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