Let's analyze the graph and the questions.

1. What is the mean?

The mean of a normally distributed variable is located at the peak of the bell curve. From the graph, the peak appears to be at 50, so the mean is 50.

2. What is the standard deviation?

The standard deviation is the measure of the spread of the data. In a normal distribution, 68% of the values lie within one standard deviation from the mean (between $\mu - \sigma$ and $\mu + \sigma$).

From the graph:

• The value at the mean (50) is clearly visible.
• One standard deviation to the right (first significant point past the mean) is at 54.
• One standard deviation to the left is at 46 (as the graph appears symmetric).

So, the standard deviation $\sigma$ is $54 - 50 = 4$.

3. What percentage of values should fall between 54 and 60?

In a normal distribution:

• 68% of values fall within one standard deviation (i.e., between 46 and 54).
• Between 54 and 60, you're looking at the right side of the curve, beyond one standard deviation. This is the area between one standard deviation and approximately 1.5 standard deviations (as 60 is about 1.5 standard deviations from the mean).

Using standard normal distribution tables, the area between the mean and one standard deviation is about 34%, and between one and 1.5 standard deviations is about 9.19%. So approximately 9.19% of values should fall between 54 and 60.

4. Which value has a z-score of -2?

The z-score is calculated as: $z = \frac{x - \mu}{\sigma}$ For a z-score of -2: $-2 = \frac{x - 50}{4}$ Solving for $x$: $x - 50 = -8 \quad \Rightarrow \quad x = 50 - 8 = 42$ So, the value with a z-score of -2 is 42.

Would you like further details or explanations on any part of this? Here are 5 related questions to expand on this:

1. How do you find the area under a normal curve for different ranges?
2. How do you calculate z-scores for values not listed on the graph?
3. Why does 68% of data fall within one standard deviation in a normal distribution?
4. How can we interpret percentages from the z-score table in normal distributions?
5. How do we estimate the probability for values beyond 2 standard deviations?

Tip: Always sketch a standard normal curve to visualize how z-scores correspond to percentages when solving related questions.