A tractor manufacturing company machines axles to extremely small tolerances. The main power axles have a mean diameter of 3.0000 inches and a standard deviation of 0.0050 inch. What is the probability that a given axle will have a diameter between 2.9915 and 3.0004 inches? Assume that errors are normally distributed.
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http://davidmlane.com/hyperstat/z_table.html
Just enter the mean, the SD and enter the values for the "between" option
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To calculate the probability that a given axle will have a diameter between 2.9915 and 3.0004 inches, we can use the concept of the standard normal distribution.
Step 1: Z-Score Calculation
First, we need to calculate the z-scores for the lower and upper limits of the diameter range.
The z-score formula is:
z = (x - μ) / σ
where x is the desired diameter, μ is the mean diameter, and σ is the standard deviation.
For the lower limit:
z_lower = (2.9915 - 3.0000) / 0.0050
For the upper limit:
z_upper = (3.0004 - 3.0000) / 0.0050
Step 2: Standard Normal Distribution
Next, we need to find the cumulative probability associated with each z-score using a standard normal distribution table or a statistical software.
The cumulative probability represents the area under the normal distribution curve up to the z-score.
Step 3: Calculate Probability
Finally, subtract the lower cumulative probability from the upper cumulative probability to get the probability of the axle having a diameter between the specified range.
P(2.9915 ≤ diameter ≤ 3.0004) = P(z_lower ≤ Z ≤ z_upper)
So essentially, to find the solution, it requires calculating the z-scores for both the lower and upper limits, finding the cumulative probabilities associated with those z-scores, and subtracting the lower cumulative probability from the upper cumulative probability to find the probability of the axle having a diameter within the specified range.