The breaking strength (in pounds) of a certain new synthetic is normally distributed, with a mean of 115 and a variance of 25. The material is considered defective if the breaking strength is less than 105 pounds. What is the probability that a single, randomly selected piece of material will be defective?
difference from mean = 115-105 = 10 below mean
variance = 25 so standard deviation = 5
so z = -10/5 = -2
it is two standard deviations below mean
for z = 2 look up in normal table
http://davidmlane.com/hyperstat/z_table.html
probability = .0228
So your graph is going to have a mean of 115 and your standard deviation is 5 (because variance is standard deviation squared). From here you can just plug in your calculator. Since the graph is "normally distributed" you can use normalcdf(left interval, right interval, mean, sd) You want to find the probability for anything less than 105 pounds so that means normalcdf(-99999,105,115,5)=.023
To find the probability that a single, randomly selected piece of material will be defective, we can use the normal distribution.
Given that the breaking strength of the material follows a normal distribution with a mean of 115 and a variance of 25, we first need to standardize the value of 105 pounds using the z-score formula:
z = (x - μ) / σ
where:
z is the standardized value (z-score),
x is the value we want to standardize (105 pounds),
μ is the mean of the distribution (115 pounds),
and σ is the standard deviation of the distribution (√variance = √25 = 5).
Plugging the values into the formula:
z = (105 - 115) / 5
z = -10 / 5
z = -2
Now that we have the z-score, we can use a standard normal distribution table (also known as z-table) or statistical software to find the probability corresponding to this z-score.
The probability of a z-score of -2 or less can be found by looking up the corresponding area to the left of -2 in the standard normal distribution table.
Consulting the standard normal distribution table, we find that the area to the left of -2 is approximately 0.0228.
Therefore, the probability that a single, randomly selected piece of material will be defective (breaking strength less than 105 pounds) is approximately 0.0228 or 2.28%.