Welcome to the final stretch, class! As we approach the end of the semester, it is time to consolidate everything we have learned. This Final Exam Review is designed to walk you through the core concepts of statistics, from descriptive measures to inferential decision-making. Below is a breakdown of the key topics covered in the attached review notes and how to approach them.

1. Descriptive Statistics and Z-Scores

We started the review by looking at enrollment data for UC campuses. Remember, when calculating parameters for a population, notation matters. The population mean is denoted by $\mu$ and the population standard deviation by $\sigma$.

A crucial concept for the final is the z-score, which tells us how many standard deviations a data point is from the mean. The formula is:

$$z = \frac{x - \mu}{\sigma}$$

In our example, calculating the z-score for the Los Angeles campus ($z = 1.14$) and Riverside ($z = -0.27$) helps us standardize and compare values across different distributions.

2. Probability and Logic

In the Adjusted Gross Income problem, we reviewed how to extract probabilities from frequency tables. Be careful with logical operators:

  • AND (Intersection): Both conditions must be true simultaneously.
  • OR (Union): Either one or both conditions can be true.
  • NOT (Complement): Everything except the specified event.

3. The Normal Distribution

Using the example of Elephant Pregnancies, we applied the Normal Model to determine percentages. To find the percentage of pregnancies lasting "between 3 and 5 years," we calculate z-scores for both boundaries and find the area under the curve between them. This connects back to our z-score formula and is a fundamental skill for the exam.

4. Sampling Error

We discussed the IRS tax return data to understand sampling error. Sampling error is simply the difference between a sample statistic (like the sample mean $\bar{x}$) and the true population parameter ($\mu$). Remember: increasing the sample size ($n$) usually decreases sampling error, but it doesn't eliminate it entirely unless you census the whole population!

5. Confidence Intervals: Z-Interval vs. T-Interval

This is often the trickiest part of the exam. You must distinguish between when to use a Z-statistic and when to use a T-statistic.

  • Use a Z-Interval (Problem 5): When the population standard deviation ($\sigma$) is known.
    Formula: $\bar{x} \pm z_{\alpha/2} \left(\frac{\sigma}{\sqrt{n}}\right)$
  • Use a T-Interval (Problem 6): When $\sigma$ is unknown and you are using the sample standard deviation ($s$). This usually applies to smaller sample sizes (like the diabetic mothers study where $n=16$).
    Formula: $\bar{x} \pm t_{\alpha/2} \left(\frac{s}{\sqrt{n}}\right)$

6. Hypothesis Testing

Finally, we looked at the Worker Fatigue study. Here, we set up a hypothesis test to see if the mean heart rate exceeded 72 bpm.

  • Null Hypothesis ($H_0$): $\mu = 72$
  • Alternative Hypothesis ($H_a$): $\mu > 72$

Because the test statistic ($z = 3.03$) fell into the rejection region (beyond the critical value of 1.645), we rejected the null hypothesis. There was sufficient evidence to support the claim.

Study Tip: Download the attached PDFs to see the handwritten step-by-step solutions for every problem. Good luck studying—you have all the tools you need to succeed!