Welcome to the final stretch of the summer semester! We have covered a tremendous amount of ground in a short time, from basic probability to complex hypothesis testing. To help you prepare for the final exam, I have compiled a detailed review based on our recent class sessions.
Below, you will find a breakdown of the key topics included in the attached class notes, along with the formulas and concepts you need to keep fresh in your mind.
1. Probability and Expected Value
Probability is the foundation of inference. In the review notes, pay close attention to the Multiplication Rule for dependent events (sampling without replacement). For example, when calculating the probability of pulling specific coins from a jar one after another, the denominator changes with each draw.
We also reviewed Expected Value ($E[x]$). Remember the card game wager problem? To determine if a game is fair or what your expected loss/gain is over time, we use the formula:
$$E[x] = \sum [x \cdot P(x)]$$Tip: Don't forget to account for the cost of playing the game when calculating your net gain ($x$)!
2. The Normal Distribution and Z-Scores
You should be comfortable navigating the Normal Curve. We practiced finding the "cutoff" values for the top or bottom percentages of a population (e.g., the top 8%). This requires using the Inverse Normal function to find a $z$-score, then solving for $x$ using:
$$x = z\sigma + \mu$$3. Sampling Distributions and The Central Limit Theorem
When we move from an individual data point to a sample mean ($\bar{x}$), our standard deviation changes. This is the Standard Error. When calculating the probability of a sample mean being greater than a specific value, remember to adjust your $z$-score formula:
$$z = \frac{\bar{x} - \mu}{\frac{\sigma}{\sqrt{n}}}$$4. Hypothesis Testing: The Core of the Exam
A significant portion of the final exam focuses on Hypothesis Testing. You must be able to distinguish between different scenarios:
- Z-Test for the Mean: Used when the population standard deviation ($\sigma$) is known.
- T-Test for the Mean: Used when $\sigma$ is unknown (we only have sample standard deviation $s$). Remember, the degrees of freedom are $df = n - 1$.
The Decision Rule:
Always set up your Null ($H_0$) and Alternative ($H_a$) hypotheses first. If your calculated test statistic falls into the rejection region (determined by your alpha level, $\alpha$), you reject the Null Hypothesis.
5. Two-Sample T-Tests (Independent Samples)
Towards the end of the notes (Pages 17-20), we tackle comparing two independent means. When assuming equal population variances, we calculate a Pooled Variance ($s_p^2$) to find our test statistic.
The formula for the $t$-statistic looks complex, but it is just comparing the difference between the sample means against the pooled standard error:
$$t = \frac{(\bar{x}_1 - \bar{x}_2) - (\mu_1 - \mu_2)}{\sqrt{s_p^2(\frac{1}{n_1} + \frac{1}{n_2})}}$$Final Study Tips
Download the attached PDF to see the handwritten solutions for every step of these problems. I recommend printing them out and trying to solve the problems yourself before checking the answer key. Make sure your calculator is set to the correct mode and that you are comfortable entering lists of data for statistical analysis.
You have put in the work all summer—finish strong! Good luck!
Class Notes | Statistics Calculators | Desmos Scientific Calculator