Respuesta :
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, [tex]\mu[/tex] = 278 and Sample size, n = 49
The z probability value is given by, Z = [tex]\frac{Xbar - \mu}{\frac{\sigma}{\sqrt{n} } }[/tex] ~ N(0,1)
Also, we are given the probability that a sample mean is less than 280 equals 0.67 i.e.;
P([tex]Xbar[/tex] < 280) = 0.67
P( [tex]\frac{Xbar - \mu}{\frac{\sigma}{\sqrt{n} } }[/tex] < [tex]\frac{280 - 278}{\frac{\sigma}{\sqrt{49} } }[/tex] ) = 0.67
P(Z < [tex]\frac{14}{\sigma}[/tex] ) = 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;
⇒ [tex]\frac{14}{\sigma}[/tex] = 0.44
⇒ [tex]\sigma[/tex] = 14/0.44 = 31.818 ≈ 31.82.
Therefore, the standard deviation of the population, [tex]\sigma[/tex] = 31.82 .
