Study Guide: Significance Tests and Normal Distributions

Significance Test

assesses the strength of evidence against a specific statement.

Null Hypothesis (H0)

The statement being tested, usually representing 'no effect,' 'no difference,' or no change.

Alternative Hypothesis (Ha)

The claim you are trying to find evidence for, usually representing 'an effect,' 'a difference,' or a change.

Test Statistic

Standardizes how far an estimate is from the parameter, often in the form (estimate - parameter) / standard deviation of the estimate.

Z-Test Statistic

For a population mean, calculated as z = (x̄ - μ₀) / (σ/√n), where μ₀ is the value from the null hypothesis.

P-Value

The probability of obtaining the sample result, or one more extreme, if the null hypothesis is true.

Statistically Significant

Data is considered statistically significant at level alpha (α) if the p-value is less than or equal to alpha.

Common Level of Significance

A common level of significance is α = 0.05.

Hypotheses

State the H0 and Ha when performing a significance test.

Conditions for Inference about a Population Mean

Includes SRS, Normality, and Independence.

Simple Random Sample (SRS)

The data must be a simple random sample from the population.

Normality

Observations from the population must have a normal distribution OR the sample size (n) must be at least 30.

Independence

The population size must be at least 10 times larger than the sample size (Population size ≥ 10 * n).

Level of Significance

A lower alpha (e.g., 0.01 instead of 0.05) requires stronger evidence to reject H0.

Statistical Significance vs Practical Importance

Statistical significance does not imply practical importance; a statistically rare outcome may not be meaningful.

Statistical Inference Validity

Statistical inference is not valid for all data sets; data must be a random sample to avoid bias.

Multiple Analyses

Performing many tests increases the chance of finding statistically significant results purely by random chance.

Significance at α = 5%

About 5% of tests could appear significant even if they are not.

Type I Error

Rejecting the null hypothesis (H0) when it is actually true (a false positive).

Type II Error

Failing to reject (Accepting) the null hypothesis (H0) when it is actually false (a false negative)

Power of a Test

The probability of correctly rejecting H0 when it is false. It represents the strength of the evidence against H0 when H0 is indeed false. Power increases the farther the truth is from H0. Increasing the sample size also increases power by lowering the chance of a Type II error

density curve

always on or above the horizontal axis and has an area of exactly 1 underneath it

density curve

describes the overall pattern of a distribution

density curve

The area under the curve above a range of values represents the proportion of all observations in that range

median

the point that divides the area under the curve in half (the equal-areas point)

mean

the balance point

symmetric density curve

the mean and median are the same and located at the center

skewed density curve

the mean is pulled away from the median in the direction of the long tail

normal distribution

a type of density curve that is mound-shaped and symmetric. It is based on a continuous variable

68-95-99.7 Rule

About 68% of observations fall within 1 standard deviation (σ) of the mean (μ)

68-95-99.7 Rule

About 95% of observations fall within 2 standard deviations (σ) of the mean (μ)

68-95-99.7 Rule

About 99.7% of observations fall within 3 standard deviations (σ) of the mean (μ)

Standardizing

converting an observation x from a distribution with mean μ and standard deviation σ means converting it to a z-score using the formula: z = (x - μ) / σ

z-score

tells you how many standard deviations an observation is from the mean

standard normal distribution

a normal distribution with a mean of 0 and a standard deviation of 1, denoted N(0, 1)

variable x

If a variable x has any normal distribution N(μ, σ), the standardized variable z will have the standard normal distribution N(0, 1)

cumulative distribution function

provides the area under the standard normal curve to the left of a given z-score

Area to the left of z (P(Z < z))

the table entry for z

Area to the right of z (P(Z > z))

1 minus the table entry for z

Area between z1 and z2 (P(z1 < Z < z2))

the difference between the table entries for z2 and z1

Inverse Normal Calculations

To find the value of x corresponding to a given area or proportion, first use Table A in reverse to find the z-score

Inverse Normal Calculations

To find the value of x corresponding to a given area or proportion, first use Table A in reverse to find the z-score. Then, use the formula x = μ + z*σ to find the value of x