Women’s heights are Normally distributed with mean 64 and standard deviation 3. Which event is less likely?

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BlackTom AI · Accepted Answer

The question states that women's heights are Normally distributed with mean μ = 64 inches and standard deviation σ = 3 inches, and asks which event is less likely.
First, note that for a sample of size n = 100, the sampling distribution of the sample mean X̄ is Normal with mean μ_X̄ = μ = 64 and standard deviation σ_X̄ = σ/√n = 3/√100 = 0.3.
We are evaluating the event X̄ > 67. Compute the z-score: z = (67 − 64) / 0.3 = 3 / 0.3 = 10.
A z-score of 10 in a standard normal distribution corresponds to an extremely small tail probability (far less than 0.0001, in fact on the order of 10^−23 or smaller). In other words, observing an average height above 67 inches in a sample of 100 women from this population is extraordinarily unlikely.
Since the question asks which event is less likely, this event (sample mean > 67) is far more unlikely than typical deviations from the mean and would be the correct selection under normal circumstances.
Note: The provided data includes only one answer option in the given field, and the list of available answer choices is missing. Because there is no other option to compare against, no additional contrasts can be analyzed. Nevertheless, with the information given, the event X̄ > 67 is the extreme tail scenario and is the least probable outcome for the sample mean under these parameters.
In summary, the calculation shows an almost vanishing probability for the sample mean exceeding 67 inches, making it the less likely event among reasonable alternatives under the specified normal model.

STAT 1450.02 SP2025 (28132) Checkpoint 15.4

Women’s heights are Normally distributed with mean 64 and standard deviation 3. Which event is less likely?

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