Lesson 20: Review for Exam 3

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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.

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
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
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.


3 Navigation

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Inference for Independence of Categorical Data
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Review for Exam 3
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Lesson 21:
Describing Bivariate Data: Scatterplots, Correlation, & Covariance