This is not designed to be a comprehensive review. There may be items on the exam that are not covered in this review. Similarly, there may be items in this review that are not tested on this exam. 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.

Unit 3 Lesson Outcomes

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Unit 3 Lesson Summaries

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.

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Lesson 16 Recap
  • 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. \[ \displaystyle {z = \frac{\textrm{value} - \textrm{mean}}{\textrm{standard deviation}} = \frac{\hat p - p}{\sqrt{\frac{p \cdot (1-p)}{n}}}} \]

Lesson 17 Recap
  • 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 Math 221 Statistics Toolbox, to perform hypothesis testing and calculate confidence intervals for problems involving one proportion.

  • 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.

Lesson 18 Recap
  • 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 Math 221 Statistics Toolbox to perform hypothesis testing and calculate confidence intervals for problems involving two proportions.

Lesson 19 Recap
  • 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.

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