Q.15 - The Practice of Statistics for AP

Question

Run a mile During World War II, able-bodied male undergraduates at the University of Illinois participated in required physical training. Each student ran a timed mile. Their times followed the Normal distribution with mean $7.11$ minutes and standard deviation $0.74$ minute. An SRS of $100$ of these students has mean time $\overline{x}=7.15$ minutes. A second SRS of size$100$ has mean $\overline{x}=6.97$ minutes. After many SRSs, the values of the sample mean x follow the Normal distribution with mean $7.11$ minutes and standard deviation$0.074$ minute.

(a) What is the population? Describe the population distribution. (b) Describe the sampling distribution of x. How is it different from the population distribution? 

Step-by-Step Solution

Answer

a).  The population distribution is a Normal distribution.

b). Average time to run a mile of $100$ students from the population. Smaller standard deviation.

Part (a) Step 1: Given Information

The values of the sample mean $x$ follow the Normal distribution with mean $7.11$ minutes and a standard deviation of $0.074$ minute. 

Part (a) Step 2: Explanation

Individuals who are being examined make up the population.

$POPULATION =12000$ able-bodied male undergraduates from the University of Illinois who took part in World War II physical training.

The population distribution is given as a Normal distribution with a mean of $7.11$ minutes and a standard deviation of $0.74$ minutes, corresponding to the time it takes for randomly selected individuals to run a mile.

Part (b) Step 1: Given Information

The values of the sample mean $x$ follow the Normal distribution with mean $7.11$ minutes and standard deviation $0.074$ minute 

Part (b) Step 2: Explanation

The sample mean $\overline{x}$values are distributed normally, with a mean of $7.11$ minutes and a standard deviation of $0.074$ minutes.

As a result $\overline{x}$'s sample distribution is the Normal distribution, with a mean of $7.11$ minutes and a standard deviation of $0.074$ minutes.

The  $\overline{x}$ sample distribution describes the average time taken by $100$ students from the population to run a mile.

The reduced standard deviation distinguishes it from the population distribution.