# Lesson 20: Review for Exam 3

You are strongly encouraged to review the readings, homework exercises, and other activities from Units 1-3 as you prepare for the exam. In particular, you should go over the Review for Exam 1 and the Review for Exam 2. Use the Index to review definitions of important terms.

## 1 Lesson Outcomes

The expectation on the exam is the following outcomes. You should be able to do:

- All of the Outcomes from Lesson 08 (Unit 1) and Lesson 15 (Unit 2)
- Calculate and interpret a sample proportion.
- Summarize categorical data with a bar or pie chart.
- Determine the mean, standard deviation and shape of a distribution of sample proportions.
- Calculate probabilities using a distribution of sample proportions.
- Confidence Intervals:
- Determining correct confidence interval for a given scenario.
- Calculate and interpret a confidence interval given a confidence level and a given parameter.
- Identify a point estimate and margin of error for a given confidence interval.
- Show the appropriate connections between the numerical and graphical summaries that support a confidence interval.
- Check the requirements for a confidence interval.
- Calculate a confidence interval for a comparison of two proportions.

- Conduct a Hypothesis Test for
- A single mean with σ known.
- A single mean with σ unknown.
- Difference of two means with dependent samples.
- The mean of differences with dependent samples.
- Several means (ANOVA).
- A single proportion.
- A comparison of two proportions.
- Test of Independence for categorical data.
- A goodness of fitness test.

- Hypothesis Testing
- Determining correct hypothesis test for a given scenario
- State the null and alternative hypothesis for the chosen test.
- Calculate the test-statistic and p-value of the hypothesis test.
- Assess the statistical significance by comparing the p-value to the α-level.
- Check the requirements for the hypothesis test.
- Show the appropriate connections between the numerical and graphical summaries that support a hypothesis test.
- Draw a correct conclusion for a hypothesis test.

## 2 Lesson Summaries

Click on the link at right for a review of the summaries from each lesson.

**Here are the summaries for each lesson in unit 3. Reviewing these key points from each lesson will help you in your preparation for the exam.**

**Pie charts**are used when you want to represent the observations as part of a whole, where each slice (sector) of the pie chart represents a proportion or percentage of the whole.

**Bar charts**present the same information as pie charts and are used when our data represent counts. A**Pareto chart**is a bar chart where the height of the bars is presented in descending order.

- $\hat p$ is a point estimator for true proportion $p$. $\displaystyle{\hat p = \frac{x}{n}}$

- The sampling distribution of $\hat p$ has a mean of $p$ and a standard deviation of $\displaystyle{\sqrt{\frac{p\cdot(1-p)}{n}}}$

- If $np \ge 10$ and $n(1-p) \ge 10$, you can conduct
**probability calculations**using the Normal Probability Applet. $\displaystyle {z = \frac{\textrm{value} - \textrm{mean}}{\textrm{standard deviation}} = \frac{\hat p - p}{\sqrt{\frac{p \cdot (1-p)}{n}}}}$

- The
**estimator**of $p$ is $\hat p$. $\displaystyle{ \hat p = \frac {x}{n}}$ and is used for both confidence intervals and hypothesis testing.

- You will use the Excel spreadsheet

- The requirements for a confidence interval are $n \hat p \ge 10$ and $n(1-\hat p) \ge 10$. The requirements for hypothesis tests involving one proportion are $np\ge10$ and $n(1-p)\ge10$.

- We can determine the sample size we need to obtain a desired margin of error using the formula $\displaystyle{ n=\left(\frac{z^*}{m}\right)^2 p^*(1-p^*)}$ where $p^*$ is a
**prior estimate**of $p$. If no prior estimate is available, the formula $\displaystyle{ \left(\frac{z^*}{2m}\right)^2}$ is used.

- When conducting hypothesis tests using two proportions, the null hypothesis is always $p_1=p_2$, indicating that there is no difference between the two proportions. The alternative hypothesis can be left-tailed ($<$), right-tailed($>$), or two-tailed($\ne$).

- For a hypothesis test and confidence interval of two proportions, we use the following symbols:

$$ \begin{array}{lcl} \text{Sample proportion for group 1:} & \hat p_1 = \displaystyle{\frac{x_1}{n_1}} \\ \text{Sample proportion for group 2:} & \hat p_2 = \displaystyle{\frac{x_2}{n_2}} \end{array} $$

- For a hypothesis test only, we use the following symbols:

$$ \begin{array}{lcl} \text{Overall sample proportion:} & \hat p = \displaystyle{\frac{x_1+x_2}{n_1+n_2}} \end{array} $$

- Whenever zero is contained in the confidence interval of the difference of the true proportions we conclude that there is no significant difference between the two proportions.

- You will use the Excel spreadsheet

- The
**$\chi^2$ hypothesis test**is a test of independence between two variables. These variables are either associated or they are not. Therefore, the null and alternative hypotheses are the same for every test:

$$ \begin{array}{1cl} H_0: & \text{The (first variable) and the (second variable) are independent.} \\ H_a: & \text{The (first variable) and the (second variable) are not independent.} \end{array} $$

- The
**degrees of freedom ($df$)**for a $\chi^2$ test of independence are calculated using the formula $df=(\text{number of rows}-1)(\text{number of columns}-1)$

- In our hypothesis testing for $\chi^2$ we never conclude that two variables are
*dependent*. Instead, we say that two variables are*not independent*.

Previous Reading: Lesson 19: Inference for Independence of Categorical Data |
This Reading: Lesson 20: Review for Exam 3 |
Next Reading: Lesson 21: Describing Bivariate Data: Scatterplots, Correlation, & Covariance |