Review for Lessons 9 to 11

Introduction Section

This page is essentially the page of formulas and notes that you can use to study for the material from Lesson 9 through 11. You will find a printable version of this in Canvas that you can print out and bring to your proctored exams. The printable version also includes the normal table which you may need for the exam as well.

Outline of Material Covered from Lesson 9 to 11 Section

Standard error of the mean (SEM) =

\[\frac{(\text{sample standard deviation})}{\sqrt{n}}\]

Standard error of a proportion (SEP) =


Standard error of the difference between sample values from two independent samples =

\[\sqrt{(\text{standard error from 1st sample})^2+(\text{standard error from 2nd sample})^2}\]

Confidence Interval Step-by-Step:

  • Step 1: What are the parameter and the statistic?
  • Step 2: *Does the normal approximation apply?
    • check: random sample or independent trials?
    • check: large enough sample?
  • Step 3: Estimate the standard deviation of the statistic (also called the standard error)
  • Step 4: Compute   \(\text{statistic} \pm z^*\) (where \(z^*\) is the standard error)

Notice the common form in this last step (formula for standard depends on type statistic).

Hypothesis Testing Step-by-Step:

  • Step 1: Ask what is the parameter of interest (e.g. is it a mean, a proportion, or the difference between means or proportions)? Write the null and alternative hypotheses as statements about this parameter.
  • Step 2: Ask what is the sample statistic and its distribution if the null hypothesis is true? If it is normally distributed, calculate the standard score.
  • Step 3: Ask how likely is what happened? Use the tables to find the P-value.
  • Step 4: Ask what can I conclude?

Hypothesis Test (also called significance tests) Caveats:

  1. Large Sample Caution: significant results based on large samples may not be of practical significance.
  2. Small Sample Caution: results that are not significant in small samples may still be of practical significance.
  3. Multiple Testing Caution: When a large number of significance tests are conducted, some individual tests may be deemed significant just by chance (false positives).

Human Subjects Issues: avoid physical or psychological harm; voluntary participation; protect vulnerable populations; ensure informed consent.

Multiplier numbers from the normal distribution
Confidence Level \(z^*\)
50% 0.67
60% 0.84
70% 1.04
80% 1.282
90% 1.645
950% 1.96
99% 2.58
99.9% 3.29