Question

The probability that a random variable is greater than or equal to z standard deviations from the mean in a standard normal distribution is p%. What can be said with certainty about the probability that the random variable is less than or equal to –z standard deviations from the mean?
The probability is less than p%.
The probability is equal to p%.
The probability is greater than p%.
The probability is not equal to p%.

Answer 1

The 2nd one ..........

Answer 2

Answer:

2. The probability is equal to p%.

Step-by-step explanation:

We have been given that the probability that a random variable is greater than or equal to z standard deviations from the mean in a standard normal distribution is p%. We are asked to choose the correct statement about the probability that the random variable is less than or equal to –z standard deviations from the mean.

We know that normal distribution curve is symmetric about mean. The z-score of data point shows that the data point is how may standard deviation above or below mean.

A positive z-score means that a data points is that many standard deviation above mean. So positive z will be above mean.

The probability that a variable is greater than or equal to z is given by p%. This probability (p%) represents the distance from z to the right end of the curve.

A negative z-score means that a data points is that many standard deviation below mean. So negative z-score $(-z)$ will be below the mean. The probability that a variable is less than or equal to -z is the distance from -z to the left end of the curve.  

Since a normal distribution curve is symmetric about the mean, therefore, the probability that a variable is less than or equal to -z will be equal to probability that a variable is greater than or equal to z.

Since probability that a variable is greater than or equal to z is given by p%, therefore, the probability that a variable is less than or equal to -z will be p% and 2nd option is the correct choice.