The Normal Distribution, Revisited - AP Statistics

1. What does a z-score represent?

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

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

   A symmetric, bell-shaped curve. Characteristic appearance of normal probability distributions.

3. What is the relationship between mean, median, and mode in a normal distribution?

   They are equal. Symmetry property of normal distributions.

4. Determine the probability of $z < -2$.

   Approximately 2.28%. Lower tail probability beyond two standard deviations.

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

   The mean is 0. Defining characteristic of the standard normal distribution.

6. Describe the tails of a normal distribution.

   The tails are asymptotic to the x-axis. Approach but never touch the horizontal axis.

7. Find the z-score for the 75th percentile.

   $z \approx 0.675$. Third quartile z-score in standard normal distribution.

8. Identify the symbol for the mean in a normal distribution.

   $\mu$ (mu). Greek letter representing population mean parameter.

9. Calculate the probability of $z > 1.96$.

   Approximately 2.5%. Upper tail beyond critical value for 95% confidence.

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

    The total area is 1. Property of all probability distributions.

11. What does the term 'standard normal distribution' mean?

    Normal distribution with $\mu = 0$, $\sigma = 1$. Normalized version with mean 0 and standard deviation 1.

12. What is the complementary probability for $z > 1.645$?

    5%. Upper tail probability beyond 1.645 standard deviations.

13. What is the probability of $z < -3$?

    Approximately 0.13%. Lower tail probability beyond three standard deviations.

14. What is the probability of $z < 0.33$?

    Approximately 62.97%. Cumulative probability up to 0.33 standard deviations.

15. Find the x-value for $z = -1$, $\mu = 100$, $\sigma = 10$.

    $x = 90$. Solving $-1 = \frac{x-100}{10}$ gives $x = 90$.

16. What proportion of data falls within 2 standard deviations in a normal distribution?

    Approximately 95%. From the empirical rule for normal distributions.

17. Determine the interquartile range (IQR) for a standard normal distribution.

    IQR = $1.349$. Distance between 75th and 25th percentiles.

18. What percentage of data is beyond $z = 3$?

    Approximately 0.13%. Combined tail areas beyond three standard deviations.

19. Identify the symbol for standard deviation in a normal distribution.

    $\sigma$ (sigma). Greek letter representing population standard deviation parameter.

20. Calculate the probability for $1 < z < 2$.

    Approximately 13.59%. Area between one and two standard deviations above mean.

21. What is the cumulative distribution function (CDF) at $z = 0$?

    0.5. At mean, exactly half the data is below.

22. Find the probability of $z = 0.84$.

    Approximately 79.93%. Cumulative probability up to 0.84 standard deviations.

23. What proportion of data falls within 3 standard deviations in a normal distribution?

    Approximately 99.7%. From the empirical rule for normal distributions.

24. What is the skewness of a normal distribution?

    The skewness is 0. Normal distributions are perfectly symmetric, not skewed.

25. Find the z-score for $x = 92$, $\mu = 100$, $\sigma = 8$.

    $z = -1$. Using $z = \frac{92-100}{8} = -1$.

26. Find the z-score for $x = 85$, $\mu = 80$, $\sigma = 5$.

    $z = 1$. Using $z = \frac{85-80}{5} = 1$.

27. State the empirical rule for a normal distribution.

    68%-95%-99.7% of data within 1, 2, 3 standard deviations. Also known as the 68-95-99.7 rule for normal distributions.

28. What is the z-score formula?

    $z = \frac{x - \mu}{\sigma}$. Standardizes values by subtracting mean and dividing by standard deviation.

29. Identify the symmetry property of a normal distribution.

    Symmetric around the mean. Mean divides distribution into two equal halves.

30. Find the z-score for the 25th percentile.

    $z \approx -0.675$. First quartile z-score in standard normal distribution.

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

    The standard deviation is 1. Standard normal has unit variance and standard deviation.

32. What z-score corresponds to the 90th percentile?

    $z \approx 1.28$. Critical value for 90% of data below this point.

33. What proportion of data falls within 1 standard deviation in a normal distribution?

    Approximately 68%. From the empirical rule for normal distributions.

34. What is the probability of $z < -1.96$ or $z > 1.96$?

    Approximately 5%. Combined tail probabilities beyond critical values.

35. Given $z = 2$, what percentage of data is below this z-score?

    Approximately 97.5%. Two standard deviations above mean in standard normal distribution.

36. Convert $z = 0.5$ to a percentage of data below.

    Approximately 69.15%. Half standard deviation above mean corresponds to this percentile.

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