Statistics Test 4 Study Guide

Confidence Interval

A range of values around a sample mean that is likely to contain the true population mean.

Interval Estimate

A range of values used to estimate a population parameter.

Point Estimate

A single value used to estimate a population parameter, usually the sample mean.

Margin of Error

The amount added to and subtracted from the sample mean to create a confidence interval.

Confidence Level

The probability that the confidence interval contains the true population mean (commonly 90%, 95%, or 99%).

Significance Level (α)

The probability of rejecting a true null hypothesis; equal to 1 minus the confidence level.

Alpha Split

In a two-tailed test or interval, the significance level α is divided equally between the two tails (α/2 each).

Population Mean (μ)

The true average value of a population.

Sample Mean (x̄)

The average of a sample used to estimate the population mean.

Standard Error

The standard deviation of the sampling distribution of a statistic.

t Distribution

A probability distribution used when the population standard deviation is unknown.

Degrees of Freedom

The number of values that are free to vary in a sample; for many problems df = n − 1.

Confidence Interval Formula

The sample mean plus or minus the margin of error.

Larger Confidence Level

Produces a wider confidence interval but increases the likelihood that the interval contains the true mean.

Smaller Confidence Level

Produces a narrower interval but decreases certainty that the interval contains the true mean.

Correct Confidence Interval Interpretation

We are a certain percent confident that the true population mean lies between the two interval values.

Incorrect Confidence Interval Interpretation

Saying that a certain percent of the data falls within the interval.

Sample Size Rule

A sample size of 30 or more is generally considered sufficient for normal approximation.

t vs z Distribution

The t distribution is wider than the normal distribution but approaches the normal distribution as sample size increases.

Hypothesis

A claim or statement about a population parameter.

Hypothesis Testing

A statistical method used to determine whether there is enough evidence to support a claim about a population.

Null Hypothesis (H₀)

The initial claim that there is no effect or no difference and always includes an equality sign.

Alternative Hypothesis (Hₐ)

The statement that contradicts the null hypothesis and represents what we want to test.

Lower-Tailed Test

A hypothesis test where the alternative hypothesis states the parameter is less than the hypothesized value.

Upper-Tailed Test

A hypothesis test where the alternative hypothesis states the parameter is greater than the hypothesized value.

Two-Tailed Test

A hypothesis test where the alternative hypothesis states the parameter is not equal to the hypothesized value.

Test Statistic

A standardized value calculated from sample data used to evaluate the null hypothesis.

Critical Value

The value that separates the rejection region from the non-rejection region in a hypothesis test.

Rejection Region

The range of values where the null hypothesis will be rejected.

Decision Rule

A rule stating when to reject or fail to reject the null hypothesis.

p-Value

The probability of observing a test statistic as extreme or more extreme assuming the null hypothesis is true.

p-Value Decision Rule

Reject the null hypothesis if the p-value is less than or equal to α.

Fail to Reject H₀

There is not enough evidence to support the alternative hypothesis.

Reject H₀

There is sufficient evidence to support the alternative hypothesis.

Type I Error

Rejecting a true null hypothesis (false positive).

Type II Error

Failing to reject a false null hypothesis (false negative).

Scatter Plot

A graph used to display the relationship between two variables.

Positive Relationship

As one variable increases, the other variable increases.

Negative Relationship

As one variable increases, the other variable decreases.

No Relationship

No clear pattern exists between the variables.

Outlier

An extreme value that can strongly affect correlation and regression.

Covariance

A measure indicating the direction of the relationship between two variables.

Correlation Coefficient (r)

A measure of the strength and direction of a linear relationship between two variables.

Range of r

The correlation coefficient ranges from −1 to +1.

Strong Correlation

When the correlation coefficient is close to −1 or +1.

Weak Correlation

When the correlation coefficient is close to 0.

Simple Linear Regression

A statistical method used to predict a dependent variable using one independent variable.

Regression Equation

ŷ = b₀ + b₁X.

Intercept (b₀)

The predicted value of Y when X equals 0.

Slope (b₁)

The change in Y for each one-unit increase in X.

Predicted Value (ŷ)

The estimated value of Y using the regression equation.

Residual

The difference between the observed value and predicted value (y − ŷ).

Least Squares Method

The method used to find the regression line that minimizes the sum of squared residuals.

Total Sum of Squares (SST)

The total variation of the dependent variable around its mean.

Sum of Squares Regression (SSR)

The portion of variation explained by the regression model.

Sum of Squares Error (SSE)

The portion of variation not explained by the regression model.

Relationship of Sums of Squares

SST = SSR + SSE.

Coefficient of Determination (R²)

The proportion of variation in the dependent variable explained by the independent variable.

R² Interpretation

The percentage of variability in the dependent variable explained by the regression model.