Probability - SAT Math

Null hypothesis: no effect or difference. Default assumption that treatment has no impact; what we test against.

Histogram. Shows distribution shape and frequency of values across intervals or bins.

Causal claim. Asserts one variable directly influences or causes changes in another.

Arrange data, find middle value(s). For odd $n$, middle value; for even $n$, average of two middle values.

Mean = 6.5. Sum all values and divide by count: $(4+5+7+10) ÷ 4 = 6.5$.

Mode = 3. The value that appears most frequently in the data set.

Mean. Extreme values pull the mean away from the center more than median or mode.

Range = 20. Difference between maximum and minimum: $22 - 2 = 20$.

Estimate range for population parameter. Provides likely bounds where true population value falls with given confidence.

Standard Deviation = 4. Standard deviation is the square root of variance: $\sqrt{16} = 4$.

Median = 7. Middle value when data is ordered from least to greatest.

Mode = 6. Value that appears most frequently in the data set.

IQR = 50. Difference between third and first quartiles: $75 - 25 = 50$.

Range = 17. Difference between maximum and minimum values: $20 - 3 = 17$.

Probability = $\frac{6}{36}$ or $\frac{1}{6}$. Six ways to make 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1) out of 36 total.

Probability = $\frac{26}{52}$ or $\frac{1}{2}$. 26 red cards (hearts and diamonds) out of 52 total cards.

Probability = $\frac{1}{6}$. One favorable outcome (rolling 3) out of six possible outcomes.

Probability = $\frac{13}{52}$ or $\frac{1}{4}$. Thirteen hearts in a standard deck of 52 cards.

P(A') = 1 - P(A). Complement probability equals 1 minus the original event's probability.

Probability = $\frac{3}{6}$ or $\frac{1}{2}$. Three even numbers (2, 4, 6) out of six possible outcomes.

Probability = $\frac{4}{52}$ or $\frac{1}{13}$. Four aces in a standard deck of 52 cards.

Probability = $\frac{1}{2}$. One favorable outcome (heads) out of two possible outcomes.

IQR = Q3 - Q1. Measures the spread of the middle 50% of data values.

$\frac{1}{4}$. 13 hearts out of 52 total cards simplifies to $\frac{1}{4}$.

$P(E) = \frac{\text{Number of favorable outcomes}}{\text{Total number of outcomes}}$. Basic probability definition using favorable over total outcomes.

$\frac{1}{6}$. One favorable outcome out of six possible outcomes.

$\frac{1}{221}$. First ace: $\frac{4}{52}$, second ace: $\frac{3}{51}$, multiply together.

$P(A \text{ and } B) = P(A) \times P(B)$. For independent events, multiply their individual probabilities.

$\frac{5}{6}$. Complement of rolling a 3: $1 - \frac{1}{6} = \frac{5}{6}$.

$\frac{1}{13}$. 2 red queens + 2 black kings = 4 favorable cards out of 52.

$0.06$. Multiply independent probabilities: $0.3 \times 0.2 = 0.06$.

$\frac{1}{2}$. Prime numbers on die: 2, 3, 5 (three out of six).

$P(A \text{ or } B) = P(A) + P(B) - P(A \text{ and } B)$. General formula accounting for overlap between events.

$\frac{11}{36}$. Use complement: $1 - P(\text{no 6s}) = 1 - \frac{25}{36} = \frac{11}{36}$.

$\frac{26}{51}$. After removing one black card, 26 red remain out of 51 total.

$\frac{2}{13}$. 8 favorable cards (4 aces + 4 kings) out of 52 total.

$P(A|B) = \frac{P(A \text{ and } B)}{P(B)}$. Probability of A given B has occurred.

$P(A') = 1 - P(A)$. The complement probability equals 1 minus the event probability.

$P(A \text{ and } B) = 0$. Mutually exclusive events cannot happen together.

Events that cannot occur simultaneously. If one occurs, the other cannot happen at the same time.

$\frac{10}{13}$. 40 non-face cards out of 52 total cards.

$\frac{1}{2}$. 26 black cards (spades and clubs) out of 52 total.

$\frac{4}{13}$. 4 kings + 13 hearts - 1 king of hearts = 16 favorable.

$0.5$. Complement of A: $1 - 0.5 = 0.5$.

