Cookies are made in a factory with an average mass of 22 grams and a standard deviation of 4 grams. some cookies are packed in a box. According to the manufacturer, the average mass of a cookie from at least 99% of the boxes is at least 20 grams. How many cookies should the manufacturer at least put in a box?
The answer has to be 22
To find out how many cookies the manufacturer should put in a box, we need to determine the minimum number of cookies that would ensure the average mass is at least 20 grams with at least 99% confidence.
First, let's find the z-score corresponding to a 99% confidence level. This can be done using a standard normal distribution table or a calculator. The z-score for a 99% confidence level is approximately 2.33.
Next, we'll use the formula for the confidence interval of the sample mean:
Confidence Interval = mean ± (z * (standard deviation / sqrt(sample size)))
We want the lower bound of the confidence interval to be at least 20 grams. Therefore, we can rearrange the formula to solve for the sample size:
sample size = (z * (standard deviation / desired margin of error))^2
Plugging in the given values:
z = 2.33 (for 99% confidence level)
standard deviation = 4 grams
desired margin of error = mean - desired average (22 grams - 20 grams = 2 grams)
sample size = (2.33 * (4 / 2))^2
sample size = 5.66^2
sample size ≈ 31.98
Since the sample size must be a whole number, we round up to the nearest whole number.
Therefore, the manufacturer should put at least 32 cookies in a box to ensure that the average mass of the cookies in at least 99% of the boxes is at least 20 grams.