Based on the normal distribution graph you provided, let's address the questions one by one.

1. What is the mean?

• From the graph, the mean is located at the peak of the bell curve, which appears to be 54. Therefore, 54 should be the correct value for the mean (not 54.14 as previously entered).

2. What is the standard deviation?

• By observing the spread of the data, one standard deviation from the mean seems to extend from 54 to 63 (or from 54 to 45 on the left). Hence, the standard deviation would be 9 (not 9.23 as entered).

3. What percentage of values should fall between 54 and 59?

• To calculate this, we need to find the z-scores for 54 and 59 and look up the area under the normal curve:
  • For $z = \frac{x - \mu}{\sigma}$:
    • $z_{54} = \frac{54 - 54}{9} = 0$
    • $z_{59} = \frac{59 - 54}{9} = 0.56$

Using a z-table, the area to the left of $z = 0$ is 0.5000, and the area to the left of $z = 0.56$ is approximately 0.7123.

Thus, the percentage of values between 54 and 59 is: $0.7123 - 0.5000 = 0.2123 \text{ or } 21.23\%$

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

• Using the z-score formula: $z = \frac{x - \mu}{\sigma}$ where $z = -2$, $\mu = 54$, and $\sigma = 9$: $-2 = \frac{x - 54}{9}$ Solving for $x$: $x = -2 \times 9 + 54 = 36$ Thus, the value corresponding to a z-score of -2 is 36.

Would you like more detailed steps or clarifications on any part of this? Here are some related questions for further practice:

1. What percentage of values fall between 45 and 54?
2. What is the z-score for a value of 60?
3. How much area lies under the normal curve for a z-score greater than 1.5?
4. How do you calculate the probability of a value being less than 49?
5. How does changing the standard deviation affect the shape of the normal curve?

Tip: For normal distributions, about 68% of values fall within one standard deviation of the mean, and 95% fall within two standard deviations.