Answer: A. Mean of sampling means [tex]=\mu=168\ Mg/dl[/tex]
Standard deviation of sampling means =[tex]2.86\ mg/dl[/tex]
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 [tex]\mu= 168[/tex] Mg/dl and standard deviation [tex]\sigma= 35[/tex] mg/dl.
Sample size : n= 150
Let [tex]\overline{x}[/tex] sample mean values.
A. The mean and the standard deviation of the distribution of the sampling means would be :
Mean of sampling means =[tex]\mu=168\ Mg/dl[/tex]
Standard deviation of sampling means = [tex]\dfrac{\sigma}{\sqrt{n}}=\dfrac{35}{\sqrt{150}}[/tex]
[tex]=\dfrac{35}{12.2474487}\\\\=2.85773803649\approx2.86\ mg/dl[/tex]
The probability that your sample has mean less than 165 would be
[tex]P(\overline{x}<165) =P(\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}<\dfrac{165-168}{\dfrac{35}{\sqrt{150}}})\\\\=P(z<-1.04)\ \ [\because \ z=\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}]\\\\=1-P(z<1.04)\ \ [\because P(Z<-z)=1-P(Z<z)]\\\\=1- 0.8508\ \ [\text{By z-table}]\\\\=0.1492[/tex]
Hence , the probability that your sample has mean less than 165 is 0.1492 .
