Answer to Question #172065 in Statistics and Probability for sean

Question

1. PICH – A – HOY is one of the best-selling cookie companies in a certain country. Assume that the weights of all packages of these PICHA-HOY cookies are normally distributed with a mean of 35 ounces and a standard deviation of 0.5 ounces. Find the probability that the mean weight, of a random sample of 20 packages of this brand of cookies will be between 34. 3 and 34. 8 ounces. Draw the curve to illustrate the problem.

Answer 1

Let $X=$ mean weight: $X\sim N(\mu, \sigma^2/n)$

Then $Z=\dfrac{X-\mu}{\sigma/\sqrt{n}}\sim N(0, 1)$

Given $\mu=35, \sigma=0.5, n=20$

$$P(34.3<X<34.8)=P(X<34.8)-P(X\leq34.3)$$

$$=P(Z<\dfrac{34.8-35}{0.5/\sqrt{20}})-P(X\leq\dfrac{34.3-35}{0.5/\sqrt{20}})$$

$$\approx P(Z<-1.789)-P(Z\leq-6.261)$$

$$\approx0.0368-0.0000=0.0368$$