Answer to Question #127863 in Statistics and Probability for Jack

Question

The thickness of books in a bookstore follows a normal distribution with the mean of 45mm and the standard
deviation of 12mm.
a) A book is obtained from the bookstore, find the probability that
i) The book is thinner than 30mm.
ii) The book’s thickness is between 34mm and 50mm.
b) 10 books are chosen from the bookstore at random, the mean thickness of that 10 books,

Answer 1

a) (i) we need P(x < 30mm)

we know that; $\mu$ = 45mm and $\delta$ = 12mm

The probability is given using the following formula

z = (x-μ)/σ

z = (30-45)/12 = 1.25

We now find P(z < -1.25) using normal table

Therefore, P(x < 30mm) = P(z < -1.25) = 0.10565

ii) We need to find P(34mm < x < 50mm)

When x = 34mm

Using the formula z = (x-μ)/σ

z = (34-45)/12 = -0.92

P(z < -0.92) = 0.17879

When x = 50mm

We use the formula

z = (x-μ)/σ

z = (50-45)/12 = 0.42

P(z < 0.42) = 0.66276

$\therefore$ P(34mm < x < 50mm) = P(-0.92 < z < 0.42)

We take the difference of the two probabilities

0.66276-0.17879 = 0.48397

b) To obtain the mean of the 10 books, we know that sample mean is an unbiased estimator of the population mean. Therefore, x̄ = μ = 45