Lecture Notes on Statistical Tests and Intervals

Flashcards on Z-tests, T-tests, and Confidence Intervals

Z-Test

is used when the population standard deviation is known and the sample size is large (n ≥ 30).

T-Test

is used when the population standard deviation is unknown, the sample size is small (n < 30), and the sample is assumed to come from a normally distributed population.

Type I Error

False Positive) occurs when the null hypothesis (H0) is rejected when it is actually true.

Type II Error

False Negative) occurs when the null hypothesis (H0) is not rejected when it is actually false.

Excel for Z-Test

In Excel, use NORM.S.DIST for cumulative distribution and NORM.S.INV for inverse (critical value) calculations.

Excel for T-Test

In Excel, use T.DIST for cumulative probability and T.INV.2T for the critical t-value.

Critical Z-Value

is used to determine the threshold for rejecting the null hypothesis at a given confidence level.

Sample Size

A Z-Test is appropriate for large sample sizes (n ≥ 30), while a T-Test is used for small sample sizes (n < 30).

Z-Test Application

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

T-Test Application

is used when the population standard deviation (σ) is unknown, the sample size is small (n < 30), and the sample is assumed to come from a normally distributed population.

Critical Z-Value in Excel

Use the function NORM.S.INV(probability) to find the z-value for a given confidence level.

Critical Z-Value Example

For a 95% confidence level (α = 0.05), use the formula =NORM.S.INV(0.025).

Critical T-Value in Excel

Use the function T.INV.2T(probability, degreesoffreedom) to find the critical t-value for a given confidence level and degrees of freedom.

Critical T-Value Example

For a 95% confidence level and 10 degrees of freedom, use the formula =T.INV.2T(0.05, 10).

T-Distribution

is used when the sample size is small and/or the population standard deviation is unknown. It is similar to the normal distribution but has heavier tails to account for increased uncertainty with small samples.

Degrees of Freedom

(df) is calculated as df = n - 1, where n is the sample size.

Critical T-Value Significance

is used to determine the threshold for statistical significance based on a given confidence level and degrees of freedom.

Finding Critical T-Value

You can find the critical t-value using a t-distribution table or Excel's T.INV.2T function.

Excel Function for T-Value

The function is T.DIST(x, degreesoffreedom, cumulative), which finds the cumulative probability for a given t-value.

T.DIST Example

To find the cumulative probability for a t-value of 2 with 10 degrees of freedom, use T.DIST(2, 10, TRUE).

Confidence Interval

gives a range of values that likely contains the population parameter.

Confidence Interval Formula

CI = x̄ ± tα/2 × (s/√n) where x̄ is the sample mean, tα/2 is the critical t-value based on the desired confidence level, s is the sample standard deviation, and n is the sample size.

Confidence Interval Components

x̄ is the sample mean, tα/2 is the critical t-value for the confidence level, s is the sample standard deviation, and n is the sample size.

Margin of Error

Critical t-value × (s/√n).

Excel Function - Margin of Error

= T.INV.2T(0.05, degreesoffreedom) × (s/√DEV.S(range)/SQRT(COUNT(range))).

Critical T-Value Dependence

depends on the desired confidence level.

Sample Size Impact

A larger sample size decreases the margin of error, resulting in a narrower confidence interval.

Sample Mean Significance

The sample mean (x̄) serves as the central point around which the confidence interval is constructed.

Sample Standard Deviation Role

The sample standard deviation (s) measures the variability of the sample data, influencing the width of the confidence interval.