In the $standardnormal$ distribution, what happens to the graph of the normal curve as the mean $\mu$ increases while the standard deviation $\sigma$ remains constant?

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

Answer: (B)

Verified step by step guidance

1. Recall that the standard normal distribution is a normal distribution with mean \(\mu = 0\) and standard deviation \(\sigma = 1\). The general normal distribution has mean \(\mu\) and standard deviation \(\sigma\).
2. Understand that the mean \(\mu\) determines the center or location of the normal curve along the horizontal axis (x-axis). Changing \(\mu\) shifts the curve left or right without altering its shape.
3. Recognize that the standard deviation \(\sigma\) controls the spread or width of the curve. Since \(\sigma\) remains constant, the shape (height and width) of the curve does not change.
4. Therefore, increasing the mean \(\mu\) moves the entire normal curve to the right along the x-axis, but the curve's height and width remain the same.
5. Conclude that the correct description is: the entire curve shifts to the right along the horizontal axis without changing its shape.