AP STAT REVIEW: Inference

Unbiased Estimator

Unbiased Estimator

when the center of the sampling distribution of the statistic is equal to the population parameter

Conditions for a 1 or 2 sample z-test or interval, one or two sample t test or t interval for mean or difference in means

Random, 10% condition, large counts

Interpreting P-Value

the probability of obtaining a test statistic as extreme or more extreme than the observed test statistic when the null hypothesis is assumed to be true

Population Distribution

distribution of responses for every individual of the population

Sample Distribution

Distribution of responses for a single sample

Sampling Distribution

distribution of all values for the statistic for all possible samples of a given size from a population

Interpret Confidence Interval

“We are C% confident that the confidence interval from to captures the [population parameter in context]

How to find the point estimate given confidence interval

Suppose confidence interval is (A,B)

The point estimate is the average of A and B.

How to find the MoE given confidence interval

Suppose confidence interval is (A,B)

MoE = B - (Point Estimate)

Calc function for one sample Z-test for p

#5: 1-PropZTest

Calc function for one sample t-test for µ

#2: T-Test

Degrees of Freedom (df) NOT for chi-squared

n-1

Test statistic formula One sample t-test for µ

t=(X BAR)-µ₀ / s/√n

Test statistic formula two-sample t-test for µ₁-µ₂

t=(X BAR₁ - X BAR₂) / √s₁²/n₁ + s₂²/n₂ OR 2-samp T-Test

Power of a Test

the probability a test will correctly reject the null hypothesis, given the alternate hypothesis is true

P(Type II Error)

P(Type II error) = 1-P(Power)

Test statistic formula for One-Sample z-test for p

z=(P HAT) - p₀ / √p₀(1-p₀)/n

Calc function for one sample t-interval for µ

#8: TInterval

CI formula for one sample z-interval for p

(P HAT) ± z* √ (P HAT) (1-P HAT)/n

CI Formula for one sample t-interval for µ

(X BAR) ± t* s/√n

Calc function for one sample z-interval for p

A: 1-PropZInt

What factors affect confidence interval width

Decreases as n increases, increases as conf level increases

Two sample z-interval for p₁-p₂ confidence interval

2-propZInt

Why do we check the condition for random?

So we can generalize to the population from which the sample was selected

Why do we check the 10% condition

so sampling without replacement is okay, and we can use the stated formula for standard deviation

Why do we check the normal/large sample condition

so the sampling distribution is approximately normal

What does rejecting the null mean?

There is statistcial evidence to support the alternative hypothesis

What does failing to reject the null mean

there is not convincing statistical evidence to support the alternative hypothesis

Confidence interval for two sample t-interval for µ₁-µ₂

2-sampTInt on calc

Type I error

when the null hypothesis is true, but is rejected

Type II erorr

when the alternate hypothesis is true, and the null is not rejected

How to choose the right inference procedure?

does the scenario describe mean(s), proportion(s), counts, or slope?

Does the scenario describe one sample, 2 sample, or paired data

Does the scenario describe a test or confidence interval?

P HAT Combined

X₁+X₂/n₁+n₂

Test statistic for two sample z-test for p₁-p₂

2-propZTest on calc

Interpret the confidence level

In repeated random sampling with the same sample size, approximately C% of confidence intervals created will capture the [population parameter in context].

Parameter

a number that describes the population

Statistic

number that describes the sample

test statistic formulat-test for slope

b-β₀/SE(sub b) (df=n-2)

Conditions for Chi-square test

Random, 10%, large counts (all expected counts ≥5)

CI formula t-interval for slope

b±t* SE(sub b) (df=n-2)

test statistic formula - chi-square test for goodness of fit

x²=Σ (observed-expected)²/expected (df = #of groups -1)

how to calculate expected counts in a chi-sguare test for homogeneity/independence

expected count = (row total)(column total) / table total

Test statistic formula chi-square test for goodness of fit

X²=Σ (observed-expected)²/expected (df=number of groups-1)

Test statistic formula chi-square test for homogeneity/independence

X²=Σ (observed-expected)²/espected (df= (rows-1)(columns-1))

Significance test conclusion

Our P-value [p=_] is LESS/GREATER than the significance level [ALPHA = _]. Therefore we REJECT/FAIL TO REJECT the null. We DO/NOT have statistical evidence for [alternate hypothesis].

How to remember how to reject null or not?

If the P-value is low, reject that hoe. If the P-value is high, the null’s that guy.

GoF Test Hypothesis

The true distribution of [VARIABLE] is equally proportional as [ ] claims

GoF Test Alt Hypothesis

The true distribution of [VARIABLE] is different than claimed.

Chi-Squared test for homogeneity hypothesis

There is no difference in the true distribution of [VAR] for [POP 1] and [POP 2].

Chi-Squared test for homogeneity alternate hypothesis

There is a difference in the true distribution of [VAR] for [POP 1] and [POP 2].

Conditions for T-Test and T-Interval for slope

Truly linear (no pattern in the residual plot), Independent (10%), Normal distribution of residuals at each X (histogram of residuals is normal-ish), Equal std. dev. of resid. for all X (no heteroskedacisity), random sample

10% condition

n ≤ 1/10 N

Large Counts Condition

np ≥ 10 and n(1-p) ≥ 10

Central Limit Theorem

n≥30