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 21.6.

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 21.6 grams and a standard deviation of 6 grams.
(Use a table or technology. Round your answers to four decimal places.)
(a) What is the probability that a randomly selected corner store purchase has more than 31 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 is the probability that both of these purchases will have more than 25 grams of fat?

Answer 1

To find the probability that a randomly selected corner store purchase has more than 31 grams of fat, calculate the z-score using the formula (x - μ) / σ and use the standard normal distribution table. The probability is approximately 0.0548.

Step-by-step explanation:

(a) To find the probability that a randomly selected corner store purchase has more than 31 grams of fat, we need to calculate the z-score and use the standard normal distribution table. The formula to calculate the z-score is given by:
z = (x - μ) / σ, where x is the value, μ is the mean, and σ is the standard deviation. Substituting the values, we get:
z = (31 - 21.6) / 6 = 1.6.
Using the standard normal distribution table or technology, we find that the probability corresponding to a z-score of 1.6 is approximately 0.0548.

(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 z-scores for both values. Using the same formula as in part (a), we get:
For 15 grams of fat: z = (15 - 21.6) / 6 = -1.1.
For 25 grams of fat: z = (25 - 21.6) / 6 = 0.5667.
Using the standard normal distribution table or technology, we can find the corresponding probabilities for these z-scores. The probability corresponding to a z-score of -1.1 is approximately 0.1359, and the probability corresponding to a z-score of 0.5667 is approximately 0.2852. To find the probability between these two values, we subtract the smaller probability from the larger probability: 0.2852 - 0.1359 = 0.1493.

(c) To find the probability that both of these purchases will have more than 25 grams of fat, we need to find the probability of one purchase having more than 25 grams of fat and then multiply it by itself. From part (b), we found that the probability of one purchase having more than 25 grams of fat is 0.2852. Multiplying this probability by itself, we get: 0.2852 * 0.2852 = 0.08142.