Exam 3 - STATS 3110

For means only!

When should you use the t-distribution?

Bell-shaped

T-distribution shape

n-1

Degrees of freedom

the t-distribution converges upon the standard normal

As the degrees of freedom increase ___________.

Get the critical value t* from the t-distribution: use df=n-1, and enter Central Probability = relavant confidence interval

Computing a confidence interval for t-distribution

Compute the test statistic and use the t-distribution to compute the P-value on the side specified by Ha

Conducting a hypothesis test for t-distribution

Computes the differences for each pair and doing a t-test on the column of paired differences

Matched pairs t-test

Analyze → Specialized Modeling → Matched Pairs → Drag both before and after columns into the Y, Paired Response Box, Click OK

Matched pairs t-test in JMP

• The original parent population needs to be bell-shaped for small sample sizes

• The sample size needs to be large enough for the Central Limit Theorem to be valid: how large depends on how much skewness is present

• The sample size still needs to be chosen randomly

T-distribution conditions

• Is done when we have two separate random samples of items or individuals

• Common in a randomized comparative experiment which randomly divides subjects into two different groups and assigns each group to a different treatment

• It also occurs when comparing random samples selected separately from two populations or comparing two separate subsets within a general population

T-Sample T-Test

Ho: μ₁-μ₂=0

Null hypothesis for two sample T-test

Ha: μ₁-μ₂ > 0 , Ha: μ₁-μ₂ < 0 , Ha: μ₁-μ₂ ≠ 0

Alternative hypothesis for two sample T-test

np & nq ≥ 10

Condition: When the samples are large, the distribution of p-hat (1) and p-hat (2) is approximately normal

E(P-hat (1) - P-hat (2))

The mean of the sampling distribution of two proportions

SE(P-hat (1) - P-hat (2))

Standard error of two proportions

Z-test

What kind of test are used for two proportions?

One success and one failure in each of the two samples

Plus 4 Adjustment

Single categorical variable with multiple levels

What kind of variable is involved in a Chi-squared goodness-of-fit test?

• Determines if the observed category counts follow the assumed model (Ho) good enough after allowing for random sampling variation

• Or if there is sufficient evidence the model does not provide a good fit

Chi-squared goodness-of-fit test

Assesses whether the observed counts fit the theoretical distribution good enough

Goodness-of-fit test

Poor Fit

Large Deviation

Good Fit

Small Deviation

Are calculated by multiplying the current sample size by the theoretical proportions

Expected counts

Describes the theoretical distribution in the target population

Null hypothesis of chi-squared goodness-of-fit test

At least one P is different

Alternative hypothesis of chi-squared goodness-of-fit test

The counted individuals should be from a random sample of the population or randomly assigned in an experiment

Randomization Condition (Chi-squared goodness-of-fit test)

The expected count for each category is at least 5

Expected Cell Frequency Condition (Chi-squared goodness-of-fit test)

Is a measure of how far observed counts are from expected counts under the null hypothesis

Chi-squared statistic

Small deviation from Ho → Insufficient evidence against Ho

Small χ²

Large deviation from Ho → Provide evidence to reject Ho

Large χ²

Computed using the chi-squared distribution with df=(number of categories -1)

• Always on the right-side since the test statistic is based on squared differences and will always be positive, never negative

P-values are __________

Determines if two categorical variables are related or associated with one another

• For two variables to be related (or associated or dependent), the subject is more likely to take a certain value in variable I than at certain values of variable 2

Test of Independence

The two variables are independent/not related/not associated

Null hypothesis for chi-squared test of independence for two categorical variables

The two variables are dependent/related/associated

Alternative hypothesis for chi-squared test of independence for two categorical variables

Use the chi-squared test of independence to see if two categorical variables are related

Two categorical variables condition (Chi-squared test of independence for two categorical variables)

Random sampling

Randomization condition (Chi-squared test of independence for two categorical variables)

Expected counts ≥ 5 in all cells

Large sample size condition (Chi-squared test of independence for two categorical variables)

Row total x column total / overall total

Calculating the expected count for a cell (Chi-squared of independence for two categorical variables)

Df=(r-1)(c-1)

Degrees of freedom (Chi-squared of independence for two categorical variables)

We can conclude that the variables are dependent

If the p-value is less than alpha (Chi-squared of independence for two categorical variables)

We would have insufficient evidence to demonstrate that they are dependent (reject Ho)

If the p-value is larger than alpha (Chi-squared of independence of two categorical variables)