Example 7.11 from Statistics, 7.3 Using the Central Limit Theorem

Using the Central Limit Theorem: example 7.11

A study was done about a medical condition that affects a certain group of people. The age range of the people was 14–61. The mean age was 30.9 years with a standard deviation of nine years.

In a sample of 25 people, what is the probability that the mean age of the people is less than 35?
Is it likely that the mean age of the sample group could be more than 50 years? Interpret the results.
In a sample of 49 people, what is the probability that the sum of the ages is no less than 1,600?
Is it likely that the sum of the ages of the 49 people are at most 1,595? Interpret the results.
Find the 95^th percentile for the sample mean age of 65 people. Interpret the results.
Find the 90^th percentile for the sum of the ages of 65 people. Interpret the results.

Work it out on paper first. Then open the solution one step at a time, and stop as soon as you can finish on your own.

1
Solution
1. P( $\bar{x}$ < 35) = normalcdf(-E99,35,30.9,1.8) = 0.9886
2. P( $\bar{x}$ > 50) = normalcdf(50, E99,30.9,1.8) ≈ 0. For this sample group, it is almost impossible for the group’s average age to be more than 50. However, it is still possible for an individual in this group to have an age greater than 50.
3. P(Σx ≥ 1,600) = normalcdf(1600,E99,1514.10,63) = 0.0864
4. P(Σx ≤ 1,595) = normalcdf(-E99,1595,1514.10,63) = 0.9005. This means that there is a 90 percent chance that the sum of the ages for the sample group n = 49 is at most 1,595.
5. The 95th percentile = invNorm(0.95,30.9,1.1) = 32.7. This indicates that 95 percent of the people in the sample of 65 are younger than 32.7 years, on average.
6. The 90th percentile = invNorm(0.90,2008.5,72.56) = 2101.5. This indicates that 90 percent of the people in the sample of 65 have a sum of ages less than 2,101.5 years.

How did it go?

Solution

Solution