Topcis

1. Significance Tests: The Basics
   • A significance test assesses the evidence provided by the data against a null hypothesis $H_0$ in favor of an **alternative hypothesis $H_a$
     • $H_0$ states no change or difference (equal to)
     • $H_a$ states what we suspect is true (there is a difference)
   • A one-sided alternative $H_a$ says that a parameter differs from the null hypothesis value in a specific direction. A two-sided alternative H_a says that a parameter differs from the null value in either direction
   • The P-value of a test is the probability (computed supposing $H_0$ is true) that the statistic will take a value at least as extreme as the observed reuslt in the direction specified by $H_a$
     • Small P-values indicate strong evidence against $H_0$.
     • To calculate a P-value, we must know the sampling distribution of the test statistic when $H_0$ is true.
   • A Type I error occurs if we reject $H_0$ when it is in fact true. A Type II error occurs if we fail to reject $H_0$ when $H_0$ is false.
     • The probability of a Type I error is the significance level $\alpha$
2. Tests about a Population Proportion
   • The conditions for a significance test of a proportion are:
     • Random: The data came from a well-designed random sample or experiment
       • 10%: When sampling without replacement, the population must be 10 times as large as the sample
     • Large Counts: The sample is large enough to satisfy $np \gt 10$ and $n(1-p) \gt 10$
   • The test statistic is calculated via $z = \frac{\hat{p}-p_0}{\sqrt{\frac{p_{0}(1-p_0)}{n}}}$
   • Confidence intervals give more information then significance tests. Mainly a set of plausible values for the true population parameter p.
   • The power of a significance test is the probability of not making a Type II error.
     • Power can be increased by increasing n, or increasing $\alpha$
   • You must decide if a Type I or Type II error is more sever as decreasing the probability of one increases the other
3. Tests about a Population Mean
   • The conditions for performing a significance test of a mean are:
     • Random: The data came from a well-designed random sample or experiment
       • 10%: When sampling without replacement, the population must be 10 times as large as the sample
     • Normal / Large Sample: The population distribution must be normal or the sample size must be $n \gt 30$
   • The test statistic is calculated via $t = \frac{\bar{x}-\mu_0}{\frac{s_x}{\sqrt{n}}}$ with $df=n-1$
   • Paired data can be analyzed by taking the difference within each pair to produce a single sample.

Formulas

Terms