Question

A normally distributed population has a mean of 250 pounds

anda standard deviation of 10 pounds. Given n= 20, what is
theprobability that this sample will have a mean value between 245
and255?

Answer 1

Answer: 0.9746

Explanation:

Given : A normally distributed population has a mean of 250 pounds and a standard deviation of 10 pounds.

i.e.
`\mu=250`
`\sigma=10`

Sample size : n= 20

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

Then, the probability that this sample will have a mean value between 245 and 255 be

`P(245<\overline{x}<255) =P((245-250)/((10)/(√(20)))<\frac{\overline{x}-\mu}{(\sigma)/(√(n))}<(255-250)/((10)/(√(20))))\\\\=P(-2.236<z<2.236)\ \ [\because \ z=\frac{\overline{x}-\mu}{(\sigma)/(√(n))}]\\\\=P(z<2.236)-P(z<-2.236)\\\\=P(z<2.236)-(1-P(z<2.236))\ \ [\because P(Z<-z)=1-P(Z<z)]\\\\=2P(z<2.236)-1\\\\= 2( 0.9873)-1\ \ [\text{By z-table}]\\\\=0.9746`

Hence , the probability that this sample will have a mean value between 245 and 255 is 0.9746 .