PSY 315 Week 2 Probability and the Normal Distribution Example

How Unusual Is That Score? Probability, z-Scores and the Normal Curve in a Student Survey

[Student Name]

University of Phoenix

PSY/315: Statistical Reasoning in Psychology

Week 2 Assignment

[Instructor Name]

[Date]

The survey, the students and all numbers are composites written for a model paper; statistical guidance comes from the sources listed.

Week 1 described what the class survey looked like. This week asks a different question: how likely or unusual is a given result? Probability gives a common scale for that question, and the normal distribution lets researchers turn a raw score into a statement about where it falls among everyone else. These tools are the base on which every significance test in later weeks is built.

The Data Again

The survey covered 120 students in three sections of an introductory course at a community college in Phoenix, Arizona. They reported hours of sleep on a typical school night, cups of caffeinated drinks per day, stress on a 1 to 10 scale and their latest exam score out of 100. Sleep had a mean of 6.4 hours with a standard deviation of 1.1; exam scores had a mean of 76.3 with a standard deviation of 11.2; caffeine was strongly right-skewed.

Simple Probability

A probability is the proportion of times an outcome occurs among all possible outcomes. Drawing one name at random from all 120, the chance of landing on someone who rated stress at 7 or higher would be 45 divided by 120, or .375. The chance of picking someone who slept fewer than six hours would be 42 divided by 120, or .35. Probabilities run from 0, impossible, to 1, certain, and Gravetter and Wallnau (2017) stress that they can be read as proportions of a population.

Joint and Conditional Probability

A cross-tabulation of sleep and stress shows how the two variables combine. Of the 42 short sleepers, 26 reported high stress; of the 78 students who slept six hours or more, 19 did. The joint probability of being both a short sleeper and highly stressed is 26 out of 120, or .22.

Conditional probability narrows the denominator to one group. Among short sleepers, the probability of high stress is 26 divided by 42, or .62. Among longer sleepers, it is 19 divided by 78, or .24. The difference is striking, but it describes an association in one sample. Stress might shorten sleep as easily as short sleep raises stress, and later weeks will test whether a gap this size could arise by chance.

Thinking in Natural Frequencies

Conditional probabilities are easy to misread when they are given as percentages. Gigerenzer and Hoffrage (1995) found that people solved problems of this type far more accurately when the information was presented as counts, such as 80 out of 100, rather than as single-event probabilities, even without training in statistics.

Consider a brief anxiety screener a campus counseling center might use. Suppose 10 percent of students have clinically significant anxiety, the screener flags 80 percent of those who do and it also flags 15 percent of those who do not. Written as counts for 1,000 students, 100 have the condition and 80 of them are flagged, while 900 do not and 135 of them are flagged anyway. Of the 215 flagged students, only 80, or about 37 percent, actually have the condition. Many readers expect a figure near 80 percent; the counts show why a positive screen should lead to a fuller assessment, not a diagnosis.

Common Errors in Judging Chance

Tversky and Kahneman (1974) described shortcuts people use when judging likelihood, such as treating a result as probable because it resembles a familiar pattern, and showed that these shortcuts produce predictable errors. One is ignoring sample size: people expect a small group to mirror the whole population as closely as a large one does. In the survey, one section of 30 might average 80 on the exam while another averages 73, and an instructor might decide the first group studied harder. The sampling section below shows that gaps like this are expected even when nothing differs.

A screener that catches 80 percent of anxious students can still be wrong about most of the students it flags.

The Normal Distribution

Many psychological measures, including the exam scores here, form a symmetric, bell-shaped curve in which most cases cluster near the mean. In a normal distribution, about 68 percent of scores lie inside one standard deviation on either side of the mean, about 95 percent within two and nearly all within three. Because the shape is fixed, the proportion of scores in any region can be found from a table or software once scores are expressed in standard units.

Converting Scores to z-Scores

A z-score tells how many standard deviations a score lies from the mean: subtract the mean and divide by the standard deviation. Jordan, a student who scored 90, has a z-score of (90 minus 76.3) divided by 11.2, or 1.22. The normal table shows that about 89 percent of scores fall below that point, so Jordan sits near the 89th percentile and roughly 11 percent of the class would be expected to score higher. A score of 60 converts to a z-score of negative 1.46, with only about 7 percent of scores below it.

The same approach works for sleep. Five hours of sleep converts to a z-score of negative 1.27, and the curve predicts that about 10 percent of students sleep less than that. The survey found 13 such students, 10.8 percent, close to the prediction.

When the Curve Does Not Fit

Caffeine is a warning. With a mean of 2.1 cups and a long right tail, the normal model would predict that some students drink fewer than zero cups, which is impossible, and it would underestimate how many drink six or more. For skewed variables, percentiles should come from the data themselves rather than from the normal table.

Sampling Distributions and the Standard Error

Individual scores vary widely, but averages of groups vary much less. If many samples of 30 students were drawn from the same population, their means would form their own distribution, the sampling distribution of the mean. Its spread, the standard error, is found by dividing the standard deviation by the root of the sample size: 11.2 divided by the square root of 30, or about 2.04 points. The central limit theorem adds that this distribution of means becomes close to normal as samples grow, even when individual scores are somewhat skewed.

This explains the section gap. A section mean of 80 is 3.7 points above 76.3, a z-score of about 1.81 on the sampling distribution, and a mean that high or higher would occur by chance in about 3.5 percent of samples. That is uncommon but not impossible, which is exactly the kind of judgment hypothesis testing formalizes next week.

Conclusion

Probability turns counts into comparable proportions, conditional probability shows how one variable changes the chances of another and z-scores locate any score on the normal curve. Natural frequencies make screening results easier to read, the standard error explains why group means are steadier than single scores and a check on shape guards against using the normal model where it does not belong.

References

Gigerenzer, G., & Hoffrage, U. (1995). How to improve Bayesian reasoning without instruction: Frequency formats. Psychological Review, 102(4), 684-704. https://doi.org/10.1037/0033-295X.102.4.684

Gravetter, F. J., & Wallnau, L. B. (2017). Statistics for the behavioral sciences (10th ed.). Cengage Learning.

Tversky, A., & Kahneman, D. (1974). Judgment under uncertainty: Heuristics and biases. Science, 185(4157), 1124-1131. https://doi.org/10.1126/science.185.4157.1124