Example 7.2 from Statistics, 7.1 The Central Limit Theorem for Sample Means (Averages)

The Central Limit Theorem for Sample Means (Averages): example 7.2

The length of time, in hours, it takes a group of people, 40 years old and older, to play one soccer match is normally distributed with a mean of 2 hours and a standard deviation of 0.5 hours. A sample of size n = 50 is drawn randomly from the population. Find the probability that the sample mean is between 1.8 hours and 2.3 hours.

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.

Solution
Let X = the time, in hours, it takes to play one soccer match.

The probability question asks you to find a probability for the sample mean time, in hours, it takes to play one soccer match.

Let $\bar{X}$ = the mean time, in hours, it takes to play one soccer match.

If μ_X = _________, σ_X = __________, and n = ___________, then X ~ N(______, ______) by the central limit theorem for means.

μ_X = 2, σ_X = 0.5, n = 50, and X ~ N $(2, \frac{0.5}{\sqrt{50}})$

Find P(1.8 < $\bar{x}$ < 2.3). Draw a graph.

$P (1.8 < \bar{x} < 2.3) = 0.9977$

normalcdf
$(1. 8,2 .3,2, \frac{.5}{\sqrt{50}}) = 0.9977$
The probability that the mean time is between 1.8 hours and 2.3 hours is 0.9977.

How did it go?

Solution

Solution