Answer to Question #313366 in Statistics and Probability for Suppose the mean n

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

We have a normal distribution, $\mu=34, \sigma=4.3, n=250.$

Let's convert it to the standard normal distribution, $z=\cfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}$ ;

$z_1=\cfrac{33.5-34}{4.3/\sqrt{250}}=-1.84, \\ z_2=\cfrac{34.5-34}{4.3/\sqrt{250}}=1.84,$

$P(33.5<\bar{X}<34.5)=\\ =P(-1.84<Z<1.84)=\\ =P(Z<1.84)-P(Z<-1.84)=$

$=0.9671-0.0329=0.9342$ (from z-table).