Question

1. -/3 points

(a) what percentage of the area under the normal curve lies to the right of μ?
%
(b) what percentage of the area under the normal curve lies between μ - 2σ and μ + 2σ?
%
(c) what percentage of the area under the normal curve lies to the right of μ + 3σ?
%

Explanation:

Step1: Recall the properties of the normal distribution

The normal distribution is symmetric about the mean \(\mu\). The total area under the normal curve is \(100\%\) or \(1\).

Step2: Solve part (a)

Since the normal curve is symmetric about \(\mu\), half of the area is to the left of \(\mu\) and half is to the right. So the percentage of the area to the right of \(\mu\) is \(\frac{100\%}{2}=50\%\)

Step3: Solve part (b)

By the empirical rule (68 - 95 - 99.7 rule), approximately \(95\%\) of the data lies within \(\mu\pm2\sigma\) in a normal distribution.

Step4: Solve part (c)

By the empirical rule, approximately \(99.7\%\) of the data lies within \(\mu\pm3\sigma\). So the area within \(\mu - 3\sigma\) and \(\mu+3\sigma\) is \(99.7\%\). The remaining area is \(100\% - 99.7\%=0.3\%\). Since the normal distribution is symmetric, the area to the right of \(\mu + 3\sigma\) is \(\frac{0.3\%}{2}=0.15\%\)

Answer:

(a) \(50\%\)
(b) \(95\%\)
(c) \(0.15\%\)