Question

The diameters of bolts produced by a certain machine are normally distributed with a mean of 0.30 inches and a standard deviation of 0.01 inches. What percentage of bolts will have a diameter greater than 0.32 inches?

a. 47.72%
b. 97.72%
c. 37.45%
d. 2.28%
e. 4.56%

Answer 1

Answer: option D is the correct answer.

Explanation:

Since the diameters of bolts produced by a certain machine are normally distributed, we would apply the formula for normal distribution which is expressed as

z = (x - µ)/σ

Where

x = diameter of bolts

µ = mean diameter

σ = standard deviation

From the information given,

µ = 0.3 inches

σ = 0.01 inches

We want to find the probability bolts that will have a diameter greater than 0.32 inches. It is expressed as

P(x > 0.32) = 1 - P(x ≤ 0.32)

For x = 0.32,

z = (0.32 - 0.3)/0.01 = 2

Looking at the normal distribution table, the probability corresponding to the z score is 0.9773

P(x > 0.32) = 1 - 0.9733. = 0.0228

The percentage of bolts will have a diameter greater than 0.32 inches is

0.0228 × 100 = 2.28%

Answer 2

d. 2.28%

Explanation:

Problems of normally distributed samples are solved using the z-score formula.

In a set with mean
`\mu` and standard deviation
`\sigma`, the zscore of a measure X is given by:

`Z = (X - \mu)/(\sigma)`

The Z-score measures how many standard deviations the measure is from the mean. After finding the Z-score, we look at the z-score table and find the p-value associated with this z-score. This p-value is the probability that the value of the measure is smaller than X, that is, the percentile of X. Subtracting 1 by the pvalue, we get the probability that the value of the measure is greater than X.

In this problem, we have that:

`\mu = 0.3, \sigma = 0.01`

What percentage of bolts will have a diameter greater than 0.32 inches?

This is 1 subtracted by the pvalue of Z when X = 0.32. So

`Z = (X - \mu)/(\sigma)`

`Z = (0.32 - 0.3)/(0.01)`

`Z = 2`

`Z = 2` has a pvalue of 0.9772

1 - 0.9772 = 0.0228 = 2.28%

So the correct answer is:

d. 2.28%