Of 311 students currently enrolled in two sections of QTM 100 in Fall, 2024, 89.7% of the students completed the course evaluation for the lecture. If the QTM 100 professor wants to conduct a hypothesis test examining if the true course evaluation completion proportion differs from 85% which is required for earning the extra credit, which of the following represents a correct standard error for this hypothesis test?
Yes, you will all receive 1 extra point on your final exam because more than 85% of the class completed the course evaluation.
Thanks for a wonderful semester, and have a fantastic break!

Question

Of 311 students currently enrolled in two sections of QTM 100 in Fall, 2024, 89.7% of the students completed the course evaluation for the lecture. If the QTM 100 professor wants to conduct a hypothesis test examining if the true course evaluation completion proportion differs from 85% which is required for earning the extra credit, which of the following represents a correct standard error for this hypothesis test? 
Yes, you will all receive 1 extra point on your final exam because more than 85% of the class completed the course evaluation. 
Thanks for a wonderful semester, and have a fantastic break!

BlackTom AI · Accepted Answer

To evaluate a hypothesis test for a proportion, the standard error under the null hypothesis p0 = 0.85 is SE = sqrt(p0 * (1 - p0) / n).

First, compute the components: p0 = 0.85, 1 - p0 = 0.15, n = 311.
The product p0 * (1 - p0) is 0.85 * 0.15 = 0.1275. Dividing by n gives 0.1275 / 311 ≈ 0.000410. Taking the square root yields SE ≈ sqrt(0.000410) ≈ 0.0203.

Now examine the options:
- Option A: (
.85
×
.15
)
311. This option shows the product p0 * (1 - p0) divided by n, i.e., 0.1275 / 311 ≈ 0.000410, but it omits the square root that is essential for a standard error. As a result, it represents the variance term’s numerator divided by n rather than the standard error itself. This makes it incorrect as the SE value.

- Option B: .85
311. This option is simply p0 divided by n, i.e., 0.85 / 311 ≈ 0.002733. This ignores the (1 - p0) factor entirely and is not a valid SE expression for a proportion test. It misuses both the probability component and the scaling.

- Option C: (
.85
×
.15
)
311. This appears to be a repetition of Option A with the same issue: it gives the product divided by n without the square root. Again, it does not produce the standard error and is therefore incorrect for SE.

- Option D: (
.897
×
.102
)
311. Here, 0.897 is the sample proportion p̂ and 0.102 is its complement (1 - p̂). Multiplying them and dividing by n mirrors the structure of a variance term, not the standard error, and uses the sample values rather than the null-proportion values. It does not reflect SE under the null hypothesis and is thus inappropriate for the standard error in this hypothesis test.

In short, the correct standard error should be computed as sqrt(p0 * (1 - p0) / n) using p0 = 0.85 and n = 311, yielding about 0.0203. None of the provided options present this sqrt form with the null-hypothesis p0; they either omit the square root or mix in alternative quantities, making them incorrect for representing the standard error of this test.

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