Final answer:
To find the number of cookies within 2 grams of the mean, use the z-score formula and calculate the probabilities. The number of cookies within this range is approximately 13,653. To find the number of rejected cookies, calculate the probabilities for cookies weighing more than 45 grams or less than 35 grams and multiply by the total number of cookies produced.
Step-by-step explanation:
To determine the number of cookies that are within 2 grams of the mean, we need to find the number of cookies that fall within the range of mean minus 2 grams to mean plus 2 grams. The range would be from 40 - 2 = 38 grams to 40 + 2 = 42 grams. To find the number of cookies within this range, we can use the z-score formula. The z-score is calculated as (value - mean) / standard deviation. So, for the lower limit, the z-score would be (38 - 40) / 2 = -1. For the upper limit, the z-score would be (42 - 40) / 2 = 1. We can then look up these z-scores in the z-table to find the corresponding probabilities. The probability for a z-score of -1 is 0.1587, and the probability for a z-score of 1 is 0.8413. To find the number of cookies within this range, we multiply these probabilities by the total number of cookies produced (20,000) and round to the nearest whole number. The number of cookies within 2 grams of the mean would be 0.8413 - 0.1587 = 0.6826 multiplied by 20,000, which is approximately 13,653 cookies.
To find the number of cookies that would be rejected, we need to find the number of cookies that weigh more than 45 grams or less than 35 grams. We can calculate the z-scores for these limits using the formula mentioned earlier. For the lower limit, the z-score would be (35 - 40) / 2 = -2.5. For the upper limit, the z-score would be (45 - 40) / 2 = 2.5. We can then look up these z-scores in the z-table to find the corresponding probabilities. The probability for a z-score of -2.5 is 0.0062, and the probability for a z-score of 2.5 is 0.9938. To find the number of rejected cookies, we multiply these probabilities by the total number of cookies produced (20,000) and round to the nearest whole number. The number of rejected cookies would be 0.9938 + 0.0062 = 1 multiplied by 20,000, which is approximately 20,000 cookies.