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
    $H_0$ $\bar{x} = \mu$ $\bar{x} \leq \mu$ $\bar{x} \geq \mu$
    $H_1$ $\bar{x} = \mu$ $\bar{x} > \mu$ $\bar{x} < \mu$

  2. determine critical values

    critical values: z-score which defines the top/bottom 2.5% of the distribution
    find with Table C1, Column C in privitera appendix
    z-value: +/- 1.96
    z-distribution is based on data from the whole population and always has the same shape

  3. calculate standard error of the mean

    $\sigma_\bar{x} = \frac{\sigma} {\sqrt{n}}$
    standard error of the mean = population standard deviation / square root of sample size
    shows the average amount a sample mean is expected to differ from the population mean based on sampling error
  4. calculate observed z-score
    $z = \frac{\bar{x} - \mu} {\sigma_\bar{x}}$
    tells us how many standard error units the sample mean is from the population mean (and in which direction)
  5. compare observed z-score to critical values
    if the observed z-score is in the rejection region (falls in the shaded region beyond the critical values [z < -1.96 OR z > 1.96]) you reject the null. if it falls within the central distribution, you fail to reject the null.
  6. obtain p-value
    the p-value demonstrates the probability of getting the observed z-score from the hypothesized distribution
    reject the null: p < .05
    fail to reject the null: p > .05
  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
      two-tailed test one-sided (upper tail) test one-sided (lower tail) test
      $H_0$ $\bar{x} = \mu$ $\bar{x} \leq \mu$ $\bar{x} \geq \mu$
      $H_1$ $\bar{x} = \mu$ $\bar{x} > \mu$ $\bar{x} < \mu$
    2. determine critical values
      critical values: t-score which defines the top/bottom 2.5% of the distribution
      in a t-test, critical values depend on the df (n - 1) used in the study
      t-values are generally > 1.96, because t-distributions are generally wider. as df approaches infinity, the narrower the distribution becomes, and the closer the t-values get to 1.96
      find with Table C2 in privitera appendix
      t-distribution is based on data from a sample, and sample data becomes more accurate as the sample size increases. (smaller sample = wider distribution, more uncertainty; larger sample = thinner distribution, looks closer to z-distribution, more certainty)
    3. calculate estimated standard error of the mean
      $\hat{s}_\bar{x} = \frac{\hat{s}} {\sqrt{n}}$
      estimated standard error of the mean = estimated population standard deviation / square root of sample size
      same as calculating the standard error of the mean in z-tests, however $\hat{s}$ is used instead of $\sigma$. in a t-test, you are given the sample standard deviation, and you must use it to estimate the population standard deviation.
    4. calculate observed t-score
      $t = \frac{\bar{x} - \mu} {\hat{s}_\bar{x}}$
      tells us how many standard error units the sample mean is from the population mean (and in which direction)
    5. compare observed t-score to critical values
      if the observed t-score is in the rejection region (falls in the shaded region beyond the calculated critical values, then you reject the null. if it falls within the central distribution, you fail to reject the null.
    6. obtain p-value
      the p-value demonstrates the probability of getting the observed t-score from the hypothesized distribution
      reject the null: p < .05
      fail to reject the null: p > .05
    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
      When the value of the population mean based on the null hypothesis is rejected, we need to specify a range of values we are relatively (95%) confident contains the 'true' population mean based on our sample mean. Our 'best guess' is the observed score.
      Calculate the confidence intervals...
      $Cl_{95} = \bar{x} \pm z_.05 \sigma_\bar{x} \leftarrow$ one sample z-test
      $Cl_{95} = \bar{x} \pm t_.05 \hat{s}_\bar{x} \leftarrow$ one sample t-test
      Important points
      • Confidence intervals can be calculated for 95% or 99%
    2. checking for errors
    quiz