In a normal distribution, what effect does increasing the $\mathrm{mean}$ have on the graph of the normal curve?

• A. The curve becomes taller and narrower.
• B. The curve becomes shorter and wider.
• C. The curve shifts upward along the vertical axis.
• D. The entire curve shifts to the right along the horizontal 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 described by the probability density function: \[ f(x) = \frac{1}{\sigma \sqrt{2\pi}} e^{-\frac{1}{2} \left( \frac{x - \mu}{\sigma} \right)^2} \]
3. When the mean \mu increases, it shifts the center of the distribution to the right along the horizontal axis (x-axis). This means the peak of the curve moves horizontally but does not affect the height or width of the curve.
4. Note that changing the mean does not affect the standard deviation \sigma, so the shape of the curve (its height and width) remains the same. The curve simply translates horizontally.
5. Therefore, increasing the mean results in the entire normal curve shifting to the right along the horizontal axis without changing its shape.