A supplier of plastic pellets claims that when the plastic is pressed into wafers of uniform thickness, they will have a mean breaking strength of 165 newtons and a standard deviation of 5 newtons. An engineer at the company randomly selects wafers from 12 batches of pellets.

a)Find the probability that the sample mean is less than 162 newtons.
b)Find the probability that the sample mean is between 164 and 166 newtons inclusive.
c)What did you assume in order to do this analysis? How should you check this assumption?

Z = (sample mean-mean)/SEm (standard error of the mean)

SEm = SD/√(n-1)

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportions related to the Z scores.

To answer these questions, we can use the concept of the sampling distribution of the mean and the properties of the normal distribution.

a) To find the probability that the sample mean is less than 162 newtons, we need to calculate the z-score and use the standard normal distribution. The formula for the z-score is:

z = (X - μ) / (σ / sqrt(n))

where X is the sample mean, μ is the population mean, σ is the population standard deviation, and n is the sample size.

In this case, X = 162 newtons, μ = 165 newtons, σ = 5 newtons, and n = 12 batches.

Calculating the z-score:

z = (162 - 165) / (5 / sqrt(12))
= -3 / (5 / 3.464)
= -3 * 3.464 / 5
= -10.392 / 5
= -2.0784

Now we can find the probability using the standard normal distribution table or a calculator. Looking up the z-score of -2.0784, we find that the corresponding probability is approximately 0.0189.

Therefore, the probability that the sample mean is less than 162 newtons is approximately 0.0189, or 1.89%.

b) To find the probability that the sample mean is between 164 and 166 newtons inclusive, we need to calculate the z-scores for both values and find the area under the curve between those two z-scores.

For 164 newtons:
z1 = (164 - 165) / (5 / sqrt(12))
= -1 / (5 / 3.464)
= -1 * 3.464 / 5
= -3.464 / 5
= -0.6928

For 166 newtons:
z2 = (166 - 165) / (5 / sqrt(12))
= 1 / (5 / 3.464)
= 1 * 3.464 / 5
= 3.464 / 5
= 0.6928

Now we can use the standard normal distribution table or a calculator to find the area between these two z-scores. Looking up the z-score of -0.6928, we find the corresponding cumulative probability to be approximately 0.2420. Similarly, looking up the z-score of 0.6928, we find the cumulative probability to be approximately 0.7579. Subtracting the smaller probability from the larger probability gives us the desired probability:

P(164 <= X <= 166) = 0.7579 - 0.2420 = 0.5159

Therefore, the probability that the sample mean is between 164 and 166 newtons inclusive is approximately 0.5159, or 51.59%.

c) To perform this analysis, we assume that the breaking strength values of the wafers are normally distributed. This assumption is based on the Central Limit Theorem, which states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the shape of the original population.

To check this assumption, you can collect more data and calculate additional sample means. If the distribution of these sample means approximates a normal distribution, it provides evidence that the assumption is valid. Additionally, you can also generate a histogram or a normal probability plot of the sample means to visually assess the normality assumption.