Events where the occurrence of one affects the probability of the other. One event's outcome changes the probability of the other.

$0.58$. Using addition rule: $0.3 + 0.4 - (0.3 \times 0.4) = 0.58$.

Events where the occurrence of one does not affect the other. One event's outcome doesn't change the other's probability.

$\frac{1}{26}$. 2 red aces (hearts and diamonds) out of 52 cards.

$\frac{11}{26}$. 13 spades + 12 face cards - 3 overlap = 22 favorable outcomes.

$\frac{1}{2}$. Three even numbers (2, 4, 6) out of six possible outcomes.

$\frac{1}{4}$. Multiply independent probabilities: $\frac{1}{2} \times \frac{1}{2} = \frac{1}{4}$.

$\frac{1}{13}$. 4 queens out of 52 cards simplifies to $\frac{1}{13}$.

$\frac{1}{12}$. 3 ways to roll 4: (1,3), (2,2), (3,1) out of 36 possibilities.

$\frac{1}{4}$. 13 diamonds out of 52 total cards.

Two events are complementary if one event is the complement of the other. One event is exactly the opposite of the other.

$\frac{5}{13}$. 13 clubs + 12 face cards - 3 club face cards = 22 favorable.

$\frac{3}{4}$. Complement of all tails: $1 - \frac{1}{4} = \frac{3}{4}$.

$\frac{1}{6}$. 6 ways to roll 7: (1,6), (2,5), (3,4), (4,3), (5,2), (6,1).

$P(A \text{ and } B) = P(A) \times P(B)$. For independent events, probability of both occurring.

A set of events that cover all possible outcomes. Events whose probabilities sum to 1 (complete sample space).

$0.5$. Apply addition rule: $0.3 + 0.4 - 0.2 = 0.5$.

$\frac{3}{8}$. Count outcomes with exactly two tails: TTH, THT, HTT.

A numerical characteristic of a population. Fixed value describing an entire population (usually unknown).

The mean is 8.6. Sum all values and divide by count: $(3+7+8+10+15)/5 = 43/5 = 8.6$.

The median is 5. Sort values first: 2, 4, 5, 7, 9; middle value is 5.

The range is 12. $15 - 3 = 12$; difference between largest and smallest values.

The mode is 8. The value 8 appears most frequently (3 times).

Probability of observed result under null hypothesis. Lower p-values suggest stronger evidence against the null hypothesis.

Both are equally strong. Both have the same absolute value, so equal strength.

A subset of the population used for analysis. Representative portion selected from the larger population for study.

A bell-shaped distribution symmetric about the mean. Most data falls within a predictable range around the center.

Results unlikely due to chance, given a threshold. Indicates results are unlikely to occur by random chance alone.

Range = Maximum - Minimum. Range measures the spread by finding the difference between extremes.

A conclusion about a population based on a sample. Uses sample data to make generalizations about the entire population.

A statement that there is no effect or difference. The baseline assumption that researchers try to reject or fail to reject.

$H_a: \text{mean} \neq 50$. Alternative hypothesis claims the parameter is not equal to the null value.

The z-score is 1. $(85-80)/5 = 1$; standardizes the value relative to the distribution.

Failure to reject a false null hypothesis. Accepting a null hypothesis when the alternative is actually true.

The claim is that the mean height > 6 feet. States that the population mean exceeds 6 feet.

The critical value is 1.96. Marks the boundary for 95% of the standard normal distribution.

95% of samples contain the true population parameter. Indicates the reliability of the interval estimation method.

The confidence interval is (46.08, 53.92). $50 \pm 1.96 \times \frac{10}{\sqrt{25}} = 50 \pm 3.92$.

The distribution of sample means approximates normality. For large samples, sample means follow a normal distribution pattern.

The entire set of individuals or items of interest. All possible subjects or objects being studied in research.

A numerical characteristic of a sample. Calculated value describing a sample (used to estimate parameters).

Bias due to non-random sample selection. Systematic error that makes samples unrepresentative of the population.

The range of error in sample estimate of population. Quantifies the uncertainty in survey or poll results.

The outlier is 100. Value that falls far outside the typical range of the data.

A graphical summary of data distribution using quartiles. Shows median, quartiles, and potential outliers in data distribution.

The strength and direction of a linear relationship. Values closer to -1 or 1 indicate stronger linear relationships.

