Unit 6: Inferences for Categorical Data: Proportions

Steps for Conducting a Significance Test

1. Parameter (p=) 2. State Hypotheses 3. State Method 4. Check Conditions 5. Plug values into test statistic 6. Find z and p-val in calc 7. Conclude test

Steps for Calculating a Confidence Interval

1. Parameters (p=) 2. State Method 3. Check Conditions 4. Plug values into confidence interval 5. Find interval on calculator 6. Interpret interval/conclude in a sentence

Null Hypothesis (H₀)

A statement of no change, no effect, or no difference. It represents the situation that is assumed to be correct unless evidence suggests otherwise

Interpreting p-value

Assuming the [Ho in context] is true, there is a [p-value] probability of getting [sample statistic] or less/more (depends on Ha) just by chance.

Concluding a test if p > α

Because the p-value of [p-value] is greater than the alpha of [alpha level], we fail to reject the HO [state HO] There is not convincing evidence that [Ha in context]

Concluding a test if p ≤ α

Because the p-value of [p-value] is less than the alpha of [alpha level], we reject the HO [state HO] There is convincing evidence that [Ha in context]

1-Prop z interval

Calculator test for a One Sample z-Interval for a population proportion

1-Prop z Test

Calculator test for a One Sample z-test for a Population Proportion

2-Prop z interval

Calculator test for a Two Sample z-Interval for the difference between two population proportions

2-Prop z test

Calculator test for a Two Sample z-test for the difference between two population proportions

Calculating Margin of Error for a one-sample proportion

Critical Value x Standard Error of the Statistic

1.64

Critical value for 90% confidence

1.96

Critical value for 95% confidence

2.33

Critical value for 98% confidence

2.58

Critical value for 99% confidence

Critical Value

Critical values (z*) represent the boundaries encompassing the middle C% of the standard normal distribution, where C% is an approximate confidence level for a proportion

Random Condition

Data should be collected using a random sample or a randomized experiment

statistic ± (critical value)(standard error of statistic)

From the AP Stats formula chart, how do you calculate a Confidence Interval?

(Statistic - Parameter)/Standard Error of the Statistic

From the AP Stats formula chart, how do you calculate a Standardized Test Statistic for a significance test?

Margin of Error

Gives how much a value of a sample statistic is likely to vary from the value of the corresponding population parameter

Fail to Reject the Ho, there IS NOT convincing evidence of the Ha

If p > α

Reject the Ho, there IS convincing of the Ha

If p ≤ α

The probability of a Type II Error DECREASES / Power of a Test INCREASES

If the Significance Level (α) of a test increases, what happens?

true parameter

If the ____ _____ value is farther from the null, the probability of a Type II error DECREASES / Power of Test INCREASES

Interpreting a Confidence Level

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

Type II Error

Occurs when the null hypothesis is false and is not rejected (false negative/under-reaction)

Type I Error

Occurs when the null hypothesis is true and is rejected (false positive/over-reaction)

Calculating a Confidence Interval

Point Estimate ± (Critical Value)(Standard Error of the Statistic) OR Point Estimate ± Margin of Error

p-value

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

Power of a Test

Probability that a test will correctly reject a false null hypothesis.

Steps to Concluding a Test

Step 1: Compare p-value to the significance level Step 2: Conclude in context (Convincing evidence or not?

two-sample z-interval for a difference between population proportions.

The appropriate confidence interval procedure for a two-sample comparison of proportions for one categorical variable

one-sample z interval for a population proportion

The appropriate procedure for estimating a population proportion from one sample of a categorical variable

one sample z-test for a proportion

The appropriate testing method for a one-sample population proportion for one categorical variable (purpose: To estimate the probability of observing a value as extreme as p-hat when given p)

two-sample z-test for a difference between two population proportions

The appropriate testing method for the difference of two population proportions for a single categorical variable

Population

The null and alternative hypotheses are always written using the ___________ proportion, not the sample.

Significance Level (α)

The predetermined probability of rejecting the null hypothesis given that it is true.

decreases

The probability of a Type II Error DECREASES / Power of a Test INCREASES if the Standard Error _______

increases

The probability of a Type II Error DECREASES / Power of a Test INCREASES if the sample size _______

Large Counts Condition (one-sample)

To check that the sampling distribution of p-hat is approximately normal, check that both the number of successes (n x p-hat) and the number of failures (n x (1-p-hat)) are at least 10 so that the sample size is large enough to support an assumption of normality

n₁pc, n₁(1-pc), n₂pc, and n₂(1-pc) are all ≥10 (or 5)

To check that the sampling distribution of p-hat₁-p-hat₂ is approximately normal, define the combined or pooled proportion and check that...

Interpreting/Concluding a Confidence Interval

We are C% confident that the interval from _____ to _____ captures the [parameter in context

Width of the confidence interval decreases

What happens to a confidence interval as the sample size increases?

Width of the confidence interval increases because to be more certain, the interval needs to be wider

What happens to the width of the confidence interval as the confidence level increases?

C.I Width = 2x Margin of Error

What is the proportion of Margin of Error to width of a population proportion interval?

Width of C.I. = 1/√2

What is the proportion of sample size to width of a population proportion interval?

CONCLUDE/INTERPRET! (In a sentence)

What must you do at the end of any Significance Test or Confidence Interval you complete?

Alternative Hypothesis (Ha)

What we are testing to be true, it is a statement that we are trying to find evidence to support

Greater than or equal to 10

When checking conditions for the difference between two proportions, you must ensure that n₁p₁, n₁(1-p₁), n₂p₂, and n₂(1-p₂) are all

both populations

When checking conditions for the difference between two proportions, you must ensure that the 10% condition is met for ________ _________

Estimating the population proportion

When do we calculate a confidence interval?

Testing a hypothesis about a population proportion

When do we conduct a significance test?

Calculating a Critical Value

inversenorm (1/2 area outside confidence level, 0, 1)