Calculating a Confidence Interval for a Population Mean

By Consumer Dummies

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 Z-distribution. When the population standard deviation is unknown, it involves a t-distribution. Calculate confidence intervals for population means in the following problems.

Sample questions

  1. 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:

    image0.jpg

    where

    image1.jpg

    is the sample mean,

    image2.jpg

    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 two-tailed confidence interval with a confidence level of 95%.

    Next, substitute the values into the formula:

    image3.jpg

    The 95% confidence interval is 8 plus/minus 0.7 (rounded to the nearest tenth), or 7.3 to 8.7 points scored.

  2. 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:

    image4.jpg

    where

    image5.jpg

    is the sample mean,

    image6.jpg

    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 two-tailed confidence interval with a confidence level of 99% is 2.58.

    Next, substitute the values into the formula:

    image7.jpg

    The confidence interval is 650 plus/minus 25.8 (rounded to the nearest tenth), or 624.2 to 678.8.

  3. 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:

    image8.jpg

    where

    image9.jpg

    is the sample mean,

    image10.jpg

    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 two-tailed confidence interval with a confidence level of 99% is 2.58.

    Next, substitute the values into the formula:

    image11.jpg

    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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