Questions: What is the difference between a standard normal distribution and a nonstandard normal distribution? Choose the correct answer below. A. The standard normal distribution has a standard deviation that is greater than the mean, while a nonstandard normal distribution has a standard deviation that is less than or equal to the mean. B. The standard normal distribution has a standard deviation that is less than or equal to the mean, while a ponstandard normal distribution has a standard deviation that is greater than the man. C. The standard normal distribution has a mean of 0 and a standard deviation of 1 , while a nonstandard normal distribution has a different value for one or both of those parameters. D. The standard normal distribution has a mean of 1 and a standard deviation of 0 , while a nonstandard normal distribution has a different value for one or both of those parameters.

Solution Steps

The standard normal distribution is defined by its mean \( \mu = 0 \) and standard deviation \( \sigma = 1 \). We calculated the probability that a sample mean falls within the range \([-1, 1]\):

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

This indicates that approximately \( 68.27\% \) of the data falls within one standard deviation of the mean in a standard normal distribution.

For a nonstandard normal distribution, we considered a mean \( \mu = 5 \) and standard deviation \( \sigma = 2 \). We calculated the probability that a sample mean falls within the range \([4, 6]\):

\[ P = \Phi(Z_{end}) - \Phi(Z_{start}) = \Phi(0.5) - \Phi(-0.5) = 0.3829 \]

This indicates that approximately \( 38.29\% \) of the data falls within this range for the nonstandard normal distribution.

The key difference between the two distributions is that the standard normal distribution has fixed parameters \( \mu = 0 \) and \( \sigma = 1 \), while the nonstandard normal distribution can have any mean and standard deviation. Therefore, the correct answer to the question is:

The standard normal distribution has a mean of \( 0 \) and a standard deviation of \( 1 \), while a nonstandard normal distribution has a different value for one or both of those parameters.

Final Answer