Questions: Decide which of the following statements are true. - There are a limited number of normal distributions. - Normal distributions are always symmetric and bell-shaped. - The inflection points for any normal distribution are one standard deviation on either side of the mean. - The y-axis is a vertical asymptote for all normal distributions.

Solution Steps

To find the Z-score for the value \( X = 1 \) in a normal distribution with mean \( \mu = 0 \) and standard deviation \( \sigma = 1 \), we use the formula:

\[ z = \frac{X - \mu}{\sigma} = \frac{1 - 0}{1} = 1.0 \]

Thus, the Z-score is \( z = 1.0 \).

Next, we calculate the probability that the sample mean falls within the range \([-1, 1]\) for a normal distribution with mean \( \mu = 0 \) and standard deviation \( \sigma = 1 \). The probability is given by:

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

This indicates that the probability of the sample mean falling within this range is \( P = 0.6827 \).

We evaluate the following statements regarding normal distributions:

1. There are a limited number of normal distributions.
   This statement is False. There are infinitely many normal distributions, each defined by different means and standard deviations.

2. Normal distributions are always symmetric and bell-shaped.
   This statement is True. Normal distributions are characterized by their symmetry and bell shape.

3. The inflection points for any normal distribution are one standard deviation on either side of the mean.
   This statement is True. The inflection points occur at \( \mu - \sigma \) and \( \mu + \sigma \).

4. The y-axis is a vertical asymptote for all normal distributions.
   This statement is False. Normal distributions do not have vertical asymptotes; they extend infinitely in both directions.

Final Answer