Introducing Statistics: Are My Results Unexpected? - AP Statistics Study Guide

Quick Summary

This guide introduces the foundational logic of statistical inference. You will learn how to formulate a testable claim about a population proportion by writing null and alternative hypotheses. By understanding how to use a simulated sampling distribution, you will be able to determine if an observed sample result is statistically unusual and calculate a p-value to quantify the strength of evidence against the null hypothesis.

Key Concepts

This topic lays the groundwork for all of hypothesis testing. The core idea is to ask: "Is the result I see in my sample so unusual that it makes me doubt my initial assumption about the population?"

1. The Logic of Hypothesis Testing: An Analogy

Think of hypothesis testing like a criminal trial in the U.S. justice system.

• The Assumption: A defendant is "innocent until proven guilty." In statistics, our starting assumption is that the null hypothesis is true. This is the "no change," "no effect," or "status quo" claim.

• The Evidence: The prosecutor presents evidence to the jury. In statistics, our evidence is the data we collect from our sample (our sample statistic).

• The Question: The jury must decide if the evidence is strong enough to reject the assumption of innocence "beyond a reasonable doubt." In statistics, we ask if our sample statistic is so far from what the null hypothesis predicts that it's unlikely to have occurred by random chance alone.

• The Verdict: The jury returns a verdict of "guilty" or "not guilty." They never declare the defendant "innocent." Similarly, we either reject the null hypothesis (we have convincing evidence for our new claim) or we fail to reject the null hypothesis (we do not have convincing evidence). We never "accept" the null hypothesis as true.

2. Formulating Hypotheses

Hypotheses are always statements about a population parameter, never a sample statistic. For this unit, our parameter is the population proportion, p.

• The Null Hypothesis (H₀):

  • This is the baseline assumption, the status quo. It's the hypothesis we are trying to find evidence against.

  • It always contains a statement of equality ($=$).

  • Form: H₀: p = p₀ (where p₀ is the hypothesized value).

  • Example: A company claims 20% of its chips are blue. The null hypothesis would be H₀: p = 0.20, where p is the true proportion of blue chips.

• The Alternative Hypothesis (Hₐ or H₁):

  • This is the claim we are trying to find evidence for. It's what we suspect might be true instead of the null.

  • It always contains a statement of strict inequality ($<$, $>$, or $\ne$).

  • One-Sided Alternatives:

    • Greater Than (>): Used when we suspect the true proportion is greater than the hypothesized value. (e.g., "Has the proportion of students who pass increased?")

    • Less Than (<): Used when we suspect the true proportion is less than the hypothesized value. (e.g., "Has a new process reduced the proportion of defective parts?")

  • Two-Sided Alternative:

    • Not Equal To (\neq): Used when we suspect the true proportion is simply different from the hypothesized value, without a specific direction. (e.g., "Has the proportion of voters who support the candidate changed from the last poll?")

Crucial Rule: You must choose your alternative hypothesis before you collect and look at your data. Using the data to choose the direction of the alternative is a form of statistical malpractice.

3. The Role of Sampling Distributions and Simulation

To decide if our sample result is "unusual," we need to know what "usual" results look like if the null hypothesis is true.

• The Model: The sampling distribution of the sample proportion (p̂) shows us every possible value of p̂ we could get from a sample of a certain size, and how often each value would occur, assuming H₀ is true.

• Simulation: We can create an approximate sampling distribution by simulating the random process many times. For example, if H₀: p = 0.50, we could flip a coin 100 times (one sample), record the proportion of heads, and repeat this process 1000 times. A dotplot of these 1000 sample proportions would show us what typical results look like when the true proportion is 0.50.

[Image: A dotplot showing the results of 100 simulations of a sample of size 50 where p=0.6. The distribution is centered at 0.6 and has a roughly symmetric, bell shape.]

4. The P-value: Quantifying "Unusual"

The p-value is the heart of hypothesis testing. It's a conditional probability that measures how surprising our result is.

• Definition: The p-value is the probability of obtaining a sample result as extreme or more extreme than the one actually observed, assuming the null hypothesis (H₀) is true.

• Interpreting "As Extreme or More Extreme": This phrase depends on your alternative hypothesis (Hₐ).

  • If Hₐ: p > p₀, "extreme" means your observed sample proportion (p̂) or greater.

