Which of the following statements are NOT correct about Standard Normal Distribution?
(A) The probability curve of the Standard Normal Distribution is a bell-shaped curve.
(B) The Standard Normal variate (Z) score describes the position of each data point in terms of its distance from the mean, when measured in standard deviation units.
(C) The Z-score is negative if the data point lies above the mean, and positive if it lies below the mean.
(D) There is a 95.45 % probability of randomly selecting a score between \(\mu - \sigma\) and \(\mu + \sigma\), when \(\sigma\) is standard deviation and \(\mu\) is mean.
Choose the correct answer from the options given below:
Answer & explanation
Correct answer: option 4
The correct answer is Option (4) → (C) and (D) only
The question asks for the statements that are NOT correct. The standard normal variate is
\( \displaystyle Z = \frac{X - \mu}{\sigma} \)
(A) The probability curve of the Standard Normal Distribution is a bell-shaped curve. — Correct. The curve is bell-shaped and symmetric about Z = 0.
(B) The Standard Normal variate (Z) score describes the position of each data point in terms of its distance from the mean, when measured in standard deviation units. — Correct. \(X - \mu\) is the distance from the mean, and dividing by \(\sigma\) measures it in standard deviations.
(C) The Z-score is negative if the data point lies above the mean, and positive if it lies below the mean. — Incorrect. Above the mean \(X - \mu \gt 0\), so Z is positive; below the mean Z is negative. The statement has the signs the wrong way round.
(D) There is a 95.45 % probability of randomly selecting a score between \(\mu - \sigma\) and \(\mu + \sigma\), when \(\sigma\) is standard deviation and \(\mu\) is mean. — Incorrect. The probability between \(\mu - \sigma\) and \(\mu + \sigma\) is about 68.27%. The value 95.45% belongs to the interval from \(\mu - 2\sigma\) to \(\mu + 2\sigma\).
The statements that are not correct are (C) and (D).
CBSE Applied Mathematics (Class XII):
Z-Score of Normal Distribution
