The breaking strengths (say, y) for one foot square samples of a particular synthetic fabric are approximately normally distributed with a mean of 2280 pounds per square inch and a standard deviation of 10.6 psi.

a) find the probability of selecting a 1 foot square sample of material at random that on testing would have a breaking strength in excess of 2240 psi.

b)what is the breaking strength value in psi that separates the strongest 20% of one foot square samples from the weakest 80%?

a) Z = (score-mean)/SD

Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability related to the Z score.

b) Start with Z value for .20 and insert i8n equation above.

To solve these problems, we will use the standard normal distribution.

a) To find the probability of selecting a sample with a breaking strength in excess of 2240 psi, we need to calculate the z-score and find the probability corresponding to that z-score using the standard normal distribution table.

Step 1: Calculate the z-score:
z = (x - μ) / σ
where x is the breaking strength (2240 psi), μ is the mean (2280 psi), and σ is the standard deviation (10.6 psi).

z = (2240 - 2280) / 10.6 = -40 / 10.6 = -3.77 (rounded to two decimal places)

Step 2: Look up the probability corresponding to the z-score:
Using a standard normal distribution table or calculator, we find that the probability of selecting a sample with a breaking strength in excess of 2240 psi is approximately 0.0002, or 0.02%.

Therefore, the probability of selecting a 1-foot square sample of material at random that on testing would have a breaking strength in excess of 2240 psi is approximately 0.02%.

b) To find the breaking strength value that separates the strongest 20% of the samples from the weakest 80%, we need to find the z-score corresponding to the 80th percentile and then calculate the breaking strength value using the z-score formula.

Step 1: Find the z-score corresponding to the 80th percentile:
Since the normal distribution is symmetric, we can find the z-score by subtracting the area to the left of the desired percentile (0.80) from 1.

z = invNorm(1 - 0.80) (using a standard normal distribution table or calculator)
= invNorm(0.20) ≈ -0.84 (rounded to two decimal places)

Step 2: Calculate the breaking strength value using the z-score:
Using the z-score formula, we have:

z = (x - μ) / σ

Rearranging the formula, we get:

x = μ + z * σ

Plugging in the values:

x = 2280 + (-0.84) * 10.6
= 2280 - 8.904
≈ 2271.096 psi

Therefore, the breaking strength value that separates the strongest 20% of one-foot square samples from the weakest 80% is approximately 2271.096 psi.

To solve these questions, we need to use the properties of the normal distribution. The normal distribution is characterized by its mean (μ) and standard deviation (σ). In this case, the mean breaking strength is 2280 psi, and the standard deviation is 10.6 psi.

a) To find the probability of selecting a sample with a breaking strength in excess of 2240 psi, we need to calculate the area under the normal curve to the right of 2240 psi.

Step 1: Standardize the value of 2240 psi.
We can calculate the z-score using the formula:
z = (x - μ) / σ
where x is the value we want to standardize, μ is the mean, and σ is the standard deviation.

In this case, z = (2240 - 2280) / 10.6 = -40 / 10.6 = -3.77

Step 2: Find the area to the right of the z-score.
Using a standard normal distribution table or a statistical calculator, you can find the area to the right of -3.77. The cumulative probability is approximately 0.0001.

Therefore, the probability of selecting a 1-foot square sample with a breaking strength in excess of 2240 psi is approximately 0.0001, or 0.01%.

b) We want to find the breaking strength value that separates the strongest 20% from the weakest 80%. In other words, we need to find the value that corresponds to a cumulative probability of 0.80.

Step 1: Find the z-score for a cumulative probability of 0.80.
Using a standard normal distribution table or a statistical calculator, find the z-score that corresponds to a cumulative probability of 0.80. The z-score is approximately 0.84.

Step 2: Calculate the breaking strength value.
Now, we can use the formula for standardization to find the breaking strength value.
z = (x - μ) / σ

In this case, we know the z-score is 0.84, and we want to find x. Rearranging the formula, we get:
x = z * σ + μ

Plugging in the values:
x = 0.84 * 10.6 + 2280 = 2288.504

Therefore, the breaking strength value in psi that separates the strongest 20% from the weakest 80% is approximately 2288.504 psi.