Question

Purchases made at small "corner stores" were studied by the authors of a certain paper. Corner stores were defined as stores that are less than 200 square feet in size, have only one cash register, and primarily sell food. After observing a large number of corner store purchases in Philadelphia, the authors reported that the average number of grams of fat in a corner store purchase was 21.8. Suppose that the variable

x = number of grams of fat in a corner store purchase
has a distribution that is approximately normal with a mean of 21.8 grams and a standard deviation of 7 grams.

a) What is the probability that a randomly selected corner store purchase has more than 29 grams of fat?
b) What is the probability that a randomly selected corner store purchase has between 15 and 25 grams of fat?
c) If two corner store purchases are randomly selected, what it the probability that both of these purchases will have more than 25 grams of fat?

Answer 1

a) The probability that a randomly selected corner store purchase has more than 29 grams of fat is approximately 14.92%. b) The probability that a randomly selected corner store purchase has between 15 and 25 grams of fat is approximately 51.27%. c) The probability that both randomly selected corner store purchases have more than 25 grams of fat is approximately 45.93%.

Step-by-step explanation:

a) To find the probability that a randomly selected corner store purchase has more than 29 grams of fat, we need to calculate the area under the normal distribution curve to the right of 29. Using the given mean of 21.8 grams and standard deviation of 7 grams, we can calculate the z-score as (29 - 21.8) / 7 = 1.03. Using a standard normal distribution table or calculator, we find that the area to the right of 1.03 is approximately 0.1492. Therefore, the probability is 0.1492 or 14.92%.

b) To find the probability that a randomly selected corner store purchase has between 15 and 25 grams of fat, we need to calculate the area under the normal distribution curve between 15 and 25. First, we calculate the z-scores for both values: z1 = (15 - 21.8) / 7 = -0.971 and z2 = (25 - 21.8) / 7 = 0.457. Using a standard normal distribution table or calculator, we find that the area to the left of -0.971 is approximately 0.1645 and the area to the left of 0.457 is approximately 0.6772. Therefore, the probability is 0.6772 - 0.1645 = 0.5127 or 51.27%.

c) To find the probability that both randomly selected corner store purchases have more than 25 grams of fat, we need to find the probability that a single purchase has more than 25 grams of fat and then multiply it by itself. Using the z-score formula, we calculate the z-score for 25 grams: (25 - 21.8) / 7 = 0.457. The probability of a single purchase having more than 25 grams of fat is 0.6772 (area to the left of 0.457). Therefore, the probability that both purchases have more than 25 grams of fat is 0.6772 * 0.6772 = 0.4593 or 45.93%.