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 21.2. 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 21.2 grams and a standard deviation of 6 grams. (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?
Guide On Rating System
Vote
(a) To find the probability that a randomly selected corner store purchase has more than 31 grams of fat, we need to find the area to the right of x = 31 under the normal distribution curve.
Using the z-score formula, we can calculate the z-score for x = 31:
z = (x - mean) / standard deviation = (31 - 21.2) / 6 ≈ 1.6333
Next, we use a standard normal distribution table or a calculator to find the probability corresponding to this z-score. The probability is the area to the right of the z-score.
Looking up the z-score of 1.6333 in a standard normal distribution table, we find that the corresponding probability is approximately 0.0516.
Therefore, the probability that a randomly selected corner store purchase has more than 31 grams of fat is approximately 0.0516.
(b) To find the probability that a randomly selected corner store purchase has between 15 and 25 grams of fat, we need to find the area between x = 15 and x = 25 under the normal distribution curve.
Using the z-score formula, we can calculate the z-scores for x = 15 and x = 25:
z1 = (15 - 21.2) / 6 ≈ -1.0333
z2 = (25 - 21.2) / 6 ≈ 0.6333
Next, we use a standard normal distribution table or a calculator to find the probability corresponding to these z-scores. We need to find the area to the right of z1 and the area to the right of z2, and subtract the two areas to get the area between z1 and z2.
Using the standard normal distribution table, we find that the area to the right of z1 is approximately 0.8485 and the area to the right of z2 is approximately 0.2643.
Therefore, the probability that a randomly selected corner store purchase has between 15 and 25 grams of fat is approximately 0.8485 - 0.2643 = 0.5842.
(c) To find the probability that both of these purchases will have more than 25 grams of fat, we need to find the probability for each purchase and multiply them together.
Using the z-score formula, we can calculate the z-score for x = 25:
z = (25 - 21.2) / 6 ≈ 0.6333
Using a standard normal distribution table or a calculator, we find that the area to the right of z is approximately 0.2643. This is the probability that one randomly selected corner store purchase has more than 25 grams of fat.
To find the probability that both purchases have more than 25 grams of fat, we multiply this probability by itself:
0.2643 * 0.2643 = 0.0699 (rounded)
Therefore, the probability that both corner store purchases will have more than 25 grams of fat is approximately 0.0699.
Got a question with my answer?
Cheapmath.com is completely free!
Asking question is free.
Getting answers is free.
+99