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Unit 6: Inferences for Categorical Data: Proportions
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Terms in this set (61)
one-sample z interval for a population proportion
The appropriate procedure for estimating a population proportion from one sample of a categorical variable
Random Condition
Data should be collected using a random sample or a randomized experiment
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
Standard error of p-hat
Margin of Error
Gives how much a value of a sample statistic is likely to vary from the value of the corresponding population parameter
Calculating Margin of Error for a one-sample proportion
Critical Value x Standard Error of the Statistic
Calculating a Confidence Interval
Point Estimate ± (Critical Value)(Standard Error of the Statistic) OR Point Estimate ± Margin of Error
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
Calculating a Critical Value
inversenorm (1/2 area outside confidence level, 0, 1)
2.58
Critical value for 99% confidence
2.33
Critical value for 98% confidence
1.96
Critical value for 95% confidence
1.64
Critical value for 90% confidence
Interpreting/Concluding a Confidence Interval
We are C% confident that the interval from _____ to _____ captures the [parameter in context
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].
Width of the confidence interval decreases
What happens to a confidence interval as the sample size increases?
Width of C.I. = 1/√2
What is the proportion of sample size to width of a population proportion interval?
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?
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)
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
Alternative Hypothesis (Ha)
What we are testing to be true, it is a statement that we are trying to find evidence to support
Population
The null and alternative hypotheses are always written using the ___________ proportion, not the sample.
(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?
statistic ± (critical value)(standard error of statistic)
From the AP Stats formula chart, how do you calculate a Confidence Interval?
Standard Error of the statistic for one proportion (p-hat)
Statistic for the difference between two proportions
Standard error of the statistic for the difference between two proportions
Pooled statistic for difference in two proportions
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.
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
CONCLUDE/INTERPRET! (In a sentence)
What must you do at the end of any Significance Test or Confidence Interval you complete?
Significance Level (α)
The predetermined probability of rejecting the null hypothesis given that it is true.
Steps to Concluding a Test
Step 1: Compare p-value to the significance level
Step 2: Conclude in context (Convincing evidence or not?
Reject the Ho, there IS convincing of the Ha
If p ≤ α
Fail to Reject the Ho, there IS NOT convincing evidence of the Ha
If p > α
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]
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]
Type I Error
Occurs when the null hypothesis is true and is rejected (false positive/over-reaction)
Type II Error
Occurs when the null hypothesis is false and is not rejected (false negative/under-reaction)
Power of a Test
Probability that a test will correctly reject a false null hypothesis.
increases
The probability of a Type II Error DECREASES / Power of a Test INCREASES if the sample size _______
The probability of a Type II Error DECREASES / Power of a Test INCREASES
If the Significance Level (α) of a test increases, what happens?
decreases
The probability of a Type II Error DECREASES / Power of a Test INCREASES if the Standard Error _______
true parameter
If the ____ _____ value is farther from the null, the probability of a Type II error DECREASES / Power of Test INCREASES
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
both populations
When checking conditions for the difference between two proportions, you must ensure that the 10% condition is met for ________ _________
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
Confidence Interval for Diff. of two Proportions
Hypotheses for C.I. for Diff. of Two Proportions
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
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...
Test statistic for a difference in 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
1-Prop z Test
Calculator test for a One Sample z-test for a Population Proportion
2-Prop z test
Calculator test for a Two Sample z-test for the difference between two population proportions
1-Prop z interval
Calculator test for a One Sample z-Interval for a population proportion
2-Prop z interval
Calculator test for a Two Sample z-Interval for the difference between two population proportions
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?
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