The breaking strength (in pounds) of a certain new synthetic is normally distributed, with a mean of 145 and a variance of 4. The material is considered defective if the breaking strength is less than 140 pounds. What is the probability that a single, randomly selected piece of material will be defective? (You may need to use the standard normal distribution table. Round your answer to four decimal places.)
145-140=5 is below mean 25=5 10/5= -2
answer 0.9988 thank you for your time and I hope you had a nice weekend.
variance of 4 means std of 2.
5 is below the mean by 2.5 std.
P(Z < -2.5) = .0062 = 0.62%
Think about it. Your answer indicates that virtually all of the pieces are defective. That would surely not ever be tolerated in a real-world situation.
Take a visit to
http://davidmlane.com/hyperstat/z_table.html
and you can play around with this Z-table stuff.
Thank you Steve
To find the probability that a single randomly selected piece of material will be defective, we need to calculate the area under the normal curve to the left of the threshold value of 140 pounds. Let's break down the steps to get the answer:
Step 1: Calculate the standard deviation σ by taking the square root of the variance.
σ = √4 = 2
Step 2: Calculate the z-score using the formula:
z = (x - μ) / σ
where x is the threshold value of 140 pounds, μ is the mean of 145 pounds, and σ is the standard deviation.
z = (140 - 145) / 2 = -2.5
Step 3: Use the standard normal distribution table or a calculator to find the probability corresponding to the z-score -2.5.
The table will give you the area to the left of -2.5, which is the probability that a randomly selected piece of material will be below 140 pounds.
Using the standard normal distribution table, we find that the closest value to -2.5 is -2.48, and the corresponding probability is 0.0062.
Step 4: Since we are interested in the area to the left of -2.5, we subtract the probability from 0.5 (since the total area under the curve is 1) to find the area to the right of -2.5.
Area to the left of -2.5 = 0.5 - 0.0062 ≈ 0.4938
Therefore, the probability that a piece of material will be defective (breaking strength less than 140 pounds) is approximately 0.4938 or 49.38%.
It's important to note that the answer you provided, 0.9988, seems to be the complement of the correct probability. The correct approach is to find the probability to the left of the threshold value, not the right.