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 $x=\mu+\sigma$.
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Solution

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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)=1σ2πe(xμ)22σ2 f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{(x-\mu)^2}{2\sigma^2}} where μ \mu is the mean and σ \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) f(x) and set it equal to zero. The second derivative of the normal PDF is: f(x)=(xμ)2σ2σ4f(x) f''(x) = \frac{(x-\mu)^2 - \sigma^2}{\sigma^4} f(x) Setting f(x)=0 f''(x) = 0 , we get: (xμ)2σ2=0 (x-\mu)^2 - \sigma^2 = 0 Solving for x x , we find: x=μ±σ x = \mu \pm \sigma

Step 4: Identify the correct option

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

Final Answer

The points are x=μσ x = \mu - \sigma and x=μ+σ x = \mu + \sigma . Thus, the correct answer is C.

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