Question

Suppose you are a member of a board of directors of a large company. The product design team has been working on a new product. They present you with a report that indicates the planned population mean weight of the units of product is 85 g, with a population standard deviation of 10 g. The company completes a test run of the production system: they create 36 units of product. What is the probability that these 36 units have a sample mean weight within 1% of the planned population mean weight? Hint: For example, suppose the planned population mean weight were, say, 1234. By "within 1% of the planned population mean weight", I mean "from 1234−(1234

∗
0.01)=1221.66 to 1234+(1234
∗
0.01)=1236.44
′′
. So, I would need to find the probability that I get a sample mean between 1221.66 and 1236.44 when I sample 36 values from the population of all unit weights.
a) Declare the random variable of interest:

Answer 1

To find the probability that the 36 units have a sample mean weight within 1% of the planned population mean weight, calculate the standard error, find the z-scores for the lower and upper limits, and use them to find the probability using a standard normal distribution table or calculator.

Step-by-step explanation:

To solve this problem, we need to calculate the probability that the sample mean weight of the 36 units of product falls within 1% of the planned population mean weight of 85g. We can do this by first calculating the standard error using the formula: standard error = population standard deviation / sqrt(sample size).

Then, we can find the z-scores for the lower and upper limits of the 1% range using the formula: z = (sample mean - population mean) / standard error. Finally, we can use the z-scores to find the probability using a standard normal distribution table or a calculator.