Which of the following statements about the normal probability curve are correct?
(A) The curve is bell-shaped and symmetric about the mean.
(B) The mean, median and mode of a normal distribution are equal.
(C) The curve meets the x-axis at the point \(x = \mu + 3\sigma\).
(D) The total area under the curve and above the x-axis is 1.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 3
The correct answer is Option (3) → (A), (B) and (D) only
The normal probability curve with mean \(\mu\) and standard deviation \(\sigma\) is a bell-shaped curve, highest at \(x = \mu\) and symmetric about the line \(x = \mu\). On both sides it comes closer and closer to the x-axis but never touches it.
So the curve is symmetric about the mean, its highest point is at the mean, it never meets the x-axis, and the total probability (area) is 1.
Checking the statements
(A) The curve is bell-shaped and symmetric about the mean. — Correct. It is symmetric about the line \(x = \mu\).
(B) The mean, median and mode of a normal distribution are equal. — Correct. Symmetry puts the median at \(\mu\), and the peak (mode) is also at \(\mu\).
(C) The curve meets the x-axis at the point \(x = \mu + 3\sigma\). — Incorrect. The curve never meets the x-axis; it only approaches it on both sides.
(D) The total area under the curve and above the x-axis is 1. — Correct. This area is the total probability, which is 1.