Question

The distribution of blood cholesterol level in the

populationof young men aged 20 to 34 years is close to normal with
meanu= 168Mg/dl and standard deviation = 35 mg/dl. You measurethe
cholesterol level of 150 young men chosen at random andcalculate
the sampling mean.
A. If you did this many times, what would be themean and the
standard deviation of the distribution of the samplingmeans.
B. What is the probability that your sample has meanless than
165.

Answer 1

Answer: A. Mean of sampling means
`=\mu=168\ Mg/dl`

Standard deviation of sampling means =
`2.86\ mg/dl`

B. The probability that your sample has mean less than 165 is 0.1492 .

Given : The distribution of blood cholesterol level in the

population of young men aged 20 to 34 years is close to normal with

mean
`\mu= 168` Mg/dl and standard deviation
`\sigma= 35` mg/dl.

Sample size : n= 150

Let
`\overline{x}` sample mean values.

A. The mean and the standard deviation of the distribution of the sampling means would be :

Mean of sampling means =
`\mu=168\ Mg/dl`

Standard deviation of sampling means =
`(\sigma)/(√(n))=(35)/(√(150))`

`=(35)/(12.2474487)\\\\=2.85773803649\approx2.86\ mg/dl`

The probability that your sample has mean less than 165 would be

`P(\overline{x}<165) =P(\frac{\overline{x}-\mu}{(\sigma)/(√(n))}<(165-168)/((35)/(√(150))))\\\\=P(z<-1.04)\ \ [\because \ z=\frac{\overline{x}-\mu}{(\sigma)/(√(n))}]\\\\=1-P(z<1.04)\ \ [\because P(Z<-z)=1-P(Z<z)]\\\\=1- 0.8508\ \ [\text{By z-table}]\\\\=0.1492`

Hence , the probability that your sample has mean less than 165 is 0.1492 .