Questions: Consider the probability that at least 42 out of 120 computers will crash in a day. Assume the probability that a given computer will crash in a day is 2%. Specify whether the normal curve can be used as an approximation to the binomial probability by verifying the necessary conditions.

Solution Steps

For a binomial distribution with parameters \( n = 120 \) and \( p = 0.02 \), we calculate the following:

• Mean \( \mu = n \cdot p = 120 \cdot 0.02 = 2.4 \)
• Variance \( \sigma^2 = n \cdot p \cdot q = 120 \cdot 0.02 \cdot 0.98 = 2.352 \)
• Standard Deviation \( \sigma = \sqrt{npq} = \sqrt{120 \cdot 0.02 \cdot 0.98} \approx 1.5336 \)

To determine if the normal curve can be used as an approximation to the binomial probability, we check the following conditions:

• Calculate \( np \): \[ np = 2.4 \]

• Calculate \( nq \): \[ nq = n \cdot q = 120 \cdot 0.98 = 117.6 \]

The conditions for using the normal approximation are:

• \( np > 5 \)
• \( nq > 5 \)

In this case:

• \( np = 2.4 \) (not greater than 5)
• \( nq = 117.6 \) (greater than 5)

Since \( np \) is not greater than 5, the normal curve cannot be used as an approximation to the binomial probability.

Final Answer