The Normal Distribution - AP Statistics

1. What is the mode of a normal distribution with mean $\nu$?

   $\nu$. Mode equals mean in normal distributions.

2. State the property of normal distribution regarding mean and spread.

   Mean determines center, standard deviation determines spread. Parameters control location and width of distribution.

3. What is the empirical rule in the context of normal distribution?

   68-95-99.7 rule. Also known as the three-sigma rule.

4. What does the Z-score represent in a normal distribution?

   Number of standard deviations from the mean. Measures distance from mean in standard deviation units.

5. What is the median of a normal distribution with mean $\nu$?

   $\nu$. Median equals mean due to symmetry.

6. Identify the symmetry property of a normal distribution curve.

   Symmetric about the mean. Normal curves are perfectly symmetric around their center.

7. Find the probability $P(-1 < Z < 1)$ in a standard normal distribution.

   0.6826. 68% rule applied to standard normal.

8. What property describes the tails of a normal distribution?

   Asymptotic. Tails approach but never touch the x-axis.

9. What is the skewness of a normal distribution?

   0 (perfectly symmetrical). No skew means perfectly balanced distribution.

10. How are the mean, median, and mode related in a normal distribution?

    They are equal. Symmetry ensures all central tendencies coincide.

11. What is the standard deviation of a standard normal distribution?

    1. Standard normal has unit variance.

12. What is the total area under a normal distribution curve?

    1. Probability density functions always integrate to 1.

13. Determine the area to the left of $Z = 0$ in a standard normal distribution.

    0.5. Standard normal is symmetric around zero.

14. What type of variable does a normal distribution model?

    Continuous random variable. Normal distributions apply to continuous data only.

15. What is the purpose of normality tests in statistics?

    Assess if data follows a normal distribution. Tests check if sample data appears normally distributed.

16. What percentage of data falls within one standard deviation of the mean in a normal distribution?

    68%. Empirical rule for one standard deviation interval.

17. What percentage of data falls within three standard deviations of the mean in a normal distribution?

    99.7%. Empirical rule for three standard deviation interval.

18. What is the mean of a standard normal distribution?

    0. Standard normal is centered at zero.

19. What percentage of data falls within two standard deviations of the mean in a normal distribution?

    95%. Empirical rule for two standard deviation interval.

20. What is the shape of a normal distribution curve?

    Bell-shaped. Classic shape of normal distribution curves.

21. What is the main use of the standard normal distribution table?

    Find probabilities for Z-scores. Standard tables give cumulative probabilities for Z-values.

22. What is the relationship between variance and standard deviation?

    Variance is the square of the standard deviation. Variance is standard deviation squared.

23. Determine the area to the right of $Z = 1.96$ in a standard normal distribution.

    0.025. Critical value leaving 2.5% in upper tail.

24. What is the significance of the Central Limit Theorem in normal distribution?

    Sample means are normally distributed. CLT ensures sampling distributions approach normality.

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