AP Statistics: One-Proportion Z-Test Study Guide

Hey there, future AP Stats master! Let's get you prepped for the exam with this super-focused guide. We'll break down the one-proportion z-test, hit all the key points, and make sure you're feeling confident. Let's do this! 💪

Hypothesis Testing: Setting the Stage

Null Hypothesis (H0)

• The null hypothesis is the statement we're trying to find evidence against. It's our initial assumption about the population parameter. Think of it as the 'status quo' or the 'no change' hypothesis.
• It's always written with an equals sign: p = [some value].

Alternative Hypothesis (Ha)

• The alternative hypothesis is what we're trying to support. It's the claim that contradicts the null hypothesis.

• It's written as either: p < [some value], p > [some value], or p ≠ [some value].

• One-tailed test: Use < or > when you're interested in a change in one direction only.

• Two-tailed test: Use ≠ when you're interested in a change in either direction.

Example

Let's say Real News Online! claims 94% of people know Baby Yoda. We poll 750 people and find 700 know him. Do we have evidence that the actual proportion is different from 94%?

• H0: p = 0.94 (The true proportion is 94%)
• Ha: p ≠ 0.94 (The true proportion is not 94%)

Image courtesy of imgflip.com

Conditions for a One-Proportion Z-Test

Before we dive into calculations, we need to make sure our data is good to go. We have three conditions to check:

Random

- If your sample is biased, your results are useless. 🙁

Independent

• We use the 10% condition to ensure independence when sampling without replacement.
• Make sure the population is at least 10 times the size of your sample.
• Example: "Since it is reasonable to believe that our population is at least 10n, we can assume our sample is independent."

Normal

• We need to prove that the sampling distribution is approximately normal to use the normal curve.
• Use the Large Counts Condition: Both the expected number of successes and failures must be at least 10. ### Example

Let's check our Baby Yoda example:

• Random: The problem states "we poll a random sample of 750 people"
• Independent: It's reasonable to believe there are at least 7500 people in the world. 😉
• Normal: 750 * 0.94 = 705, 750 * 0.06 = 45. Both are greater than 10. 😊

Calculating Test Statistics

Now for the fun part: crunching the numbers! 📱

Z-Score

• The z-score tells us how many standard deviations our sample proportion is from the hypothesized population proportion.
• It's calculated using this formula:

$$z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}$$

Where: - $\hat{p}$ is the sample proportion - $p_0$ is the hypothesized population proportion - n is the sample size

• For our example: - $\hat{p}$ = 700/750 = 0.9333 - $p_0$ = 0.94 - n = 750

$$z = \frac{0.9333 - 0.94}{\sqrt{\frac{0.94(1-0.94)}{750}}} = -0.769$$

P-Value

• The p-value is the probability of getting a sample proportion as extreme as (or more extreme than) ours, assuming the null hypothesis is true.
• We use the z-score and the normal distribution to find the p-value.
• For our example, the p-value is the probability of getting a z-score of -0.769 or lower. Using normalcdf(-100000, -0.769, 0, 1) on a calculator, we get a p-value of 0.221. 🅿️

• Your calculator can do all of this for you! 🍀
• Go to Stats Tests and select 1-Prop Z-Test.
• Enter your data, and your calculator will give you the z-score and p-value automatically! 👏

Final Exam Focus

Alright, let's talk about what to really focus on for the exam. These are the areas that pop up most often:

• Hypothesis Formulation: Make sure you can write clear null and alternative hypotheses. Pay close attention to whether it's a one-tailed or two-tailed test.
• Conditions: Always check and state the random, independent, and normal conditions. Don't skip this step!
• Interpretation: Understand what the p-value means and how to use it to make a conclusion about the null hypothesis.
• Type I and Type II Errors: Be able to identify and explain the consequences of making a Type I or Type II error in the context of the problem.
• Context is Key: Always relate your findings back to the original problem. What does your conclusion mean in the real world?

• Time Management: Don't get bogged down on one problem. If you're stuck, move on and come back later.

• Common Pitfalls: Double-check your calculations, especially the z-score formula. Be careful with calculator inputs.

• FRQ Strategies: Show all your work, even if you use your calculator. Label everything clearly. Write in context.

Practice Questions

Practice Question

Multiple Choice Questions

1. A researcher is testing the hypothesis $H_0: p = 0.4$ versus $H_a: p \neq 0.4$. A random sample of 100 observations yields a sample proportion of 0.48. Which of the following is the correct test statistic?

(A) $z = \frac{0.48 - 0.4}{\sqrt{\frac{0.4(0.6)}{100}}}$ (B) $z = \frac{0.4 - 0.48}{\sqrt{\frac{0.48(0.52)}{100}}}$ (C) $t = \frac{0.48 - 0.4}{\sqrt{\frac{0.4(0.6)}{100}}}$ (D) $t = \frac{0.4 - 0.48}{\sqrt{\frac{0.48(0.52)}{100}}}$ (E) $z = \frac{0.48 - 0.4}{\sqrt{\frac{0.48(0.52)}{100}}}$

2. A polling organization is interested in estimating the proportion of voters who support a particular candidate. They take a random sample of 500 voters and find that 260 support the candidate. What is the standard error of the sample proportion?

(A) 0.000499 (B) 0.000500 (C) 0.0223 (D) 0.0499 (E) 0.0500

3. A company claims that 60% of its customers are satisfied with their service. A random sample of 200 customers is taken, and 110 report being satisfied. What is the p-value of the hypothesis test of $H_0: p = 0.6$ versus $H_a: p < 0.6$?

(A) 0.001 (B) 0.002 (C) 0.004 (D) 0.008 (E) 0.016

Free Response Question

A researcher wants to investigate whether the proportion of adults who own a smartphone is different from 85%. They take a random sample of 400 adults and find that 328 own a smartphone.

(a) State the null and alternative hypotheses for this test. (b) Check if the conditions for conducting a one-proportion z-test are met. (c) Calculate the test statistic and the p-value for this test. (d) What is the conclusion of the test at a significance level of 0.05? (e) Describe a Type II error in the context of this problem.

Scoring Guidelines

(a) 1 point

• H0: p = 0.85
• Ha: p ≠ 0.85

(b) 3 points

• Random: The sample is stated as random.
• Independent: It is reasonable to assume that there are at least 4000 adults in the population.
• Normal: np = 400 * 0.85 = 340 and n(1-p) = 400 * 0.15 = 60. Both are greater than 10. (c) 2 points
• Test statistic: $z = \frac{0.82 - 0.85}{\sqrt{\frac{0.85(0.15)}{400}}} = -1.75$
• P-value: 2 * normalcdf(-100000, -1.75, 0, 1) = 0.080

(d) 1 point

• Since the p-value (0.080) is greater than the significance level (0.05), we fail to reject the null hypothesis. There is not sufficient evidence to conclude that the proportion of adults who own a smartphone is different from 85%.

(e) 1 point

• A Type II error would mean that we fail to reject the null hypothesis when it is false. In this context, it would mean that we conclude that the proportion of adults who own a smartphone is 85% when it is actually different from 85%.

You've got this! Keep practicing, and you'll be ready to rock the AP Stats exam! 🎉