The breaking strength (in pounds) of a certain new synthetic is normally distributed, with a mean of 166 and a variance of 4. The material is considered defective if the breaking strength is less than 160 pounds. What is the probability that a single, randomly selected piece of material will be defective? (Give the answer to two decimal places.
variance = 4, so standard deviation = 2 (because squareroot of 4 is 2)
160 - 166 = -6 = -3 standard deviation.
You take out a normal distribution table and you find out that a standard deviation of -3 or less happens only 0.13% of the time.
Answer : 0.13%
Thanks a million!!!
To find the probability that a randomly selected piece of material will be defective, we need to calculate the probability that the breaking strength is less than 160 pounds.
First, we need to standardize the breaking strength using the z-score formula:
z = (X - μ) / σ
Where:
X = 160 (value we want to find)
μ = 166 (mean breaking strength)
σ = √variance = √4 = 2
Plugging in the values, we get:
z = (160 - 166) / 2
z = -3 / 2
z = -1.5
Next, we need to look up the corresponding probability from the standard normal distribution table.
From the table, we find that the probability of a z-score less than -1.5 is approximately 0.0668.
Therefore, the probability that a single, randomly selected piece of material will be defective is approximately 0.07.
To find the probability that a randomly selected piece of material will be defective, we need to calculate the area under the normal distribution curve that is less than 160 pounds.
Step 1: Calculate the standard deviation (σ) by taking the square root of the variance: σ = √4 = 2.
Step 2: Standardize the value of 160 pounds using the formula: z = (x - μ) / σ, where x is the given value, μ is the mean, and σ is the standard deviation.
z = (160 - 166) / 2
= -6 / 2
= -3.
Step 3: Use a Z-table or a calculator with a normal distribution function to find the area to the left of z = -3. This represents the probability that the breaking strength is less than 160 pounds.
The Z-table provides the cumulative probability up to a certain z-value. By looking up -3 in the Z-table, we can find the corresponding cumulative probability.
From the Z-table, the cumulative probability for z = -3 is approximately 0.0013.
Step 4: Convert the cumulative probability to the probability of selecting a defective piece of material. Since we are interested in the area to the left of z = -3, this represents the probability of selecting a defective piece.
Therefore, the probability of a single randomly selected piece of material being defective is approximately 0.0013 or 0.13% (rounded to two decimal places).
Note: In cases where exact Z-table values are not available, a calculator or software with a normal distribution function can be used to find the probability directly using the given values of mean, standard deviation, and desired cutoff point.