Consider the probability that exactly 38 out of 542 computers will crash in a day. Choose the best description of the area under the normal curve that would be used to approximate binomial probability.

A. Area to the right of 37.5
B. Area to the right of 38.5
C. Area to the left of 37.5
D. Area to the left of 38.5
E. Area between 37.5 and 38.5

Question

16-10-2023
Mathematics

High School

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Consider the probability that exactly 38 out of 542 computers will crash in a day. Choose the best description of the area under the normal curve that would be used to approximate binomial probability.

A. Area to the right of 37.5
B. Area to the right of 38.5
C. Area to the left of 37.5
D. Area to the left of 38.5
E. Area between 37.5 and 38.5

Answer :

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TaylorSmith474 TaylorSmith474 · Answer 1 · 2025-03-25T06:13:14-04:00

Final answer:

To approximate the binomial probability of exactly 38 out of 542 computers crashing using a normal distribution, the continuity correction factor is applied, leading to the area between 37.5 and 38.5 under the normal curve as the best description (option e).

Explanation:

To approximate the probability of exactly 38 out of 542 computers crashing using a normal distribution, we apply the continuity correction factor. The continuity correction is used to adjust for the fact that the normal distribution is continuous whereas the binomial distribution is discrete. When approximating binomial probabilities with a normal curve, we need to account for the entire range of values that round to the target value (in this case, 38 computers crashing).

For exactly 38 computers crashing, we would look for the area under the normal curve between 37.5 and 38.5. Therefore, the best description of the area under the normal curve to approximate this binomial probability is the area between 37.5 and 38.5, which translates to option e. This approach is informed by the fact that it's the area under the curve for the entire interval that rounds to 38, thereby better approximating the probability of the discrete outcome in the binomial setting.