Q.6.54 - Elementary Statistics

Question

According to Table II, the area under the standard normal curve that lies to the left of $1.96$ is $0.975$. Without further consulting Table II, determine the area under the standard normal curve that lies to the left of $-1.96$. Explain your reasoning.

Step-by-Step Solution

Answer

The area to the left of $1.96$ under the standard normal curve is $0.975$.

Step 1: Given Information

The area under the standard normal curve that lies to the left of $1.96$ is $0.975$.

Step 2: Explanation

The histogram of the distribution is substantially bell-shaped when the variable is approximately distributed normally, as seen below:

The area under the standard normal curve is one, and the curve is symmetric around $0$.

Step 3: Explanation

The area under the standard normal curve is one, and the curve is symmetric around $0$.

As a result, the area to the right of the conventional normal curve has increased$1.96$ equals $1$ minus the area to the left of $1.96$.

As a result, the area under $1.96$'s standard normal curve to the right will be $1-0.9750=0.0250$.

The area to the left of the standard normal curve is symmetric because it is $-1.96$ will be equal to the area to the right of $-1.96$.

As a result, $0.0250$ will be the area to the left of the usual normal curve $-1.96$.