Questions: The diameters of ball bearings are distributed normally. The mean diameter is 107 millimeters and the standard deviation is 5 millimeters. Find the probability that the diameter of a selected bearing is greater than 115 millimeters. Round your answer to four decimal places.

Solution Steps

To find the probability that the diameter of a selected bearing is greater than 115 mm, we first calculate the Z-score using the formula:

\[ z = \frac{X - \mu}{\sigma} \]

where:

• \( X = 115 \) mm (the value we are checking),
• \( \mu = 107 \) mm (the mean diameter),
• \( \sigma = 5 \) mm (the standard deviation).

Substituting the values, we have:

\[ z = \frac{115 - 107}{5} = \frac{8}{5} = 1.6 \]

Thus, the Z-score for a diameter of 115 mm is \( z = 1.6 \).

Next, we need to find the probability that the diameter is greater than 115 mm. This can be expressed as:

\[ P(X > 115) = P(Z > 1.6) = 1 - P(Z \leq 1.6) \]

Using the cumulative distribution function \( \Phi(z) \), we can express this as:

\[ P(X > 115) = \Phi(\infty) - \Phi(1.6) \]

Since \( \Phi(\infty) = 1 \), we have:

\[ P(X > 115) = 1 - \Phi(1.6) \]

From standard normal distribution tables or calculators, we find:

\[ \Phi(1.6) \approx 0.9452 \]

Thus, the probability becomes:

\[ P(X > 115) = 1 - 0.9452 = 0.0548 \]

Final Answer