Questions: A factory produces plate glass with a mean thickness of 4 mm and a standard deviation of 1.1 mm. A simple random sample of 100 sheets of glass is to be measured, and the mean thickness of the 100 sheets is to be computed. What is the probability that the average thickness of the 100 sheets is less than 4.13 mm? Round your answers to 5 decimal places.

A factory produces plate glass with a mean thickness of 4 mm and a standard deviation of 1.1 mm. A simple random sample of 100 sheets of glass is to be measured, and the mean thickness of the 100 sheets is to be computed.

What is the probability that the average thickness of the 100 sheets is less than 4.13 mm? Round your answers to 5 decimal places.

Transcript text: A factory produces plate glass with a mean thickness of 4 mm and a standard deviation of 1.1 mm . A simple random sample of 100 sheets of glass is to be measured, and the mean thickness of the 100 sheets is to be computed. What is the probability that the average thickness of the 100 sheets is less than 4.13 mm ? Round your answers to 5 decimal places. $\square$ A

Solution

Solution Steps

Step 1: Define the Problem

We are tasked with finding the probability that the average thickness of a sample of 100 sheets of glass is less than 4.13 mm. The population from which the sample is drawn has a mean thickness $ \mu = 4 $ mm and a standard deviation $ \sigma = 1.1 $ mm.

Step 2: Calculate the Standard Error

The standard error (SE) of the sample mean is calculated using the formula: \[ SE = \frac{\sigma}{\sqrt{n}} = \frac{1.1}{\sqrt{100}} = \frac{1.1}{10} = 0.11 \]

Step 3: Calculate the Z-Score

To find the Z-score for the upper bound of 4.13 mm, we use the formula: \[ Z = \frac{X - \mu}{SE} = \frac{4.13 - 4}{0.11} = \frac{0.13}{0.11} \approx 1.18182 \]

Step 4: Determine the Probability

Using the Z-score, we can find the probability that the average thickness is less than 4.13 mm. This is represented as: \[ P(X < 4.13) = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.18182) - \Phi(-\infty) \] From the calculations, we find: \[ P(X < 4.13) \approx 0.88136 \]