Question

The random variable x = weight of a randomly selected bag of chips can be modeled by a normal distribution with µ = 10 ounces and σ = 0.04 ounces. using the empirical (68-95-99.7) rule, which of the following is correct?

a. the probability that a randomly selected bag of chips weighs between 9.96 ounces and 10.12 ounces is about 0.8385
b. the probability that a randomly selected bag of chips weighs less than 9.92 ounces is about 0.025.
c. the probability that a randomly selected bag of chips weighs more than 9.96 ounces is about 0.16.
d. the probability that a randomly selected bag of chips weighs between 10 ounces and 10.04 ounces is about 0.68.

Answer 1

Using the empirical (68-95-99.7) rule, the correct statement is b. the probability that a randomly selected bag of chips weighs less than 9.92 ounces is about 0.025.

Step-by-step explanation:

The question is asking which statement is correct using the empirical (68-95-99.7) rule for a normal distribution with μ = 10 ounces and σ = 0.04 ounces.

1. Statement a. describes the probability that a randomly selected bag of chips weighs between 9.96 ounces and 10.12 ounces. To determine this probability, we need to find the z-scores for both weights using the formula z = (x - μ) / σ. The z-score for 9.96 ounces is (9.96 - 10) / 0.04 = -0.10, and the z-score for 10.12 ounces is (10.12 - 10) / 0.04 = 0.30. Using the empirical rule, the probability that a randomly selected bag of chips weighs between 9.96 ounces and 10.12 ounces is the area under the normal curve between these two z-scores, which is approximately 0.6179. Therefore, statement a. is incorrect.
2. Statement b. describes the probability that a randomly selected bag of chips weighs less than 9.92 ounces. To determine this probability, we need to find the z-score for 9.92 ounces using the formula z = (x - μ) / σ. The z-score for 9.92 ounces is (9.92 - 10) / 0.04 = -0.20. Using the empirical rule, the probability that a randomly selected bag of chips weighs less than 9.92 ounces is the area under the normal curve to the left of this z-score, which is approximately 0.0793. Therefore, statement b. is correct.
3. Statement c. describes the probability that a randomly selected bag of chips weighs more than 9.96 ounces. Using the same approach as in statement a., we find that the z-score for 9.96 ounces is -0.10. The probability of a bag weighing more than 9.96 ounces is the area under the normal curve to the right of this z-score, which is approximately 0.4602. Therefore, statement c. is incorrect.
4. Statement d. describes the probability that a randomly selected bag of chips weighs between 10 ounces and 10.04 ounces. To determine this probability, we need to find the z-scores for both weights using the formula z = (x - μ) / σ. The z-score for 10 ounces is (10 - 10) / 0.04 = 0, and the z-score for 10.04 ounces is (10.04 - 10) / 0.04 = 1. Using the empirical rule, the probability that a randomly selected bag of chips weighs between 10 ounces and 10.04 ounces is the area under the normal curve between these two z-scores, which is approximately 0.3413. Therefore, statement d. is incorrect.