Answer to Question #307883 in Statistics and Probability for jane

Question

Suppose the mean number of days to germination of a variety of seeds is 34, with a standard deviation of 4.3 days. What is the probability that the mean germination time of a sample of 250 seeds will be within 0.5 days of the population mean?

Answer 1

Let X be a random variable representing the mean germination time of a sample of 250 seeds, then, according to the central limit theorem, X ~ $N(34, {\frac {4.3^2} {\sqrt {250}}})=N(34, 1.17)$

$P(34-0.5<X<34+0.5)=P(33.5<N(34, 1.17)<34.5)=P(33.5<34+1.08N(0,1)<34.5)=P(-0.463<N(0,1)<0.463)=P(N(0,1)<0.463)-P(N(0,1)<-0.463)=0.67832-0.32168=0.35644$