showing relationship: observational studies surveys showing

causation: controlled experiment survey is used to analyze the construct

Median is robust Mode – measure of the center   cut tail: lower 25% upper 25% Boxplots – IQR,max-min=range   how spread out of the chart small bin size to have as more deatails as possible

probability density function we’re never 100% sure


central limit theorem – the distribution of sample means is approximately normal.


95% confidence interval for the mean

A confidence interval gives an estimated range of values which is likely to include an unknown population parameter, the estimated range being calculated from a given set of sample data.

If independent samples are taken repeatedly from the same population, and a confidence interval calculated for each sample, then a certain percentage (confidence level) of the intervals will include the unknown population parameter. Confidence intervals are usually calculated so that this percentage is 95%, but we can produce 90%, 99%, 99.9% (or whatever) confidence intervals for the unknown parameter.

The width of the confidence interval gives us some idea about how uncertain we are about the unknown parameter (see precision). A very wide interval may indicate that more data should be collected before anything very definite can be said about the parameter.

Confidence intervals are more informative than the simple results of hypothesis tests (where we decide “reject H0” or “don’t reject H0”) since they provide a range of plausible values for the unknown parameter.

A confidence interval for a mean specifies a range of values within which the unknown population parameter, in this case the mean, may lie. These intervals may be calculated by, for example, a producer who wishes to estimate his mean daily output; a medical researcher who wishes to estimate the mean response by patients to a new drug; etc.


margin of error

95% of sample means fall within 1.96 standard errors from the population mean.

98% of sample means fall within 2.33 standard errors from the population mean.


levels of likelihood

critical region

if falls into the crtical region, it can be concluded that most likely we do not get this sample mean by chance.

the critical region defines unlikely values if the null hypothesis is true.

z-critical value

when we do the statical test, we set up our own criteria to make a decision


two-tailed test



we reject the null hypothesis when p value is less than the a value.

cohen’s d:

standardized mean difference that measures the distance between means in standardized units

margin of error

dependent t-test for paired samples:

same subject take the test twice,


  • two conditions: each subject is assigned two condition in random order
  • pre-test, post-test
  • growth over time- longitudinal study 

statistical significance

  • reject the null
  • results are not likely due to chance – sampling error

“statistically significant” finding

When a statistic is significant, it simply means that you are very sure that the statistic is reliable. It doesn’t mean the finding is important or that it has any decision-making utility.

To say that a significant difference or relationship exists only tells half the story. We might be very sure that a relationship exists, but is it a strong, moderate, or weak relationship? 

After finding a significant relationship, it is important to evaluate its strength. Significant relationships can be strong or weak. Significant differences can be large or small. It just depends on your sample size.

One-Tailed and Two-Tailed Significance Tests

When your research hypothesis states the direction of the difference or relationship, then you use a one-tailed probability. For example, a one-tailed test would be used to test these null hypotheses: Females will not score significantly higher than males on an IQ test.

A two-tailed test would be used to test these null hypotheses: There will be no significant difference in IQ scores between males and females.

Procedure Used to Test for Significance

Whenever we perform a significance test, it involves comparing a test value that we have calculated to some critical value for the statistic. It doesn’t matter what type of statistic we are calculating (e.g., a t-statistic, a chi-square statistic, an F-statistic, etc.), the procedure to test for significance is the same.

  1. Decide on the critical alpha level you will use (i.e., the error rate you are willing to accept).
  2. Conduct the research.
  3. Calculate the statistic.
  4. Compare the statistic to a critical value obtained from a table.

If your statistic is higher than the critical value from the table:

  • Your finding is significant.
  • You reject the null hypothesis.
  • The probability is small that the difference or relationship happened by chance, and p is less than the critical alpha level (p < alpha ).


The formula for calculating margin of error is made for two-tailed tests, i.e. while calculating margin of error, we only take one side of t=0 into account. If you remember, when we were doing that example, for calculating margin of error on a two-tailed test, we didn’t take twice the t-critical (or t-critical positive minus t-critical negative), we took only one t-critical. Now, when we are calculating t-critical for a one tailed test, how can we use the same margin of error formula, to remove the mental burden of remembering another formula? So, we assume that it’s a two-tailed test. But how does that work out? It works out as the total critical area in both tails for the same alpha value would equal the critical area in a one-tailed test. Let’s say we take -1.711 as our t-critical for calculating margin of error, then this is same as doing a two-tailed test, but now the total alpha value changes to 0.05*2=0.1. Now to keep the total alpha value (or the total critical area same for both tests) as 0.05, we think the test as a two-tailed test and use the same formula for calculating the margin of error we used for a two-tailed test. Does that make sense?

when do we use t-test rather z-test?

Z-test and t-test are basically the same; they compare between two means to suggest whether both samples come from the same population. There are however variations on the theme for the t-test. If you have a sample and wish to compare it with a known mean (e.g. national average) the single sample t-test is available. If both of your samples are not independent of each other and have some factor in common, i.e. geographical location or before/after treatment, the paired sample t-test can be applied. There are also two variations on the two sample t-test, the first uses samples that do not have equal variances and the second uses samples whose variances are equal.



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