Let's analyze the normal distribution graph and answer the questions.

From the image:

• The distribution is centered at 89, which is the mean.
• The distances between consecutive points (68, 75, 82, 89, 96, 103, 110) show a gap of 7, which indicates the standard deviation.

Now let's fill in the blanks:

a. The mean is 89.

b. The standard deviation is 7.

For the remaining questions:

• 68% of the data lies within 1 standard deviation of the mean. So, the values are between:

  • $89 - 7 = 82$
  • $89 + 7 = 96$

  Therefore, c. 68% of the data values are between 82 and 96.

• 95% of the data lies within 2 standard deviations of the mean. So, the values are between:

  • $89 - 2(7) = 75$
  • $89 + 2(7) = 103$

  Therefore, d. 95% of the data values are between 75 and 103.

• 99.7% of the data lies within 3 standard deviations of the mean. So, the values are between:

  • $89 - 3(7) = 68$
  • $89 + 3(7) = 110$

  Therefore, e. 99.7% of the data values are between 68 and 110.

Would you like more details or explanations on any part?

Here are 5 related questions for further practice:

1. How would you calculate the mean and standard deviation if the values were not given graphically?
2. What is the empirical rule, and how does it apply to normal distributions?
3. If the standard deviation were smaller, how would the graph of the normal distribution change?
4. How does a skewed distribution differ from a normal distribution?
5. What is the probability of a data point falling more than 3 standard deviations from the mean in a normal distribution?

Tip: The normal distribution is symmetric, so its mean, median, and mode are all equal.