Question

jerry asked 500 people to run as fast as possible for 15 seconds. he then recorded the distance each of the 500 people ran. he found that the distances were normally distributed with a mean of 82 meters and a standard deviation of 14 meters.
for each pair of values, select whether value a is greater, value b is greater, or the values are equal.
value a: the mean distance of runners within 1 standard deviation of the mean.
value b: the mean distance of all the runners.
value a: the number of people who ran between 68 and 82 meters.
value b: the number of people who ran between 54 and 68 meters.
value a: the number of people who ran less than 82 meters.
value b: the number of people who ran distances within 1 standard deviation of the mean.

Explanation:

First Pair:

Step1: Recall properties of normal distribution

In a normal distribution, the mean of a subset within 1 - standard deviation of the mean is the same as the population mean.
Let \(\mu = 82\) (population mean) and \(\sigma=14\). The interval within 1 - standard deviation of the mean is \(\mu-\sigma\) to \(\mu + \sigma\) (i.e., \(82 - 14=68\) to \(82 + 14 = 96\)). The mean of a normal distribution is symmetric about \(\mu\). The mean of the values in the interval \([\mu-\sigma,\mu+\sigma]\) is \(\mu\).

Second Pair:

Step1: Use the 68 - 95 - 99.7 rule (empirical rule)

The 68 - 95 - 99.7 rule states that for a normal distribution:

• The proportion of data within \(\mu-\sigma\) and \(\mu\) is the same as the proportion of data within \(\mu\) and \(\mu+\sigma\).
• The interval \(68\) to \(82\) (\(\mu-\sigma\) to \(\mu\)) and \(54\) to \(68\) (\(\mu - 2\sigma\) to \(\mu-\sigma\)).
• The proportion of data in the interval \((\mu-\sigma,\mu)\) is \(0.34\) (from the 68 - 95 - 99.7 rule: \(P(\mu-\sigma<X<\mu+\sigma)=0.68\), so \(P(\mu-\sigma<X<\mu)=0.34\)) and the proportion of data in the interval \((\mu - 2\sigma,\mu-\sigma)\) is \(0.135\) (since \(P(\mu - 2\sigma<X<\mu+2\sigma)=0.95\), so \(P(\mu - 2\sigma<X<\mu-\sigma)=\frac{0.95 - 0.68}{2}=0.135\)).
• If \(n = 500\), the number of people in the interval \(68\) to \(82\) is \(n_1=500\times0.34 = 170\) and the number of people in the interval \(54\) to \(68\) is \(n_2=500\times0.135=67.5\)

Third Pair:

Step1: Apply the 68 - 95 - 99.7 rule

• The proportion of data less than \(\mu\) (where \(\mu = 82\)) is \(0.5\) (since the normal distribution is symmetric about \(\mu\)).
• The proportion of data within \(\mu-\sigma\) and \(\mu+\sigma\) is \(0.68\).

Answer:

	Value A is Greater	Value B is Greater	Equal
Second Pair	\(\square\)
Third Pair		\(\square\)