# Reviewing for Lessons 10 to 12

### Introduction

This page is essentially the page of formulas and notes that you can use to study for the material from Lesson 10 through 12. 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.

This page contains much more information than the printable version. Click on the 'Tell Me More...' links to access the basic idea behind, examples and further references for the topics listed on this page.

### Outline of Material Covered from Lesson 10 to 12

**Standard error of the mean** (SEM) =

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

**Standard error of a proportion **(SEP) =

\[\sqrt{\frac{\hat{p}(1-\hat{p})}{n}}\].

**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 is 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 or 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**:

**Large Sample Caution**: significant results based on large samples may not be of practical significance.**Small Sample****Caution**: results that are not significant in small samples may still be of practical significance.**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 |