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 greater than 0.57?
That would be greater than two standard deviations from the mean.
You need a computation tool or table for the "error" or normal distribution function.
Using http://psych.colorado.edu/~mcclella/java/normal/accurateNormal.html , I get 2.3%
For more ab9out the subject, see
http://en.wikipedia.org/wiki/Standard_deviation
To find the percentage of dowels with a diameter greater than 0.57 inches, we need to calculate the z-score and then use a standard normal distribution table.
Step 1: Calculate the z-score
The z-score measures the number of standard deviations an observation is from the mean. It can be calculated using the formula:
z = (x - μ) / σ
where x is the value we want to find the z-score for, μ is the mean, and σ is the standard deviation.
In this case, x = 0.57 inches, μ = 0.55 inches, and σ = 0.01 inches. Plugging these values into the formula, we get:
z = (0.57 - 0.55) / 0.01 = 2
Step 2: Use the standard normal distribution table
The standard normal distribution table gives the cumulative probability (area under the curve) to the left of a given z-score. Since we are interested in the percentage of dowels with a diameter greater than 0.57 inches, we need to find the area to the right of the z-score.
Looking up the z-score of 2 in the standard normal distribution table, we find that the area to the left is 0.9772.
Step 3: Calculate the percentage
To find the area to the right of the z-score, we subtract the area to the left from 1:
area to the right = 1 - 0.9772 = 0.0228
Multiplying this by 100 gives us the percentage:
percentage = 0.0228 * 100 = 2.28%
Therefore, approximately 2.28% of the dowels will have a diameter greater than 0.57 inches.