Suppose an oil exploration company purchases drill bits that have a life span that is approximately normally distributed, with a mean equal to 80 hours and a standard deviation equal to 10 hours.

To find the probability that a drill bit will fail before 60 hours of use, we can use the Z-score formula for a normal distribution. The Z-score tells us how many standard deviations a particular value is from the mean.

Given Data:

Z-score Calculation:

The Z-score is calculated using the formula: \[Z = \frac{X - \mu}{\sigma}\]

Plugging in the values: \[Z = \frac{60 - 80}{10} = \frac{-20}{10} = -2\]

Finding the Probability:

Next, we look up the Z-score of -2 in the standard normal distribution table or use a calculator with a normal distribution function. The Z-score of -2 corresponds to a cumulative probability of approximately 0.0228.

Interpretation:

This means that the probability that a drill bit will fail before 60 hours of use is approximately 2.28%³.

To find the probability that a drill bit will last between 70 hours and 90 hours, we can use the Z-score formula for a normal distribution.

Given Data:

Z-score Calculation:

First, we calculate the Z-scores for 70 hours and 90 hours.

  1. Z-score for 70 hours: \[Z_1 = \frac{70 - 80}{10} = \frac{-10}{10} = -1\]

  2. Z-score for 90 hours: \[Z_2 = \frac{90 - 80}{10} = \frac{10}{10} = 1\]

Finding the Probabilities:

Next, we look up the Z-scores in the standard normal distribution table or use a calculator with a normal distribution function.

Probability Between 70 and 90 Hours:

To find the probability that a drill bit will last between 70 and 90 hours, we subtract the cumulative probability at \(Z_1\) from the cumulative probability at \(Z_2\):

\[P(70 < X < 90) = P(Z_2) - P(Z_1) = 0.8413 - 0.1587 = 0.6826\]

Interpretation:

The probability that a drill bit will last between 70 hours and 90 hours is approximately 68.26%¹.