As the $\mathrm{mean}$ increases in a $\mathrm{normal}\mathrm{distribution}$, what happens to the graph of the $\mathrm{normal}\mathrm{curve}$?

• A. The curve becomes wider and shorter.
• B. The curve becomes taller and narrower.
• C. The curve shifts upward along the vertical axis.
• D. The curve shifts to the right along the $\mathrm{horizontal}\mathrm{axis}$ without changing its shape.

Answer: (D)

Verified step by step guidance

1. Recall that a normal distribution is defined by two parameters: the mean (\( \mu \)) and the standard deviation (\( \sigma \)). The mean determines the center of the distribution, while the standard deviation controls the spread or width of the curve.
2. Understand that the graph of the normal distribution is a bell-shaped curve centered at the mean \( \mu \). The formula for the normal distribution's probability density function is: \[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} \left( \frac{x - \mu}{\sigma} \right)^2} \]
3. When the mean \( \mu \) increases, it shifts the entire curve along the horizontal axis (the \( x \)-axis) because the center of the distribution moves to the right. This does not affect the shape of the curve since the standard deviation \( \sigma \) remains the same.
4. The height and width of the curve are controlled by the standard deviation \( \sigma \). Since \( \sigma \) is unchanged, the curve neither becomes wider nor taller; it simply moves horizontally.
5. Therefore, the correct interpretation is that increasing the mean shifts the normal curve to the right along the horizontal axis without changing its shape.