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Quiz: Standardization and Normal Distributions

Test your understanding of z-scores, the normal distribution, and the Empirical Rule with these review questions.

1. A student scored 72 on a test where the mean was 80 and the standard deviation was 8. What is this student's z-score?

  1. -8
  2. -1
  3. 0.5
  4. 1
Show Answer

The correct answer is B. Using the z-score formula: \( z = \frac{x - \mu}{\sigma} = \frac{72 - 80}{8} = \frac{-8}{8} = -1 \). This means the student scored exactly one standard deviation below the mean. The negative sign indicates the score is below average.

Concept Tested: Calculating Z-Scores

2. What does it mean when a data value has a z-score of 0?

  1. The value is an outlier
  2. The value equals the standard deviation
  3. The value equals the mean
  4. The value cannot be compared to other values
Show Answer

The correct answer is C. A z-score of 0 means the value is exactly at the mean. When you substitute into the z-score formula, if x equals the mean, the numerator becomes zero, making the entire z-score equal to zero. This is true regardless of the standard deviation.

Concept Tested: Interpreting Z-Scores

3. According to the Empirical Rule (68-95-99.7 Rule), what percentage of data in a normal distribution falls within two standard deviations of the mean?

  1. 68%
  2. 95%
  3. 99.7%
  4. 100%
Show Answer

The correct answer is B. The Empirical Rule states that for normal distributions, approximately 68% of data falls within 1 standard deviation, 95% within 2 standard deviations, and 99.7% within 3 standard deviations of the mean. This leaves about 5% in the tails (2.5% in each tail).

Concept Tested: Empirical Rule (68-95-99.7 Rule)

4. Which of the following is NOT a property of all normal distributions?

  1. They are symmetric about the mean
  2. The mean, median, and mode are all equal
  3. The total area under the curve equals 1
  4. They have a standard deviation of 1
Show Answer

The correct answer is D. Normal distributions can have any positive standard deviation. Only the standard normal distribution has a standard deviation of 1. All normal distributions share the other properties: symmetry about the mean, equal mean/median/mode, and total area of 1 under the curve.

Concept Tested: Parameters of Normal Distribution

5. Student A scored 85 on a test with mean 78 and standard deviation 7. Student B scored 92 on a different test with mean 88 and standard deviation 8. Which student performed better relative to their class?

  1. Student A, because 85 is closer to the mean
  2. Student A, because z = 1.0 is greater than z = 0.5
  3. Student B, because 92 is a higher raw score
  4. They performed equally well
Show Answer

The correct answer is B. Student A's z-score: \( z = \frac{85-78}{7} = 1.0 \). Student B's z-score: \( z = \frac{92-88}{8} = 0.5 \). Student A performed one full standard deviation above the mean, while Student B was only half a standard deviation above. Raw scores can't be compared directly across different tests; z-scores allow fair comparison.

Concept Tested: Comparing with Z-Scores

6. On a normal probability plot (QQ plot), what pattern indicates that the data is approximately normally distributed?

  1. Points follow a curved pattern
  2. Points are scattered randomly
  3. Points follow approximately a straight line
  4. Points cluster at the extremes
Show Answer

The correct answer is C. A normal probability plot graphs each data value against the z-score it would have if the data were perfectly normal. When data is approximately normal, the points fall along a straight line. Curved patterns indicate skewness, while scattered points suggest the data doesn't follow any standard distribution.

Concept Tested: Normal Probability Plot and Assessing Normality

7. If the area under a normal curve to the left of z = 1.5 is 0.9332, what is the area to the right of z = 1.5?

  1. 0.0668
  2. 0.4332
  3. 0.5000
  4. 0.9332
Show Answer

The correct answer is A. Since the total area under any density curve equals 1, the area to the right equals 1 minus the area to the left: 1 - 0.9332 = 0.0668. This means about 6.68% of values in a standard normal distribution are greater than z = 1.5.

Concept Tested: Finding Normal Probabilities

8. Heights of adult women are normally distributed with mean 64.5 inches and standard deviation 2.5 inches. What height marks the 90th percentile (the top 10%)?

  1. 61.3 inches
  2. 64.5 inches
  3. 67.7 inches
  4. 70.0 inches
Show Answer

The correct answer is C. For the 90th percentile, we need the z-score with area 0.90 to its left, which is approximately z = 1.28. Converting back to the original scale: x = 64.5 + (1.28)(2.5) = 64.5 + 3.2 = 67.7 inches. This is an inverse normal calculation where we work backwards from probability to value.

Concept Tested: Inverse Normal Calculations

9. A density curve is drawn so that the area between x = 10 and x = 20 is 0.35. What does this value represent?

  1. The mean of the distribution is 0.35
  2. 35% of observations fall between 10 and 20
  3. The height of the curve at x = 15 is 0.35
  4. The standard deviation is 0.35
Show Answer

The correct answer is B. For density curves, area under the curve represents proportion or probability. An area of 0.35 between x = 10 and x = 20 means that 35% of all observations in this distribution fall within that range. This is a fundamental property of all density curves.

Concept Tested: Density Curve and Area Under Curve

10. SAT scores are approximately normal with mean 1060 and standard deviation 195. Using the Empirical Rule, approximately what percentage of students score between 865 and 1255?

  1. 34%
  2. 47.5%
  3. 68%
  4. 95%
Show Answer

The correct answer is C. First, calculate how many standard deviations these boundaries are from the mean: 1060 - 195 = 865 and 1060 + 195 = 1255. These are exactly one standard deviation below and above the mean. According to the Empirical Rule, 68% of data falls within one standard deviation of the mean.

Concept Tested: Applying the Empirical Rule