Topcis

1. Comparing Two Proportions
   • The sampling distribution of $\hat{p}_1-\hat{p}_2$ has the following properties:
     • Shape: Approximately normal if the sample are large enough that large counts is met for both samples
     • Center: Mean at $p_1-p_2$
     • Spread: As long as each sample is no more than 10% of its population, the standard deviation is $\sigma_{p_d}=\sqrt{\frac{\hat{p}_{1}(1-\hat{p}_{1})}{n_1}+\frac{\hat{p}_{2}(1-\hat{p}_{2})}{n_2}}$
   • If the conditions for estimating or testing a claim for a one sample proportion are met for each distribution, a test can be preformed with the difference.
   • A two-sample z interval for $p_1-p_2$ is performed by the formula $(\hat{p}_1-\hat{p}_2) \pm z^{*}\sqrt{\frac{\hat{p}_{1}(1-\hat{p}_{1})}{n_1}+\frac{\hat{p}_{2}(1-\hat{p}_{2})}{n_2}}$
   • A two-sample z test for $p_1-p_2$ is performed by first finding $\hat{p}_C$ and then $z=\frac{(\hat{p}_1-\hat{p}_2)-0}{\sqrt{\frac{\hat{p}_{C}(1-\hat{p}_{C})}{n_1}+\frac{\hat{p}_{C}(1-\hat{p}_{C})}{n_2}}}$
2. Comparing Two Means
   • The sampling distribution of $\bar{x}_1-\bar{x}_2$ has the following properties:
     • Shape: Normal if both population distributions are normal, otherwise approximately normal if both samples are large enough according to the CLT
     • Center: Mean at $\mu_1-\mu_2$
     • Spread: As long as each sample is no more then 10% of its population, the standard deviation is $\sqrt{\frac{\sigma^{2}_1}{n_1}+\frac{\sigma^{2}_2}{n_2}}$
   • If the conditions for estimating or testing a claim for a one sample mean are ment for each distribution, a test can be preformed with the difference
   • Since the population standard deviation is often never known, a t distribution should be used. To avoid using a long formula, df should equal the smallest of $n_1-1$ and $n_2-1$

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