The diameters of a wooden dowel produced by a new machine are normally distributed with a mean of 0.55 inches and a standard deviation of 0.01 inches. What percent of the dowels will have a diameter less than 0.57?

I filled in the data in my favourite applet

http://davidmlane.com/hyperstat/z_table.html
and it gave me 97.7%

To find the percentage of dowels with a diameter less than 0.57 inches, we can use the concept of the standard normal distribution.

1. Begin by standardizing the value of 0.57 inches using the formula for z-score: z = (x - μ) / σ
Here, x represents the value we want to standardize (0.57 inches), μ is the mean (0.55 inches), and σ is the standard deviation (0.01 inches).

z = (0.57 - 0.55) / 0.01
= 2 / 0.01
= 200

2. Next, we need to find the area under the standard normal curve to the left of this z-score. This represents the percentage of values that are less than 0.57 inches.
We can use a standard normal distribution table or a calculator to find this area.

Using a standard normal distribution table, we find that the area to the left of a z-score of 2 is approximately 0.9772.

3. Convert the decimal to a percentage by multiplying it by 100.
0.9772 * 100 = 97.72

Therefore, approximately 97.72% of the dowels produced by the machine will have a diameter less than 0.57 inches.