  • If Hₐ: p < p₀, "extreme" means your observed sample proportion (p̂) or less.

  • If Hₐ: p \neq p₀, "extreme" means as far away from p₀ (in either direction) as your p̂ is, or farther.

[Image: A normal curve representing a sampling distribution centered at p₀. For a "greater than" test, the observed p̂ is on the right, and the p-value is the shaded area to its right.]

5. Making a Conclusion

The p-value provides the evidence. We compare it to a pre-determined threshold called the significance level (α), which is usually 0.05 unless stated otherwise.

• Small P-value (e.g., p-value < α):

  • Our observed result is very unlikely to occur by random chance if the null hypothesis were true.

  • This provides strong evidence against H₀ and in favor of Hₐ.

  • Conclusion: We reject the null hypothesis (H₀).

• Large P-value (e.g., p-value \ge α):

  • Our observed result is plausible; it could have easily occurred by random chance if the null hypothesis were true.

  • This does not provide convincing evidence against H₀.

  • Conclusion: We fail to reject the null hypothesis (H₀).

Key Vocabulary

• Null Hypothesis (H₀): The claim of "no difference" or "no effect," stated using a population parameter. It is the assumption we begin with, and it always contains an equality sign.

• Alternative Hypothesis (Hₐ): The claim for which we are trying to find evidence. It is a statement of inequality (<, >, or \neq) about a population parameter.

• P-value: The probability of getting a sample statistic as extreme or more extreme than the one observed, calculated under the assumption that the null hypothesis is true. A small p-value indicates an unusual result.

• Parameter: A numerical value that describes a characteristic of a population (e.g., p, the true proportion).

• Statistic: A numerical value that describes a characteristic of a sample (e.g., p̂, the sample proportion). We use statistics to make inferences about parameters.

• Sampling Distribution: A theoretical probability distribution of a statistic obtained from all possible samples of a specific size from a population. It shows what values the statistic can take and how often.

Calculator Tech (TI-84)

While formal hypothesis tests have dedicated calculator functions (covered in later topics), you can use the calculator to run simulations that help you estimate p-values. The key function is $randBin()$.

$randBin(n,p,[numberoftrials])$: Simulates binomial experiments.

• $n$: The sample size.

• $p$: The probability of success on one trial (the value from H₀).

• $[numberoftrials]$: How many times you want to repeat the experiment.

Example: A basketball player claims to make 80% of her free throws. In a sample of 50 shots, she makes 32. Is this result unusual? Let's simulate 200 samples of 50 shots, assuming she is an 80% shooter.

1. Go to the MATH menu: Press MATH.

2. Navigate to PRB: Arrow over to the PRB menu.

3. Select randBin(): Choose 7:randBin(.

4. Enter the arguments:$randBin(50,0.80,200)$

   • $n=50$ (sample size)

   • $p=0.80$ (hypothesized proportion from H₀)

   • $200$ (we want to simulate 200 different sets of 50 shots)

5. Store the results: Press STO-> and then 2nd -> 1 to store the results in list L1. Your screen should say randBin(50, 0.80, 200) -> L1. Press ENTER.

6. Analyze the results: You can now view the list L1 (STAT -> Edit...) or create a histogram/dotplot of L1 to see the distribution of the number of made shots in your 200 simulated samples. To find the p-value, you would count how many of the 200 simulated results were 32 or less.

How to Show Work on the FRQ

For this introductory topic, Free Response Questions will focus on your ability to correctly set up the first step of a hypothesis test: defining the parameter and stating the hypotheses. Use this clear, two-step template to earn full credit.

Template for Stating Hypotheses:

1. Define the Parameter:

   • Let p = the true proportion of [describe the population and success attribute in context].

2. State the Hypotheses:

   • H₀: p = [value from the null hypothesis]

   • Hₐ: p [<, >, or \neq] [value from the null hypothesis]

Example FRQ Application:

Prompt: "A recent report claimed that 28% of high school students have a part-time job. A school counselor suspects the proportion is actually lower at her school. She takes a random sample of students to investigate."

Your Answer:

1. Parameter: Let p = the true proportion of all students at the counselor's high school who have a part-time job.

2. Hypotheses:

   • H₀: p = 0.28

   • Hₐ: p < 0.28

This structure is precise, uses correct notation, and is grounded in the context of the problem, ensuring you meet all scoring guidelines.

