Questions: True or false: The mean of a normal distribution has no effect on its spread. Explain your answer. Choose the correct answer below. A. True. The spread of a normal distribution is completely determined by its standard deviation. B. False. The spread of a normal distribution is completely determined by its mean. C. True. The spread of a normal distribution is completely determined by its median. D. False. The spread of a normal distribution is determined by both its mean and standard deviation.

Solution Steps

For the given range of \([-1, 1]\), we calculate the Z-scores corresponding to the lower and upper bounds:

\[ Z_{start} = \frac{-1 - \mu}{\sigma} = \frac{-1 - 0}{1} = -1.0 \]

\[ Z_{end} = \frac{1 - \mu}{\sigma} = \frac{1 - 0}{1} = 1.0 \]

Using the Z-scores, we find the probability that the sample mean falls within the specified range:

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(1.0) - \Phi(-1.0) = 0.6827 \]

The mean (\(\mu\)) of a normal distribution shifts the center of the distribution but does not affect its spread. The spread is determined solely by the standard deviation (\(\sigma\)). Therefore, the statement that the mean has no effect on the spread is true.

Final Answer