Questions: Complete the statement below. The points at x= and x= are the inflection points on the normal curve. What are the two points? A. The points are x=μ-2σ and x=μ+2σ. B. The points are x=μ-3σ and x=μ+3σ. C. The points are x=μ-σ and x=μ+σ.

Complete the statement below. The points at x= and x= are the inflection points on the normal curve.

What are the two points? A. The points are x=μ-2σ and x=μ+2σ. B. The points are x=μ-3σ and x=μ+3σ. C. The points are x=μ-σ and x=μ+σ.

Transcript text: Complete the statement below. The points at $\mathrm{x}=$ $\qquad$ and $x=$ $\qquad$ are the inflection points on the normal curve. What are the two points? A. The points are $x=\mu-2 \sigma$ and $x=\mu+2 \sigma$. B. The points are $x=\mu-3 \sigma$ and $x=\mu+3 \sigma$. C. The points are $x=\mu-\sigma$ and $x=\mu+\sigma$.

Solution

Solution Steps

Step 1: Understand the concept of inflection points on a normal curve

Inflection points on a normal curve are the points where the curve changes its concavity. For a normal distribution, these points occur where the second derivative of the probability density function (PDF) equals zero.

Step 2: Recall the PDF of a normal distribution

The probability density function (PDF) of a normal distribution is given by: \[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} \] where $ \mu $ is the mean and $ \sigma $ is the standard deviation.

Step 3: Find the second derivative of the PDF

To find the inflection points, we need to compute the second derivative of $ f(x) $ and set it equal to zero. The second derivative of the normal PDF is: \[ f''(x) = \frac{(x-\mu)^2 - \sigma^2}{\sigma^4} f(x) \] Setting $ f''(x) = 0 $, we get: \[ (x-\mu)^2 - \sigma^2 = 0 \] Solving for $ x $, we find: \[ x = \mu \pm \sigma \]

Step 4: Identify the correct option

The inflection points occur at $ x = \mu - \sigma $ and $ x = \mu + \sigma $. Therefore, the correct option is: C. The points are $ x = \mu - \sigma $ and $ x = \mu + \sigma $.