Question

statistical modeling
practice on the empirical rule and normal distri

1. in a normal distribution, what percent of the values lie...

a) below the mean?
b) above the mean?
c) within one standard deviation of the mean?
d) within two standard deviations of the mean?
e) within three standard deviations of the mean?

Explanation:

Part (a)

Step1: Recall Normal Distribution Symmetry

A normal distribution is symmetric about its mean ($\mu$). The total area under the normal curve (representing total probability or percentage of values) is 100% or 1. Since it's symmetric, the area below the mean is half of the total area.

Step2: Calculate Percentage Below Mean

Half of 100% is $ \frac{100\%}{2} = 50\% $.

Part (b)

Step1: Use Symmetry of Normal Distribution

Similar to part (a), due to symmetry about the mean, the area above the mean is also half of the total area under the curve.

Step2: Calculate Percentage Above Mean

Half of 100% is $ \frac{100\%}{2} = 50\% $.

Part (c)

Step1: Recall the Empirical Rule (68 - 95 - 99.7 Rule)

The empirical rule for normal distributions states that about 68% of the data lies within 1 standard deviation ($\sigma$) of the mean (i.e., between $\mu - \sigma$ and $\mu + \sigma$).

Step2: Apply the Empirical Rule

From the empirical rule, the percentage of values within one standard deviation of the mean is 68%.

Part (d)

Answer:

s:
a) $\boldsymbol{50\%}$
b) $\boldsymbol{50\%}$
c) $\boldsymbol{68\%}$
d) $\boldsymbol{95\%}$
e) $\boldsymbol{99.7\%}$