# Symbols, Meanings & Formulas

Adapted from ocw.smithw.org

 Name Symbol Population Sample Sample Size $N$ $n$ Mean $\mu$ ("mu") $\bar{x}$ ("x-bar") Standard Deviation $\sigma$ (lower case "sigma") $s$ Variance $\sigma^2$ (lower case "sigma" squared) $s^2$ Proportion $\pi$ $p$ Correlation Coefficient $\rho$ (lower case "rho") $r$

 Name Symbol Concept Summation $\Sigma$ (upper case "sigma") Null Hypothesis $H_0$ prediction of no difference $\rightarrow \bar{x} = \mu$ Alternate Hypothesis $H_1$ prediction of a difference $\rightarrow \bar{x} \neq \mu$ Power $1 - \beta$ the probability that a test of significance will reject the null hypothesis Type 1 Error $\alpha$ (lower case "alpha") false positive Type 2 Error $\beta$ (lower case "beta") false negative

# one sample z-test, one sample t-test

For one sample z-tests, the population standard deviation is known and the critical values are always +/- 1.96.

For one sample t-tests, the population standard deviation is unknown and the critical values depend on the degrees of freedom: (df = n - 1).

1. # state hypotheses

 two-tailed test one-sided (upper tail) test one-sided (lower tail) test $H_0$ $\bar{x} = \mu$ $\bar{x} \leq \mu$ $\bar{x} \geq \mu$ $H_1$ $\bar{x} = \mu$ $\bar{x} > \mu$ $\bar{x} < \mu$

3. # calculate standard error of the mean

4. calculate observed z-score
5. compare observed z-score to critical values
6. obtain p-value
7. conclude whether to reject the null, or fail to reject the null. however, there is still the possibility of an error - type 1 and type 2 errors will be covered next.

# one sample t-test

• population variance unknown
• critical values depend on the degrees of freedom (df = n - 1)
1. state hypotheses
2. determine critical values
3. calculate estimated standard error of the mean
4. calculate observed t-score
5. compare observed t-score to critical values
6. obtain p-value
7. conclude whether to reject the null, or fail to reject the null. however, there is still the possibility of an error - type 1 and type 2 errors will be covered next.

# non-directional (two-tailed) vs. directional (one-tailed) tests

non-directional (two-tailed)
• alpha (.05) split equally between two tails
• critical values are larger
• directional (one-tailed)
• alpha (.05) is all in one tail
• critical values larger
• more 'powerful' test
• # further steps for z-tests & t-tests

1. confidence intervals
2. checking for errors
quiz