PSY 3200 Unit #3 AS: Calculating Percentages You Will Be Giv ✓ Solved

PSY 3200 Unit #3 AS: Calculating Percentages You will be give

You will be given a series of questions regarding a normal distribution. You will be asked to either determine the percentage above or below particular raw scores, or to calculate the raw score that will correspond to a particular percentage. You will be asked to calculate either raw scores or percentages. For each question write out your calculation, the appropriate Z score, what on the curve you should be shading, the exact percentage from the normal curve table, and the final answer. A group of students takes a Statistics Exam where the average was M = 90 and the standard deviation was SD = 7.8.

Answer the following questions regarding this distribution using your normal curve table. Depending on the problem, be sure to identify the raw scores, Z scores, what in particular you shaded when creating a curve and the final answer:

  1. If you score a 99, what percentage of the population scored above you?
  2. What raw score is needed to be in the top 8%?
  3. What raw score is needed to reach the 14th percentile?
  4. What range of raw scores makes up the middle 40%?
  5. If you score an 80, what percentage of the population scored above you?
  6. What percentage of the population lands between the raw scores of 82 and 107?
  7. If you score an 85, what percentage of the population scored above you?
  8. What raw score is needed to be in the bottom 16%?
  9. If you score a 96, what percentage of the population scored above you?
  10. What raw score is needed to be in the top 17%?
  11. What raw score is needed to reach the 90th percentile?
  12. What percentage of the population lands between the raw scores of 83 and 100?
  13. If you score an 85, what percentage of the population below you?
  14. What raw score is needed to be in the bottom 28%?
  15. What percentage of the population lands between the raw scores of 86 and 105?

Paper For Above Instructions

This paper aims to solve a series of problems related to a normal distribution where the average score (M) is 90 and the standard deviation (SD) is 7.8. In these calculations, we will be using the Z-score formula:

Z = (X - M) / SD

Where:

  • X = raw score
  • M = mean (90)
  • SD = standard deviation (7.8)

1. If you score a 99, what percentage of the population scored above you?

First, we calculate the Z-score for a raw score of 99:

Z = (99 - 90) / 7.8 = 1.15

Using the Z-table, a Z-score of 1.15 corresponds to a percentile of about 87.66%. Therefore, the percentage that scored above is:

100% - 87.66% = 12.34%

2. What raw score is needed to be in the top 8%?

For the top 8%, we look for a Z-score that corresponds to 92% (100% - 8%). The Z-table gives us a Z-score of approximately 1.41.

Using the Z-score formula, we can find the raw score:

X = Z SD + M = 1.41 7.8 + 90 = 101.00 (rounded to two decimal places)

3. What raw score is needed to reach the 14th percentile?

A Z-score corresponding to the 14th percentile is approximately -1.08.

Calculating the raw score:

X = -1.08 * 7.8 + 90 = 84.56 (rounded to two decimal places)

4. What range of raw scores makes up the middle 40%?

The middle 40% leaves 30% in the tails (15% each side). This corresponds to Z-scores of approximately -1.04 and +1.04.

Calculating the raw scores:

Lower bound: X = -1.04 * 7.8 + 90 = 84.87

Upper bound: X = 1.04 * 7.8 + 90 = 95.13

Range: 84.87 to 95.13.

5. If you score an 80, what percentage of the population scored above you?

Z = (80 - 90) / 7.8 = -1.28. From the Z-table, a Z-score of -1.28 corresponds to approximately 10.40%. Thus, the percentage scoring above 80 is:

100% - 10.40% = 89.60%

6. What percentage of the population lands between the raw scores of 82 and 107?

For 82: Z = (82 - 90) / 7.8 = -1.03 (corresponds to 15.87% below).

For 107: Z = (107 - 90) / 7.8 = 2.18 (corresponds to 98.59% below).

Percentage between 82 and 107: 98.59% - 15.87% = 82.72%.

7. If you score an 85, what percentage of the population scored above you?

Z = (85 - 90) / 7.8 = -0.64 (corresponds to about 26.72%). Thus, the percentage above 85 is:

100% - 26.72% = 73.28%

8. What raw score is needed to be in the bottom 16%?

For the bottom 16%, we look for a Z-score of approximately -0.99. Calculating the raw score:

X = -0.99 * 7.8 + 90 = 84.22

9. If you score a 96, what percentage of the population scored above you?

Z = (96 - 90) / 7.8 = 0.77 (corresponds to approximately 77.34%). The percentage above 96 is:

100% - 77.34% = 22.66%

10. What raw score is needed to be in the top 17%?

This corresponds to a Z-score of approximately 0.95:

X = 0.95 * 7.8 + 90 = 97.41

11. What raw score is needed to reach the 90th percentile?

The corresponding Z-score is approximately 1.28:

X = 1.28 * 7.8 + 90 = 101.00

12. What percentage of the population lands between the raw scores of 83 and 100?

For 83: Z = (83 - 90) / 7.8 = -0.90 (corresponds to roughly 18.81%). For 100: Z = (100 - 90) / 7.8 = 1.28 (corresponds to 90.72%). Thus, the percentage between 83 and 100 is:

90.72% - 18.81% = 71.91%

13. If you score an 85, what percentage of the population below you?

As calculated in previous problems, the percentage scoring below 85 is 26.72%.

14. What raw score is needed to be in the bottom 28%?

For bottom 28%, we find the corresponding Z-score of approximately -0.58:

X = -0.58 * 7.8 + 90 = 86.52.

15. What percentage of the population lands between the raw scores of 86 and 105?

For 86: Z = (86 - 90) / 7.8 = -0.51. For 105: Z = (105 - 90) / 7.8 = 1.92.

Corresponding percentages: 30.85% (for 86) and 97.09% (for 105). Thus, the percentage between 86 and 105 is:

97.09% - 30.85% = 66.24%

References

  • Aron, A., Coups, E.J., & Aron, E. (2013). Statistics for Psychology (6th ed.).
  • Nolan, S., & Heinzen, T. (2017). Statistics for the Behavioral Sciences (4th ed.).
  • Lecture: Z Scores and Probability.
  • American Psychological Association. (2020). Publication Manual of the American Psychological Association (7th ed.).
  • 30. Gibbons, J.D. (2015). Statistical Inference. SIAM.
  • Ghasemi, A. & Zahediasl, S. (2012). Normality Tests for Statistical Analysis: A Guide for Non-Statisticians. International Journal of Endocrinology and Metabolism, 10(2), 486–489.
  • DeCarlo, L.T. (1997). On the Meaning and Use of Kurtosis. Psychological Bulletin, 122(1), 1–5.
  • Field, A. (2013). Discovering Statistics Using IBM SPSS Statistics (4th ed.). Sage.
  • Tabachnick, B.G., & Fidell, L.S. (2013). Using Multivariate Statistics (6th ed.). Pearson.
  • Cohen, J. (1988). Statistical Power Analysis for the Behavioral Sciences (2nd ed.). Lawrence Erlbaum Associates.

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