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.
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)=σ2π1e−2σ2(x−μ)2
where μ is the mean and σ 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)=σ4(x−μ)2−σ2f(x)
Setting f′′(x)=0, we get:
(x−μ)2−σ2=0
Solving for x, we find:
x=μ±σ
Step 4: Identify the correct option
The inflection points occur at x=μ−σ and x=μ+σ. Therefore, the correct option is:
C. The points are x=μ−σ and x=μ+σ.
Final Answer
The points are x=μ−σ and x=μ+σ. Thus, the correct answer is C.