Question

The average price of a television on a certain Web site is ​$840. Assume the price of these televisions follows the normal distribution with a standard deviation of $160.

Complete parts a through d below.

a. What is the probability that a randomly selected television from the site sells for less than ​$700?

​(Round to four decimal places as​ needed.)

b. What is the probability that a randomly selected television from the site sells for between

​$400 and ​$500?

​(Round to four decimal places as​ needed.)

c. What is the probability that a randomly selected television from the site sells for between

​$900 and ​$1,000​?

​(Round to four decimal places as​ needed.)

d. There are 13 televisions on the site. How many televisions are within a ​$800

​budget? There are nothing televisions on the site that are within a ​$800 budget.

​(Round down to the nearest whole​ number.)

Answer 1

SOLUTION:

From given data,

The average price of a television on a certain Web site is ​$840. Assume the price of these televisions follows the normal distribution with a standard deviation of $160.

Where,

= $840

Standard deviation = = $160

Let x shows the price.

(a). What is the probability that a randomly selected television from the site sells for less than ​$700?

z-score for x = $ 700 is

z = (x - ) / = (700 - 840) / 160 = -140 / 160 = - 0.875

P(x < 700) = P(z < - 0.875)

P(x < 700) = 0.1907

(b). What is the probability that a randomly selected television from the site sells for between ​$400 and ​$500?

z-score for x = $ 400 is

z = (x - ) / = (400 - 840) / 160 = -440 / 160 = -2.75

z-score for x = $ 500 is

z = (x - ) / = (500 - 840) / 160 = -340 / 160 = -2.125

P(400 < x < 500) = P(-2.75 < z < -2.125)

P(400 < x < 500) = P(z < -2.125) - P(z < -2.75)

P(400 < x < 500) = 0.01679 - 0.00298

P(400 < x < 500) = 0.0138

(c). What is the probability that a randomly selected television from the site sells for between ​$900 and ​$1,000​?

z-score for x = $ 900 is

z = (x - ) / = (900 - 840) / 160 = 60 / 160 = 0.375

z-score for x = $ 1000 is

z = (x - ) / = (1000 - 840) / 160 = 160 / 160 = 1

P(900 < x < 1000) = P(0.375 < z < 1)

P(900 < x < 1000) = P(z < 1) - P(z < 0.375)

P(900 < x < 1000) = 0.15865- 0.35383

P(900 < x < 1000) = -0.1951

(d). There are 13 televisions on the site. How many televisions are within a ​$800 ​budget? There are nothing televisions on the site that are within a ​$800 budget.

z-score for x = $ 800 is

z = (x - ) / = (800 - 840) / 160 = -40 / 160 = - 0.25

P(x < 800) = P(z < -0.25)

P(x < 800) = 0.40129