Inferences & Claims From Statistics - SAT Math

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

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

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

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

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

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

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

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

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

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

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

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

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

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

$s = \sqrt{\frac{1}{n-1} \sum (x_i - \bar{x})^2}$. Measures spread by calculating square root of variance with $n-1$ denominator.

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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.

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.

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

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 population. Fixed value describing an entire population (usually unknown).

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.

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

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.

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

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

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