Question

In a study of exercise, a large group of male runners walk on a treadmill for six minutes. After this exercise, their heart rates vary with mean 8.6 beats per five seconds and standard deviation 1.1 beats per five seconds. This distribution takes only whole-number values, so it is certainly not Normal. (Round your answers to four decimal places.)

(a) Let x be the mean number of beats per five seconds after measuring heart rate for 24 five-second intervals (two minutes). What is the approximate distribution of x according to the central limit theorem?

mean=	beats per five seconds
standard deviation=	beats per five seconds

(b) What is the approximate probability that x is less than 8?

(c) What is the approximate probability that the heart rate of a runner is less than 100 beats per minute? (Hint: Restate this event in terms of x.)

Answer 1

SOLUTION:

From given data,

In a study of exercise, a large group of male runners walk on a treadmill for six minutes. After this exercise, their heart rates vary with mean 8.6 beats per five seconds and standard deviation 1.1 beats per five seconds. This distribution takes only whole-number values, so it is certainly not Normal.

standard deviation = = 1.1 beats per five seconds

mean = = 8.6 beats per five seconds

(a) Let x be the mean number of beats per five seconds after measuring heart rate for 24 five-second intervals (two minutes). What is the approximate distribution of x according to the central limit theorem

Central Limit Theorem:

says that If simple random samples of size n is selected from a distribution with mean and standard deviation = ,then the distribution of sample means approach a normal distribution, as sample size increases, irrespective of parent distribution of population . The greater the sample size, the better is the approximation (typically n> 30).

Sample mean follows normal distribution

Mean of sampling means:

Standard deviation sample mean or Standard error:

SE = =

=  

0.224537 = 0.225

(b) What is the approximate probability that x is less than 8?

P(X < 8)

Normal Distribution, =8.6, =1.1, n = 24

P(X<8)= Area to the left of 8

we convert this to standard normal using

  

= -2.672171

   - 2.67

P(X < 8) = P(Z < -2.67)

= 0.0038 (from z -table)

P(Z < -2.67) : in a z-table having area to the left of z , locate -2.6 in the left most column.

move across row to the right under column 0.07 and get value 0.0038

P(X < 8) = 0.0038 = 0.38%

(c) What is the approximate probability that the heart rate of a runner is less than 100 beats per minute? (Hint: Restate this event in terms of x.)

We cannot calculate probability about a single runner because distribution is unknown