STAT 2ND EXAM

Normal Distribution Properties

Bell-shaped, symmetric about the mean, and the total area under the curve is exactly 1.

Central Limit Theorem (CLT)

If n≥30, the sampling distribution of the mean is approximately normal regardless of the population's shape.

Standard Error

The standard deviation of a sampling distribution; it measures how much a statistic varies from sample to sampe

T-Distribution

Similar to the standard normal but more variable (flatter); used when the population standard deviation (σ) is unknown and n<30.

Consistent Estimator

As the sample size (n) increases, the value of the estimator approaches the true value of the parameter.

CI Interpretation

You are (1−α)100% confident that the interval estimator encloses the true mean, NOT that there is a "probability" it falls there after the sample is drawn.

Confidence Level vs. Width

For a fixed sample size, as the confidence coefficient increases (e.g., 95% to 99%), the width/length of the interval also increases.

Null Hypothesis (H0)

A statement of no difference or no change; it is the hypothesis the researcher aims to test and often doubts.

Alternative Hypothesis (H1)

The claim the researcher believes to be true and wishes to prove (can be directional or non-directional).

Type I Error (α)

Occurs when you reject the null hypothesis (H0) when it is actually true.

Type II Error (β)

Occurs when you fail to reject the null hypothesis (H0) when it is actually false.

Level of Significance (α)

The maximum probability allowed for committing a Type I error.

P-value Decision Rule

If P-value ≤α, Reject H0. If P-value >α, Do Not Reject H0.

Critical Region

The range of test values that indicates a significant difference and leads to rejecting the null hypothesis

Two-Tailed Test

Used when the alternative hypothesis (H1) contains a "not equal to" (=) sign (non-directional).

One-Sample Z-Test Rule

Use this when n≥30 OR when the population is normal and σ is known.

statistical test

a procedure that uses sample data to decide whether or not to reject the null hypothesis.

One-Tailed Test

Definition: This test specifies a one-directional difference (either "greater than" or "less than") for the parameter being studied. Situational Example: A chemist creates an additive to increase battery life. Since she only cares if the life is longer than the current 36 months, her hypotheses are: H0:μ=36 H1:μ>36

Two-Tailed Test

Definition: This test does not specify a direction. It simply looks to see if the parameter has changed or is different. Situational Example: A researcher wants to know if a medication affects pulse rate. They don't know if it will increase it, decrease it, or leave it unchanged—they just want to know if it is different from the average of 82. Their hypotheses are: H0:μ=82 H1:μ/=82 (not equal to 82)

sampling distribution

The probability distribution of a statistic

statistic

any number (like a mean or standard deviation) that comes only from a sample . It is called a "random variable" because every time you pull a new random sample, that number will change.

parameter

a value that is fixed and represents the whole group (like μ)

Sampling Distribution

a probability distribution (a graph or list) of all the possible values a statistic could take if you pulled every possible sample of the same size from a population . The most common one is the sampling distribution of the mean, which is just the distribution of all possible sample averages (Xˉ) (x bar).

Standard Error

The standard deviation of the sampling distribution.

Two Areas of Statistical Inference

1. Estimation  Point estimation  Interval estimation 2. Hypothesis Testing

Point Estimation

Providing a single number as the best guess for a population value. "best estimate" or a single specific value (like the sample mean Xˉ).

Interval Estimation

Providing a range of values (an interval) where the population value is expected to fall. gives a range with a margin of error (e.g., "between 10 and 15 grams")

Use the Z-test when

the population standard deviation (σ) is known or when the sample size is large (n≥30)

Use the T-test when

σ is unknown and the sample size is small (n<30).

The three methods used to test hypotheses

1. The traditional method 2. The P-value method 3. The confidence interval method