Question at position 51 The time to complete a standardized exam is approximately Normal with a mean of 70 and a standard deviation of 10 minutes. How much time should be given to complete the exam so that 80% of the students will complete the exam in the time given?78.4 minutes92.8 minutes84.0 minutes61.6 minutes79.8 minutes

Question

BlackTom AI · Accepted Answer

We’re told the exam times follow a Normal distribution with mean 70 minutes and standard deviation 10 minutes. The question asks for the time t so that 80% of students finish within that time, i.e., P(X ≤ t) = 0.80.

To solve, convert the 80th percentile to a z-score using the standard Normal distribution. The z-value corresponding to 0.80 is approximately 0.8416. Then translate back to the original scale: t = mu + z*sigma = 70 + (0.8416)(10) ≈ 70 + 8.416 ≈ 78.416 minutes.

Now evaluate each option:
- 78.4 minutes: This is essentially the computed value (rounded to one decimal place). It yields t ≈ 78.4, which gives about 80% of students finishing by that time. This matches the calculation.
- 92.8 minutes: This would correspond to a much higher percentile than 80% (roughly z ≈ 1.28, giving around 90th percentile), so it overestimates the time needed for 80% to finish.
- 84.0 minutes: While larger than the mean, 84 minutes corresponds to z ≈ 1.4, which gives well above 80% (closer to around 91st percentile). This would overshoot the desired 80th percentile.
- 61.6 minutes: This is below the mean by about 8.4 minutes (z ≈ -0.84), which would place the 80th percentile far below the mean and yield a percentile less than 50%, not 80%.
- 79.8 minutes: This is slightly higher than 78.4 but still near the mean; however, it does not correspond to the 80th percentile as precisely as 78.4 does, and the intended calculation targets the 80th percentile specifically. In a precise multiple-choice setting, 78.4 minutes is the closest correct value.

In summary, the correct choice aligns with finding the 80th percentile of N(70, 10^2), which is about 78.4 minutes, while the other options either overestimate or underestimate the required time for 80% completion.

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