If a dataset follows a normal distribution with mean 5 and standard deviation 3, which code calculates the proportion of observations that is below 4?

Question

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To determine the proportion of observations below 4 for a normal distribution with mean μ = 5 and standard deviation σ = 3, we use the cumulative distribution function F(x) = P(X ≤ x) = pnorm(x, μ, σ).

Option 1: pnorm(3, 4, 5) uses x = 3, with a mean of 4 and a standard deviation of 5. This does not correspond to the given distribution (μ = 5, σ = 3) and also evaluates at the wrong value, so it is incorrect for the problem as stated.

Option 2: pnorm(4, 3, 5) uses x = 4 but with μ = 3 and σ = 5. Again, both the parameters μ and σ do not match the dataset in the question, so this option does not compute P(X ≤ 4) for the specified distribution.

Option 3: pnorm(5, 4, 3) uses x = 5 with μ = 4 and σ = 3. While the standard deviation matches the problem, the mean does not, and we are not evaluating P(X ≤ 4) but P(X ≤ 5) under the wrong distribution, so this is incorrect.

Option 4: pnorm(4, 5, 3) correctly computes the probability that X ≤ 4 for X ~ N(μ = 5, σ = 3). This directly corresponds to the requested proportion (below 4) in the specified normal distribution, since we plug in x = 4, μ = 5, and σ = 3 into the normal CDF formula.

In summary, the correct approach is to evaluate the normal CDF at x = 4 with the given mean and standard deviation, which is exactly what pnorm(4, 5, 3) does. The other options mix up either the x-value or the distribution parameters, leading to results that do not reflect the problem's conditions.

MATH1062/1005/1023 (ND) MATH1062/1005 Data Quiz 4

If a dataset follows a normal distribution with mean 5 and standard deviation 3, which code calculates the proportion of observations that is below 4?

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