From -1 to 1. Perfect positive correlation is 1, perfect negative is -1.

A graph showing the relationship between two variables. Each point represents one observation with two variable values.

The frequency distribution of a data set. Shows how often different values or ranges occur in data.

A distribution that is not symmetric. Data is pulled more toward one tail than the other.

Variance = $\frac{\text{sum of squared deviations}}{n-1}$. Measures how spread out data points are from the mean.

A scatter plot shows the relationship between two variables. Points reveal correlation patterns.

It displays quantitative data to show distribution details. Separates tens and ones digits clearly.

An outlier is a data point significantly different from others. Falls far from typical data pattern.

IQR = 10. $Q^3 - Q^1 = 15 - 5 = 10$

Range = 14. $19 - 5 = 14$

Mode = 6. 6 appears most frequently (3 times).

Median = 3. Ordered: 1,2,3,4,5; middle is 3.

A histogram displays the frequency distribution of data. Bars show count in each interval.

Negatively skewed: tail extends to the left. Most data on right, few low values.

A symmetric distribution is mirrored around its center. Both sides look identical when folded.

$Variance = \frac{\sum (x_i - \text{mean})^2}{n}$. Square deviations from mean, then average.

The standard deviation measures data spread around the mean. Shows how far values deviate from center.

The median is least affected by outliers. Extreme values don't shift median much.

Mean = 5.75. $(4+8+6+5) ÷ 4 = 5.75$

Positively skewed: tail extends to the right. Most data on left, few high values.

A box plot displays the distribution, median, and quartiles of a data set. Five-number summary in visual format.

The mode is the most frequently occurring value. Count frequency of each value.

Variance = 0.25. Mean is 2.5, deviations squared and averaged.

Skewness indicates the asymmetry of a distribution. Measures if distribution leans left or right.

It represents the frequencies of data points in a line graph. Connected dots show frequency distribution.

It shows cumulative totals of data points up to each class. Running total increases at each point.

IQR = Q3 - Q1. Third quartile minus first quartile.

Standard deviation = 0.5. Mean is 4.5, deviations squared and averaged.

A distribution with two distinct peaks or modes. Two separate frequency peaks exist.

The median is the middle value when data is ordered. Sort values first, then find center.

A uniform distribution has equal frequency across values. All intervals have same probability.

$Mean = \frac{\sum x_i}{n}$. Sum all values, divide by count.

Range = Maximum value - Minimum value. Subtract smallest from largest value.

A dot plot uses dots to represent frequency of data values. Each dot equals one data point.

$z = \frac{x - \text{mean}}{\text{standard deviation}}$. Standardizes distance from mean in units.

The median is the best measure for skewed data. Outliers don't affect median much.

It refers to the variability or dispersion of data points. How scattered data points are.

Median = 5. Ordered: 3,5,9; middle value is 5.

The range is the difference between the highest and lowest values. Span from minimum to maximum value.

Range = 2. $10 - 8 = 2$

Mean absolute deviation = 1. Average distance from mean is 1.

Quartiles divide data into four equal parts. 25%, 50%, 75% split points.

z = 2. $(10-8) ÷ 1 = 2$ standard deviations.

In skewed data, mean is pulled toward the tail. Median stays centered, mean shifts.

$Geometric mean = \sqrt[n]{x_1 \cdot x_2 \cdot ... \cdot x_n}$. Multiply all values, take nth root.

The mean is most affected. Outliers pull the mean toward extreme values.

The shape is right-skewed. Long tail extends toward higher values on the right.

The median is 4. Middle value when ordered: 2, 3, 4, 7, 9.

The mean is 6. Sum all values (30) and divide by count (5).

Correct: 'The sample result supports the hypothesis.'. 'Supports' is proper; samples provide evidence but never absolute proof.

1/6. One specific outcome (rolling 4) out of six equally likely outcomes.

Drawing conclusions about a population based on a sample. Uses sample data to make educated guesses about the entire population.

Probability = (Number of favorable outcomes) / (Total number of possible outcomes). This is the fundamental definition of probability in statistics.

P-value. Measures probability that observed results occurred by random chance alone.

1. All possible outcomes together form the complete sample space.

$P(A \text{ and } B) = P(A) \times P(B)$. For independent events, multiply individual probabilities.

1/4. 13 hearts out of 52 total cards in a standard deck.