% confidence, the average points scored by all intramural basketball players is between 7.2 and 8.8 points.
(E) With 95% confidence, the average points scored by all intramural basketball players is between 7.6 and 8.4 points.
Answer: A. With 95% confidence, the average points scored by all intramural basketball players is between 7.3 and 8.7 points.
Use the formula for finding the confidence interval for a population when the standard deviation is known:
where
is the sample mean,
is the population standard deviation, n is the sample size, and z* represents the appropriate z*value from the standard normal distribution for your desired confidence level. The data has to come from a normal distribution, or n has to be large enough (a standard rule of thumb is at least 30 or so), for the central limit theorem to apply.
The z*value is 1.96 for a twotailed confidence interval with a confidence level of 95%.
Next, substitute the values into the formula:
The 95% confidence interval is 8 plus/minus 0.7 (rounded to the nearest tenth), or 7.3 to 8.7 points scored.

On the SAT Math test, a random sample of the scores of 100 students in a high school had a mean of 650.
The standard deviation for the population is 100. What is the confidence interval if 99% is the confidence level?
Answer: The 99% confidence interval for the average SAT math score for all students at the high school is between 624.2 and 678.8.
Use the formula for finding the confidence interval for a population when the standard deviation is known:
where
is the sample mean,
is the population standard deviation, n is the sample size, and z* represents the appropriate z*value from the standard normal distribution for your desired confidence level. The data has to come from a normal distribution, or n has to be large enough (a standard rule of thumb is at least 30 or so), for the central limit theorem to apply.
The z*value for a twotailed confidence interval with a confidence level of 99% is 2.58.
Next, substitute the values into the formula:
The confidence interval is 650 plus/minus 25.8 (rounded to the nearest tenth), or 624.2 to 678.8.

An apple orchard harvested ten trees of apples. From a random sample of 50 apples, the mean weight of an apple was 7 ounces.
The population standard deviation is 1.5 ounces. What is the confidence interval if 99% is the confidence level?
Answer: The 99% confidence interval for the average weight of all apples from the ten trees is between 6.5 and 7.5 ounces.
Use the formula for finding the confidence interval for a population when the standard deviation is known:
where
is the sample mean,
is the population standard deviation, n is the sample size, and z* represents the appropriate z*value from the standard normal distribution for your desired confidence level. The data has to come from a normal distribution, or n has to be large enough (a standard rule of thumb is at least 30 or so), for the central limit theorem to apply.
The z*value for a twotailed confidence interval with a confidence level of 99% is 2.58.
Next, substitute the values into the formula:
The confidence interval is 7 plus/minus 0.5 (rounded to the nearest tenth), or 6.5 to 7.5 ounces.
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