Z Scores 101
A student scores 78 on a test with mean 70 and SD 4. What is their z-score?
2.0
In a normal distribution, what percentile is approximately z = –1.00?
Approx. the 15.9th percentile - using the Z-score and your Table A, remember that percentile means <=, so we're finding the area to the left of -1.00. We find the z-score on the table and see that the area is 0.1587, so about the 15.9th percentile.
The total area under the density curve for a normal distribution is what?
1
For a normal distribution with mean 75 and standard deviation of 12, shade the middle 95% of the distribution. Between which two values will this middle region fall?
51 and 99 - remember, if they fall within the standard deviations, we can fall back on our empirical rule! 2 standard deviations on either side is 51 below the mean and 99 above the mean!
How many dogs does Ms. Wentworth have?
4! Reaper, Grim, Scythe, and Azazel!
If a z-score is negative, what does this tell you about the data point compared to the mean?
The data point is below the mean - if z = -1.2, then it is 1.2 standard deviations below the mean.
About what percent of values fall within 1 standard deviation of the mean in a normal distribution?
A distribution of reaction times is N(250,40). What proportion of reaction times are faster than 210 ms?
0.1587. Remember, we are being asked about the area under the curve (proportion), so we need the z-score. z = (210-250)/40 = -1.0. We want to know about times FASTER than 210 ms, so that would be to the LEFT of our z-score (smaller times means faster). Using our table, we see that area is 0.1587. Calculator: normalcdf(-10000, -1.0, 250,40)
Sketch N(50,5). Shade the region for x > 60. About what percent is shaded?
About 2.3% - we see that we're searching for the area above 60 (which we determine the z-score to be 2.0), so we'll use the tail part of Table A and find that area to be 0.0228.
Who played V in V for Vendetta?
Hugo Weaving - duh.
The mean = 84 and SD = 6. Which score has z = 2.0?
96 - because if the z-score is 2.0, then it is 2 standard deviations above the mean. 84 + 6 = 90, and 90 + 6 = 96
What z-score corresponds to the 90th percentile?
A distribution of recorded commute times for employees at Company A is N(32,5). What commute time is at the 10th percentile?
Approx. 25. 6 minutes. The 10th percentile gives us a z-score of ≈ −1.28 → x = 32 + (−1.28)(5) ≈ 25.6 minutes.
What is true for all normal distributions regarding their area?
The total area under a normal curve = 1.
What are the names of Hades minions in Hercules?
Pain and Panic - "If...if is good."
The 90th percentile of a normal distribution corresponds most closely to which z-score?
1.28 - find the area of 0.9 on Table A(continued), and it corresponds to a z-score of 1.28.
A runner finishes 1.25 SD below the mean. About what percentile are they in?
Approx. 10.6th percentile. Remember, a z-score tells us how many standard deviations from the mean in a normal distribution we are. So, our z-score is 1.25. We know percentile means <=, so we're finding a left tail area. The corresponding area to that z-score is approx. 0.1056.
A sample of 100 boxes of cereal weights' are N(16,0.3). What proportion is greater than 15.5 oz?
0.9525 (or about 95.3%) - we're finding the area (proportion), so we first find our z-score: 15.5-16 = -0.5/0.3 = -1.67. Use the table and we're trying to find the big area (below the mean and all the way to the right) - oh no! Our table shows it going the other way and the values are positive? That's ok because of symmetry! So, our z-score of -1.67 will be positive on this and the area associated with it is 0.9525.
The distribution of sale prices for four-year-old Harley Davidson touring motorcycles is approximately normally distributed with a mean of $14,000 and a standard deviation of $4000. Akira plans to spend between $8000 and $12,000 on a motorcycle. What proportion of the motorcycles of this type are within his budget?
0.2417 - if we draw this out we see we're looking for a boundary so we'll have to find the bigger area and then subtract the smaller area. Or, we can use our calculator - normalcdf(8000, 12000, 14000, 4000) or standardized: normalcdf(-1.5, -0.5, 0, 1)
Who was the founder of house Slytherin?
Salazar Slytherin - duh.
In a normal distribution, approximately what percent of observations are more than 2 standard deviations away from the mean (in either direction)?
5% - because 95% of the observations fall WITHIN 2 standard deviations, so only 5% falls outside of (more than) 2 standard deviations.
Here are the exam scores for the 15 students in Mrs. Stevenson’s statistics class:
72 75 75 78 81 81 85 89 90 90 90 91 95 95 98
Zaphod is at the 80th percentile of the distribution. What score did Spencer earn on the exam?
91. There are 15 data values - 15*0.8 = 12. So, Zaphod's score is 12th on the list, which is a score of 91.
A sample of reading test scores for third-graders in City A are N(75,10), while the sixth-graders' scores are N(82,11). Eileen the 3rd grader scores 78, and Vicki the 6th grader scores 84. Who is higher relative to their grade?
Eileen is higher than Vicki. Find each z-score. Eileen's z-score is (78-75)/10 = 0.3 and Vicki's z-score is (84-82)/11 = 0.18. So, Eileen's score is farther from the mean, meaning that her grade is higher relative to her grade's distribution.
An SAT section has scores ~N(500, 100). Shade the area representing the middle 50% of scores. What are the approximate cutoff scores for this range?
433 and 567. The middle 50% is between the 25th and 75th percentiles - -0.67 z-score for the 25th percentile, and +0.67 for the 75th percentile. Let's convert those back to SAT scores! Mean +- sd(z-score) = 500 +- 100(+-0.67) = 433 and 567.
"The last ever Dolphin message, misinterpreted as a surprisingly sophisticated attempt to do a double backwards somersault through a hoop while whistling the 'Star Spangled Banner'", actually means what?
"So long, and thanks for all the fish." - from the best book ever written, The Hitchhiker's Guide to the Galaxy, by Douglas Adams.