Questions: The area under a normal curve distribution is equal to 1 Equal to 100 almost equal to 1 equal to .5

Solution Steps

The area under a normal curve distribution is a fundamental property of the normal distribution. It is known that the total area under the curve is equal to 1. This can be expressed mathematically as:

\[ \int_{-\infty}^{\infty} f(x) \, dx = 1 \]

where \( f(x) \) is the probability density function of the normal distribution.

To confirm this property, we calculate the probability that the sample mean falls within the entire range of the normal distribution, from negative infinity to positive infinity. The probability can be expressed as:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(\infty) - \Phi(-\infty) \]

where \( \Phi \) is the cumulative distribution function (CDF) of the standard normal distribution.

From the properties of the CDF:

\[ \Phi(\infty) = 1 \quad \text{and} \quad \Phi(-\infty) = 0 \]

Thus, we have:

\[ P = 1 - 0 = 1.0 \]

Final Answer