The diameters of pencils produced by a certain machine are normally distributed with a mean of 0.30 inches and a standard deviation of 0.01 inches. In a random sample of 460 pencils, approximately how many would you expect to have a diameter less than 0.293 inches?

A. 123
B. 118
C. 112
D. 111
My asnwer is C

I got 111.32, so I guess it depends on how you want to round it.

To find the approximate number of pencils with a diameter less than 0.293 inches, we can use the concept of the standard normal distribution.

First, we need to convert the given diameter value into a z-score. The z-score represents how many standard deviations an observed value is from the mean. It is calculated using the formula:

z = (x - μ) / σ

Where:
x = observed value (0.293 inches)
μ = mean (0.30 inches)
σ = standard deviation (0.01 inches)

Plugging in the values, we get:
z = (0.293 - 0.30) / 0.01
z = -0.007 / 0.01
z = -0.7

Next, we need to find the area under the standard normal distribution curve to the left of the z-score value of -0.7. This area represents the proportion of pencils with a diameter less than 0.293 inches.

Using a standard normal distribution table or a calculator, we can find the corresponding area. In this case, the area to the left of -0.7 is approximately 0.2419.

Finally, to find the approximate number of pencils, we multiply the proportion by the sample size:

Number of pencils = Proportion * Sample size
Number of pencils = 0.2419 * 460
Number of pencils ≈ 111

So, approximately 111 pencils would be expected to have a diameter less than 0.293 inches. Therefore, the correct answer is D. 111.