Please answer in details with explanation

1)

A quality control engineer at a bottling plant observed that the amount of soda pop in

each 32-ounce bottle follows approximately a

normal distribution with a mean of 32.2

ounces and a standard deviation of 0.3 ounces. (Assuming that the standard deviation of a single bottle

always remains fixed at 0.3 ounces, even if mean change for questions follow)

(a)

What fraction of bottles contains less than 32 ounces?

(b)

The plant manager finds the fraction in part (a) unacceptable. What would the

mean number of ounces per bottle have to be to ensure that at most 2% of the

bottles have fewer than 32 ounces? You can assume that the standard deviation

remains fixed at 0.3 ounces. (plz answer with explanation using trial-and-error or convert the

probability to a standard-normal random variable.)

(c)

Find the probability that if a customer buys a carton of 4 bottles, the average of

the 4 bottles will be less than 32 ounces.

(d)

What would the mean number of ounces per bottle have to be to ensure that at

most 2% of the time, the average in a 4-bottle pack has fewer than 32 ounces?

2)

Heights of women are normally distributed with mean 63.7 inches and standard deviation

2.47 inches. Find the height, to the nearest inch, that is the 80th percentile. Explaining the

answer in the context of this problem.