Did I Get This – When to Use Z-Interval (Means)
Below are four different situations in which a confidence interval for μ (mu) is called for.
Situation A: In order to estimate μ (mu), the mean annual salary of high-school teachers in a certain state, a random sample of 150 teachers was chosen and their average salary was found to be $38,450. From past experience, it is known that teachers’ salaries have a standard deviation of $5,000.
Situation B: A medical researcher wanted to estimate μ (mu), the mean recovery time from open-heart surgery for males between the ages of 50 and 60. The researcher followed the next 15 male patients in this age group who underwent open-heart surgery in his medical institute through their recovery period. (Comment: Even though the sample was not strictly random, there is no reason to believe that the sample of “the next 15 patients” introduces any bias, so it is as good as a random sample). The mean recovery time of the 15 patients was 26 days. From the large body of research that was done in this area, it is assumed that recovery times from open-heart surgery have a standard deviation of 3 days.
Situation C: In order to estimate μ (mu), the mean score on the quantitative reasoning part of the GRE (Graduate Record Examination) of all MBA students, a random sample of 1,200 MBA students was chosen, and their scores were recorded. The sample mean was found to be 590. It is known that the quantitative reasoning scores on the GRE vary normally with a standard deviation of 150.
Situation D: A psychologist wanted to estimate μ (mu), the mean time it takes 6-year-old children diagnosed with Down’s Syndrome to complete a certain cognitive task. A random sample of 12 children was chosen and their times were recorded. The average time it took the 12 children to complete the task was 7.5 minutes. From past experience with similar tasks, the time is known to vary normally with a standard deviation of 1.3 minutes.
Here is another set of situations:
Situation A: A marketing executive wants to estimate the average time, in days, that a watch battery will last. She tests 50 randomly selected batteries and finds that the distribution is skewed to the left, since a couple of the batteries were defective. It is known from past experience that the standard deviation is 25 days.
Situation B: A college professor desires an estimate of the mean number of hours per week that full-time college students are employed. He randomly selected 250 college students and found that they worked a mean time of 18.6 hours per week. He uses previously known data for his standard deviation.
Situation C: A medical researcher at a sports medicine clinic uses 35 volunteers from the clinic to study the average number of hours the typical American exercises per week. It is known that hours of exercise are normally distributed and past data give him a standard deviation of 1.2 hours.
Situation D: A high-end auto manufacturer tests 5 randomly selected cars to find out the damage caused by a 5 mph crash. It is known that this distribution is normal. Assume that the standard deviation is known.
This document is linked from Population Means (Part 3).