Question

In a study, a large group of male runners walk on a treadmill for 6 minutes. Their heart rates vary according to the Normal distribution N(104, 12.5). However, the heart rates for non-runners is N(130, 17). Answer the following questions providing all the details.

1. What percent of the runners have heart rates above 140?

2. What percent of the non-runners have heart rates above 140? what do you conclude?

3. Find the median, first and third quartiles of the distribution of runner’s heart rates.

4. What range of non-runner’s heart rates covers the middle 68% of their Normal distribution?

Answer 1

The heart rate of runners have a normal distribution with mean = 104 and standard deviation = 12.5

and the heart rate for non-runners have a normal distribution with mean = 130 and standard deviation = 17

1. Percent of the runners have heart rates above 140, that is first found P(X > 140)

Convert x into z score, the formula of z score is

P(X > 140) becomes P(Z > 2.88)

The probability for z = 2.88 using the z score table is 0.9980

Z score always provides the less than probability, to find the more than probability just subtract the less than from 1.

1 - 0.9980 = 0.0020

0.0020 * 100 = 0.2%

Therefore, the percent of the runners have heart rates above 140 is 0.2%

2. Percent of the runners have heart rates above 140 that is P(X > 140)

Convert x into z score, the formula of z score is

P(X > 140) becomes P(Z > 0.59)

The probability for z = 0.59 using the z score table is 0.7224

Z score always provides the less than probability, to find the more than probability just subtract the less than from 1.

1 - 0.7224 = 0.0020

0.2776 * 100 = 27.76%

Therefore, the percent of the non-runners have heart rates above 140 is 27.76%

3. Median, first quartile and third quartile for runner's heart rate.

Median divides the data into 2 equal parts, 50% of data falls below the median

P(X < median) = 0.50

For a normal distribution, the mean, median and mode are the same.

Therefore, Median = mean = 104.

The first quartile (Q1): The first quartile has 25% data below it.

Using the area 0.25 below the first quartile, first find the z score.

Search 0.25 or it's the closest value in the middle body of the table and then take the corresponding z score.

The z score corresponding to area 0.25 is -0.67

To find the first quartile, use the formula of z score and solve for x

That is

First quartile = 95.625

Third quartile: The third quartile has 75% data below it. First search area 0.75 or more closest area in the middle body of the table and then take the z score.

The z score corresponding to area 0.75 is 0.67

To find the first quartile, use the formula of z score and solve for x

That is

Third quartile = 112.375

4. Range of non-runners heart rate covers the middle 68% for the normal distribution.

For bell-shaped distribution, 68% of data falls within 1 standard deviation from the mean.

113 and 147

68% of the non-runners have heart rate within the range 113 and 147.