Question

The weights of packages of cookies produced by a certain manufacturer are normally distributed with a mean of 202 grams and a standard deviation of 4 grams. in a random sample of 300 packages of cookies, about how many packages do we expect to contain between 199 and 207 grams?

(a) 200
(b) 100
(c) 225
(d) 175
(e) 150

Answer 1

To find the number of packages expected to contain between 199 and 207 grams, we calculate the z-scores for these weights using the z-score formula and then find the corresponding probabilities. Multiplying the probability by the sample size gives us the expected number of packages within the weight range.

Step-by-step explanation:

To find the number of packages expected to contain between 199 and 207 grams, we need to calculate the z-scores for these weights and then use the z-table to find the corresponding probabilities.

The z-score formula is z = (x - μ) / σ. Where x is the weight, μ is the mean (202 grams), and σ is the standard deviation (4 grams).

For 199 grams, the z-score is (199 - 202) / 4 = -0.75. For 207 grams, the z-score is (207 - 202) / 4 = 1.25.

Using the z-table or a calculator, we find that the probability of a z-score between -0.75 and 1.25 is approximately 0.6357. Multiplying this probability by the sample size (300), we can expect about 190.71 packages to be within the weight range of 199 to 207 grams. Rounding to the nearest whole number, the answer is approximately 191 packages.