Question

Suppose a random sample of size 49is drawn from a population with a mean value of 278. If the probability that a sample mean is less than 280equals 0.67, what is the standard deviation of the population

Answer 1

Answer:

Standard deviation of the population = 31.82

Step-by-step explanation:

We are given that a random sample of size 49 is drawn from a population with a mean value of 278 i.e.;

Population mean, $\mu$ = 278   and    Sample size, n = 49

The z probability value is given by, Z = $\frac{Xbar - \mu}{\frac{\sigma}{\sqrt{n} } }$ ~ N(0,1)

Also, we are given the probability that a sample mean is less than 280 equals 0.67 i.e.;

              P($Xbar$ < 280) = 0.67

              P( $\frac{Xbar - \mu}{\frac{\sigma}{\sqrt{n} } }$ < $\frac{280 - 278}{\frac{\sigma}{\sqrt{49} } }$ ) = 0.67

              P(Z < $\frac{14}{\sigma}$ ) = 0.67

Now, looking at z table we find that the probability area of 0.67 is given at the critical value of x of 0.44 which means;

                     ⇒   $\frac{14}{\sigma}$ = 0.44

                     ⇒ $\sigma$ = 14/0.44 = 31.818 ≈ 31.82.

Therefore, the standard deviation of the population, $\sigma$ = 31.82 .