Question

Normal distribution help? will medal and fan

Answer 1

Suppose a population is normally distributed, with a mean of 100 and a standard deviation of 10.

1. What percent of the data falls below the value 100?

2. What percent of the data falls between 80 and 120?

3. What percent of the data is above 90?

Answer 2

By standardizing:
\[ Z=\frac{X-\mu}{\sigma }\] In your question, \(\mu=100, \sigma=10\)

1): The question asks \(P(X < 100)\) . Without calculation, if you know that the mean is 100 and the ask you what percentage of data lies below 100, you should know this right away :) This is because the mean lies directly in the very middle of your data if it has a normal distribution. Even if you did not realize that, the mechanical way to proceed is to standardize:
\[ P(X<100)=P\left( \frac{X-100}{10}<\frac{100-100}{10}\right)=P(Z<0)\]

2) \[\begin{align}P(80 < X < 120) &= P\left( \frac{80-100}{10}<\frac{X-100}{10}<\frac{120-100}{10}\right)\\ &=P(-2<Z<2)\\&=P(Z<2)-P(Z<-2)\end{align}\]

3) \[P(X>90)=P \left( \frac{X-100}{10}>\frac{90-100}{10}\right) =P(Z>-1)=1-P(Z<-1)\]