Questions and Answers
What are the appropriate hypotheses for a significance test about a population parameter?
Interpret a P-value in context.
The P-value represents the probability of obtaining evidence for the alternative hypothesis as strong or stronger than observed, assuming the null hypothesis is true.
What conclusion should be drawn if the P-value is small?
Reject the null hypothesis and conclude there is evidence for the alternative hypothesis.
What happens in a Type I error?
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What happens in a Type II error?
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The significance level alpha represents the probability of making a ______ error.
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What are the conditions for performing a significance test about a population proportion?
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How is the standardized test statistic calculated?
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The power of a significance test is the probability of finding ______ for the alternative hypothesis when it is true.
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What are the appropriate hypotheses for a significance test about the difference between two proportions?
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What conditions need to be met for performing a test about a difference between two proportions?
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Calculate the standardized test statistic for a test about a difference between two proportions.
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What should be included in the conclusion of a significance test about the difference between two proportions?
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Study Notes
Hypotheses for Significance Test
- Appropriate hypotheses should clearly state the null (H0) and alternative (Ha) hypothesis, as well as the parameter of interest.
- Example: H0: p = 0.2 and Ha: p >, <, or not equal to 0.2; where p is the true proportion within a specific context.
Interpreting P-value
- A P-value indicates the probability of obtaining evidence for the alternative hypothesis as strong or stronger than the observed evidence when the null hypothesis is true.
- For instance, if the null hypothesis states a mean daily calcium intake of 1300 mg, a P-value indicates the chances of obtaining a sample mean of a certain value under random sampling.
Drawing Conclusions from P-value
- If the P-value is small (typically < alpha level), one can reject the null hypothesis, indicating strong evidence for the alternative hypothesis.
- If the P-value is not small, one fails to reject the null, suggesting insufficient evidence for the alternative hypothesis.
Type I and Type II Errors
- A Type I error occurs when the null hypothesis is rejected when it is actually true (false positive).
- Example consequence: Acceptance of bad product (like potatoes) leading to financial losses.
- A Type II error occurs when the null hypothesis is not rejected when the alternative is true (false negative).
- Example consequence: Allowing poor quality potatoes into production, damaging customer satisfaction and sales.
Error Probabilities
- Type I error is represented by the significance level alpha.
- Type II error probability is represented as 1 minus the power of the test.
Conditions for Testing Population Proportions
- Randomness is established by ensuring the data comes from a random sample.
- The 10% condition must show that n and n(1-p) are both greater than 10.
Standardized Test Statistic and P-value Calculation
- The standardized test statistic helps measure how far a sample statistic is from what is expected if the null hypothesis is true, using standard deviation (SD) units.
- Calculation: z = (statistic - parameter) / SD. Use the normal cumulative distribution function (normalcdf) for determining P-values.
Conducting a Significance Test
- State the hypothesis, significance level, and parameter of interest (e.g., “At a significance level of 0.05, p”).
- Plan by selecting the suitable inference method after checking randomness, 10% condition, and large counts.
- Execute the test by gathering sample statistics, calculating the z-score, and finding the P-value.
- Conclude by comparing the P-value with alpha and making a decision regarding the null hypothesis.
Understanding Power of a Test
- Power reflects the probability of correctly rejecting the null hypothesis when the alternative hypothesis is true.
- For example: If the true mean of an item is known, power can indicate the likelihood of finding significant evidence for the alternative hypothesis.
- Increasing sample size, significance level, and the distance between parameter values boosts the power of a test.
Hypotheses for Differences Between Two Proportions
- Null hypothesis (H0): p1 - p2 = 0, indicating no difference between two proportions.
- Alternative hypothesis (Ha): p1 - p2 is not equal to 0, indicating a difference.
Conditions for Two Proportions Testing
- Random sampling or random assignment is required for the groups being compared.
- The 10% condition mandates that both n1 and n2 satisfy n1(1-p-hatc) > 10 and n2(1-p-hatc) > 10.
Differences Between Proportions Calculations
- Calculate p-hatc as the combined proportion: p-hatc = (x1 + x2) / (n1 + n2).
- The standardized statistic for the difference is calculated as z = (p-hat1 - p-hat2) / sqrt[p-hatc(1 - p-hatc)(1/n1 + 1/n2)].
Performing a Difference of Proportions Test
- Define hypotheses, significance level, and parameter context.
- Use a two-sample z-test and ensure all conditions are met.
- Execute by calculating sample statistics, z-scores, and P-values.
- Conclude by comparing the P-value to the chosen alpha level, determining evidence for the difference in population proportions.
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Description
Prepare for your AP Statistics exam with these flashcards focused on Chapter 9. This quiz covers key concepts such as formulating hypotheses and interpreting P-values within the context of significance testing. Enhance your understanding and boost your exam readiness!
