Introducing Statistics: Why Be Normal - AP Statistics

1. Find the z-score for $x = 15$, given $\text{μ} = 15$ and $\text{σ} = 5$.

   0. Using $z = \frac{15-15}{5} = 0$.

2. What is the function used to standardize a normal variable?

   $z = \frac{x - \text{μ}}{\text{σ}}$. Z-score formula converts any normal variable to standard normal.

3. What is a z-score?

   Standard score indicating deviation from mean. Measures how many standard deviations a value is from the mean.

4. What does the area under a normal distribution curve represent?

   Probability. Area under any probability density curve represents probability.

5. Find the area between $z = -2$ and $z = 2$ in a standard normal distribution.

   0.9545. This represents 95% of the distribution within two standard deviations.

6. What percentage of data falls within one standard deviation in a normal distribution?

   68%. First part of the 68-95-99.7 empirical rule.

7. Which z-score corresponds to the 5th percentile?

   -1.645. Critical z-value that leaves 5% in the lower tail.

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

   They are equal. Perfect symmetry makes all three measures of central tendency identical.

9. What does a z-score of 0 indicate?

   Value equals the mean. Zero z-score means the observation equals the population mean.

10. Find the area to the left of $z = 1$ in a standard normal distribution.

    0.8413. Standard normal table gives area left of $z = 1$.

11. Find the z-score for $x = 12$, given $\text{μ} = 10$ and $\text{σ} = 4$.

    0.5. Using $z = \frac{12-10}{4} = 0.5$.

12. State the property of symmetry in a normal distribution.

    Symmetrical about the mean. Equal areas on both sides of the mean create perfect symmetry.

13. Find the probability of $z < -2$ in a standard normal distribution.

    0.0228. Standard normal table gives area left of $z = -2$.

14. Find the z-score for $x = 7$, $\text{μ} = 5$, $\text{σ} = 1$.

    2. Using $z = \frac{7-5}{1} = 2$.

15. Find the z-score for $x = 5$, given $\text{μ} = 3$ and $\text{σ} = 2$.

    1. Using $z = \frac{5-3}{2} = 1$.

16. Find the area to the left of $z = -2$ in a standard normal distribution.

    0.0228. Standard normal table gives area left of $z = -2$.

17. Find the area to the right of $z = -1$ in a standard normal distribution.

    0.8413. Area to the right of $z = -1$ equals area to the left of $z = 1$.

18. What is the probability that a z-score is exactly 0 in a continuous normal distribution?

    0. Continuous distributions assign zero probability to any single point.

19. What is the effect of decreasing standard deviation on a normal curve?

    Narrows and steepens the curve. Smaller σ concentrates data more tightly around the mean.

20. What is the kurtosis of a normal distribution?

    3. Mesokurtic distribution with standard baseline kurtosis value.

21. What percentage of data falls within three standard deviations in a normal distribution?

    99.7%. Third part of the 68-95-99.7 empirical rule.

22. What is the cumulative distribution function (CDF) used for?

    To find probabilities. CDF gives probability that random variable is less than or equal to x.

23. Identify the range of the majority of data points in a normal distribution.

    Within $\mu \pm 3\sigma$. Three standard deviations captures 99.7% of all data points.

24. Identify the empirical rule for normal distributions.

    68-95-99.7 rule. Also known as the empirical rule for normal distribution percentages.

25. Find the probability of $z > 2$ in a standard normal distribution.

    0.0228. Using $1 - 0.9772 = 0.0228$ from standard normal table.

26. How does a normal distribution relate to the Central Limit Theorem?

    Sample means are normally distributed. CLT states that sample means approach normality regardless of population shape.

27. Identify the skewness of a normal distribution.

    Zero. Perfect symmetry means no skewness in either direction.

28. Identify the z-score for the mean in a standard normal distribution.

    0. The mean has zero deviation from itself by definition.

29. Find the area between $z = -1$ and $z = 1$ in a standard normal distribution.

    0.6826. This represents the middle 68% of the distribution.

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

    1. Standard normal distribution has unit variance and standard deviation.

31. What is the notation for a normal distribution with mean $\text{μ}$ and standard deviation $\text{σ}$?

    $N(\text{μ}, \text{σ})$. Standard notation where μ is mean and σ is standard deviation.

32. What is the effect of increasing standard deviation on a normal curve?

    Flattens and widens the curve. Larger σ spreads data more widely around the mean.

33. Which z-score corresponds to the 95th percentile?

    1.645. Critical z-value that leaves 5% in the upper tail.

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

    Bell-shaped. The normal curve resembles a bell due to its symmetric, unimodal shape.

35. Why is the normal distribution important in statistics?

    Many variables are approximately normal. Central Limit Theorem and many natural phenomena follow normal distributions.

36. What is the key characteristic of a normal distribution's tails?

    Asymptotic. Tails approach but never touch the horizontal axis.

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

    0. Standard normal distribution is centered at zero by definition.

38. Calculate the z-score for $x = 10$, $\text{μ} = 8$, $\text{σ} = 1.5$.

    1.33. Using $z = \frac{10-8}{1.5} = 1.33$.

39. What percentage of data falls within two standard deviations in a normal distribution?

    95%. Second part of the 68-95-99.7 empirical rule.

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