Practice Problems

Problem 1:

A pharmaceutical company develops a new drug to reduce the side effects of a medical treatment. They claim the new drug is more effective than the old drug, which was known to be effective in 60% of patients. Researchers conduct a clinical trial to test the company's claim. State the null and alternative hypotheses the researchers should use for their study.

Solution:

This problem asks for the hypotheses to test if the new drug is more effective. This indicates a "greater than" alternative hypothesis.

1. Define the Parameter:

   Let p = the true proportion of all patients who experience a reduction in side effects when taking the new drug.

2. State the Hypotheses:

   The "status quo" or baseline is the effectiveness of the old drug, which is 60%. The company claims the new drug is better.

   • H₀: p = 0.60

   • Hₐ: p > 0.60

Problem 2:

A student believes their favorite YouTuber's claim that 30% of their subscribers are international. The student surveys a random sample of 80 subscribers and finds that 30 of them (p̂ = 30/80 = 0.375) are international. To see if this result is unusual, the student's teacher runs a simulation of 100 samples of size 80, assuming the YouTuber's claim (p = 0.30) is true. The dotplot below shows the number of international subscribers in each of the 100 simulated samples.

[Image: A dotplot titled "Simulation of International Subscribers (n=80, p=0.30)". The dots are centered around 24 (which is 80 * 0.30). The distribution ranges from about 15 to 33. There are dots at the following values at or above 30: one dot at 30, two dots at 31, and one dot at 33.]

Based on the simulation, what is the approximate p-value? What conclusion should the student make?

Solution:

The student is testing if their sample result of 30 international subscribers is unusual, assuming the true proportion is 30%. The hypotheses would be H₀: p = 0.30 and Hₐ: p \neq 0.30 (since the student is just checking if the claim is plausible, not if it's specifically higher or lower). However, for calculating the p-value from this specific result, we look at how extreme 30 is.

1. Identify the Observed Result: The student observed 30 international subscribers.

2. Calculate the Approximate P-value from the Simulation: The p-value is the probability of getting a result as extreme or more extreme than 30, assuming p=0.30. We look at the dotplot to see how many of the 100 simulated samples resulted in 30 or more international subscribers.

   • Number of dots at 30: 1

   • Number of dots at 31: 2

   • Number of dots at 32: 0

   • Number of dots at 33: 1

   • Total "extreme" results = 1 + 2 + 0 + 1 = 4.

   Since there are 100 total simulations, the approximate probability (p-value) is 4/100 = 0.04.

   (Note: For a two-sided test, we would also consider results as extreme in the other direction, but this simulation doesn't show results low enough to be as far from the center as 30 is. We'll stick with the one-sided calculation which is what is typically expected from a dotplot like this.)

3. Make a Conclusion:

   The approximate p-value is 0.04. Assuming a standard significance level of α = 0.05, our p-value (0.04) is less than α (0.05). This means our observed result of 30 international subscribers is statistically unusual if the true proportion were really 30%. Therefore, we have convincing evidence to reject the null hypothesis. The student should conclude that there is strong evidence that the true proportion of international subscribers is not 30%.

Common Mistakes to Avoid

• Using Statistics in Hypotheses: Never write hypotheses with p̂ (the sample statistic). Hypotheses are always about p (the population parameter). Writing H₀: p̂ = 0.50 is an immediate and critical error.

• Misinterpreting the P-value: The p-value is NOT the probability that H₀ is true. It is the probability of seeing your data (or more extreme data) given that H₀ is true. This is a subtle but crucial distinction.

• "Accepting" the Null Hypothesis: If you have a large p-value, you conclude that you "fail to reject H₀." You never "accept H₀" or claim that "H₀ is true." A lack of evidence against a claim is not the same as evidence for that claim.

• Forgetting to Define the Parameter: On an FRQ, always define the parameter (p) in the context of the problem before stating your hypotheses. Omitting this step will lose you points.

• Choosing the Alternative Hypothesis After Seeing the Data: You must formulate your hypotheses based on the question being asked before you look at your sample data. If you use your sample result to decide whether to use a < or > sign in Hₐ, you are invalidating the entire statistical procedure.