Question

The weight of an energy bar is approximately normally distributed with a mean of 42.40 grams and a standard deviation of 0.035 grams. Complete parts a through e below.

a. What is the probability that an individual energy bar weighs less than 42.375 grams?
0.239 (Round to three decimal places as needed.)

b. What is the probability that an individual energy bar weighs more than 42.425 grams?
0.076 (Round to three decimal places as needed.)

c. If a sample of 25 energy bars is selected, what is the probability that the sample mean weight is less than 42.375 grams?
0.000 (Round to three decimal places as needed.)

d. Explain the difference in the results of a and c.
Part a refers to an individual bar, which can be thought of as a sample with a sample size of 1. Therefore, the standard error of the mean for an individual bar is the same as the standard error of the sample in c with a sample size of 25. This leads to a probability in part a that is the same as the probability in part c. Type integers or decimals. Do not round.

Answer 1

The question deals with normal distribution and the central limit theorem, focusing on the probabilities of weights of individual energy bars and the sample mean weight of a group of bars.

Step-by-step explanation:

Understanding Normal Distribution Problems

The question provided involves understanding the concept of a normally distributed random variable and the difference between individual data points and sample means. In these scenarios, the central limit theorem plays a significant role. For an individual energy bar's weight, the probability of a specific weight is determined directly from the normal distribution with the provided mean and standard deviation. On the other hand, for a sample of energy bars, the distribution of the sample mean will have the same mean but a smaller standard deviation, known as the standard error, which is the standard deviation divided by the square root of the sample size.

Calculating the probability of individual weights (parts a and b) uses the normal distribution directly, whereas calculating the probability for the sample mean (part c) requires adjustment for sample size, which significantly affects the resulting probability.

The provided responses contradict the principles of the central limit theorem, which dictates that the standard error should decrease with an increase in sample size, leading to a reduced probability of extreme values for the mean. Therefore, a clarification is necessary to explain these principles correctly.