AP Statistics Ch 12: Significant Tests in Practice

Part 1: Population Mean

• use a t-score for test statistic
• t = (x-bar - mean value claimed) / (standard deviation from data / sqrt(sample size))
• simplifies to t = (x-bar - mean value claimed) / (s/sqrt(n))
• measures how far x-bar is from the claimed mean; measured in standard deviation
• types of alternative hypothesis
• meana > mean0
• meana < mean0
• meana does not equal to mean0 (this is two-sided)
• still need to have Normality, SRS, and Independence to use it
• for paired t-tests, carry out the same steps, only this time, the parameter is the difference between the two samples
• Skewed curve is the most serious of non-Normality
• larger sample size (over 30)>>>> Central Limit Theorem; can count as Normal
• high power is important; detects deviation from the claimed parameter value
• is probability that test will reject the null hypothesis when the mean is a certain alternative value
• 1 - Probability of Type 2 error

Part 2: Population Proportion

• sample must be Normal, Independent, and from SRS
• Will use z-test
• z = (p-hat - p0)/ (sqrt((p0(1 - p0))/n)
• p-hat is the p from the alternative hypothesis
• p0 is the p from the null hypothesis
• n is the sample size
• types of alternative hypothesis
• pa > p0
• pa < p0
• pa does not equal to p0 (this is two-sided)
• aka large-sample test
• based on Normal curve; p gets more accurate as the sample size increases
• confidence intervals tell us if our test is good enough to detect deviations; if confidence interval really large, then the power of the test is small >>> hard to detect deviations, consider increasing sample size and other actions that make the power larger
• also tells us how far above or below the value reported in the null hypothesis the real value really is; good to include in conclusion
• standard error for confidence interval is
• sqrt((pa(1 - pa))/n
• pa is value reported in sample (which is the data)
• n is the sample size
• doing this will mean that significance level of two-tailed significance test is not exactly the 1 - significance value of confidence interval of two-tailed significance test, but it is close enough