AP Statistics

Standard Deviation

The context typically varies by SD from the mean of mean.

Example: The height of power forwards in the NBA typically varies by 1.52 inches from the mean of 80.1 inches.

Percentile

percentile % of context are less than or equal to value.

Example: 75% of high school student SAT scores are less than or equal to 1200.

z-score

Specific value with context is z-score standard deviations above/below the mean.

Example: A quiz score of 71 is 1.43 standard deviations below the mean. (z = -1.43)

Describe a distribution

Be sure to address shape, center, variability, and outliers (in context).

Example: The distribution of student height is unimodal and roughly symmetric. The mean height is 65.3 inches with a standard deviation of 8.2 inches. There is a potential upper outlier at 79 inches and a gap between 60 and 62 inches.

Correlation (r)

The linear association between x-context and y-context is weak/moderate/strong (strength) and positive/negative (direction).

Example: The linear association between student absences and final grades is fairly strong and negative. (r = −0.93)

Residual

The actual y-context was residual above/below the predicted value when x-context = #.

Example: The actual heart rate was 4.5 beats per minute above the number predicted when Matt ran for 5 minutes.

y-intercept

The predicted y-context when x = 0 context is y-intercept.

Example: The predicted time to checkout at the grocery store when there are 0 customers in line is 72.95 seconds.

slope

The predicted y-context increases/decreases by slope for each additional x-context.

Example: The predicted heart rate increases by 4.3 beats per minute for each additional minute jogged.

Standard Deviation of Residuals (s)

The actual y-context is typically about s away from the value predicted by the LSRL.

Example: The actual SAT score is typically about 14.3 points away from the value predicted by the LSRL.

Coefficient of Determination (r²)

About r²% of the variation in y-context can be explained by the linear relationship with x-context.

Example: About 87.3% of variation in electricity production is explained by the linear relationship with wind speed.

Describe the relationship

Be sure to address strength, direction, form and unusual features (in context).

Example: The scatterplot reveals a moderately strong, positive, linear association between the weight and length of rattlesnakes. The point at (24.1, 35,7) is a potential outlier.

Probability P(A)

After many many context, the proportion of times that context A will occur is about P(A).

Example: P(heads) = 0.5. After many many coin flips, the proportion of times that heads will occur is about 0.5.

Conditional Probability P (A|B)

Given context B, there is a P(A|B) probability of context A.

Example: P(red car | pulled over) = 0.48. Given that a car is pulled over, there is a 0.48 probability of the car being red.

Expected Value (Mean, μ):

If the random process of context is repeated for a very large number of times, the average number of x-context we can expect is expected value. (decimals OK).

Example: If the random process of asking a student how many movies they watched this week is repeated for a very large number of times, the average number of movies we can expect is 3.23 movies.

Binomial Mean

After many, many trials the average # of success context out of n is After many, many trials the average # of success context out of n is μ.

Example: After many, many trials the average # of property crimes that go unsolved out of 100 is 80.

Binomial Standard Deviation

The number of success context out of n typically varies by σ from the mean of μ.

Example: The number of property crimes that go unsolved out of 100 typically varies by 1.6 crimes from the mean of 80 crimes.

Standard Deviation of Sample Proportions

The sample proportion of success context typically varies by σ from the true proportion of p.

Example: The sample proportion of students that did their AP Stats homework last night typically varies by 0.12 from the true proportion of 0.73.

Standard Deviation of Sample Means

The sample mean amount of x-context typically varies by σ from the true mean of μ

Example: The sample mean amount of defective parts typically varies by 5.6 parts from the true mean of 23.2 parts.

Confidence Interval (A, B)

We are % confident that the interval from A to B captures the true parameter context.

Example: We are 95% confident that the interval from 0.23 to 0.27 captures the true proportion of flowers that will be red after cross-fertilizing red and white.

Confidence Level

If we take many, many samples of the same size and calculate a confidence interval for each, about confidence level % of them will capture the true parameter in context

Example: If we take many, many samples of size 20 and calculate a confidence interval for each, about 90% of them will capture the true mean weight of a soda case.

p-value

Assuming H0 in context, there is a p-value probability of getting the observed result or less/greater/more extreme, purely by chance.

Example: Assuming the mean body temperature is 98.6 °F (H0: μ = 98.6), there is a 0.023 probability of getting a sample mean of 97.9 °F or less, purely by chance.

Conclusion for a Significance Test

Because p-value p-value < / > significant level we reject / fail to reject H0. We do / do not have convincing evidence for Ha in context.

Example: Because the p-value 0.023 < 0.05, we reject H0. We do have convincing evidence that the mean body temperature is less than 98.6 °F (Ha: μ < 98.6).

Type 1 Error

The H0 context is true, but we find convincing evidence for Ha context.

Example: The mean body temperature is actually 98.6 °F, but we find convincing evidence the mean body temperature is less than 98.6 °F.

Type II Error

The Ha context is true, but we don’t find convincing evidence for Ha context.

Example: The mean body temperature is actually less than 98.6 °F, but we don’t find convincing evidence that the mean body temperature is less than 98.6 °F.

Power

If Ha context is true at a specific value there is a power probability the significance test will correctly reject H0

Example: If the true mean body temperature is 97.5 °F, there is a 0.73 probability the significance test will correctly reject H0: μ = 98.6

Standard Error of the Slope

The slope of the sample LSRL for x-context and y-context typically varies from the slope of the population LSRL by about SEb

Example: The slope of the sample LSRL for absences and final grades typically varies from the slope of the population LSRL by about 1.2 points/absence.