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AP Statistics
Unit 7 Overview: Means
7.1 Introducing Statistics: Should I Worry About Error?
7.2 Constructing a Confidence Interval for a Population Mean
7.3 Justifying a Claim About a Population Mean Based on a Confidence Interval
7.4 Setting Up a Test for a Population Mean
7.5 Carrying Out a Test for a Population Mean
7.6 Confidence Intervals for the Difference of Two Means
7.7 Justifying a Claim About the Difference of Two Means Based on a Confidence Interval
7.8 Setting Up a Test for the Difference of Two Population Means
7.9 Carrying Out a Test for the Difference of Two Population Means
7.10 Skills Focus: Selecting, Implementing, and Communicating Inference Procedures
7.11 Multiple Choice Questions
7.12 MC Answers and Review

📊ap statistics review

7.10 Skills Focus: Selecting, Implementing, and Communicating Inference Procedures

Verified for the 2025 AP Statistics exam•Last Updated on June 18, 2024

image courtesy of:  pixabay.com

Multiple Choice

When given a multiple choice question regarding inferential procedures, it is almost always going to pertain to selecting the correct procedure, interpreting a p-value or drawing a conclusion given a p-value. 🎨

Selecting Correct Procedure

When asked to select a correct procedure, the best way of approaching the problem is to ask yourself 2 questions: 🤔

  1. Am I dealing with means (t-_) or proportions (z-__)?
  2. Do I have one or two samples? This will help you to determine if you are running a one-/two-sample/proportion- t-/z-test/interval. These two questions are the guiding factor in answering what type of test/interval we will run. 

Special Cases

A matched pairs t-test (also known as a dependent samples t-test) is used to compare the means of two related groups, where each subject in one group is paired with a subject in the other group. This type of test is often used in experimental studies where every unit receives both treatments (e.g. a drug and a placebo) and the differences between the treatments are measured. 

On the other hand, a two-sample t-test is used to compare the means of two independent groups. This test is appropriate when the subjects in one group are not related to the subjects in the other group (e.g. men and women). Don't confuse matched pairs t-test with a two-sample t-test!

As for multiple proportions, a chi-square test may be necessary in some cases -- we'll talk about this in the next unit more. In general, a chi-square test is a statistical test that is used to compare observed frequencies with expected frequencies in a contingency table. It is often used to test hypotheses about categorical data, such as the relationship between different groups or the association between two variables. If you have more than two proportions that you want to compare, a chi-square test may be the appropriate statistical test to use.

Interpreting P-Value

When asked to interpret a p-value, remember that it is the probability of obtaining your given sample from the sampling distribution of that particular sample size, given that the true mean/proportion is what the null hypothesis claims. 🅿️

Example

In a hypothesis test where the H0: p = 0.2 and the Ha: p < 0.2, we collect a sample of 100 where our p-hat is 0.15. Our significance test reveals a p-value of 0.11. Interpret this p-value.

A p value of 0.11 tells us that the probability of obtaining a sample of 100 where the success rate was 0.15 or lower would happen approximately 11% of the time, given our normal sampling distribution when n=100.

Drawing Conclusions

In drawing a conclusion from an inference procedure, we are generally comparing a p-value to a significance level. You can follow the chart below in making your decision:

p < alpha: Reject the H0We have significant evidence of the Ha (in context).
p > alpha: Fail to Reject the H0We do not have significant evidence of the Ha (in context).

We never "accept" a H0 or Ha! 🙅 

Free Response

When dealing with free response questions requiring inferential procedures, we usually see one of the following 2 prompts: 💬

  1. Do the data give convincing evidence... (Significance Test)
  2. Construct and interpret a ___% confidence interval (Confidence Interval) Both of these stems can follow the SPDC Template outlined below:

(1) State the Parameters/Hypotheses

  • When performing a confidence interval, here is where you should state what the parameter(s) are for our population(s) that we are estimating.
  • When performing a significance test, this is the place in the problem when you should write the hypotheses for your questions. Also, label and identify your parameters.- Remember, your Ho will always have an equal sign and your Ha will always have some form of inequality (<, > or not equal to)

(2) Plan the Problem

  • This is where we check our three conditions for inference: Random, Independent and Normal. This is basically the same from confidence interval or significance test, but varies based on the type of data (categorical or quantitative). 📝

(3) Do the Math

  • Start out by identifying the test/interval you are performing. This is usually which function you are selecting in the STATS menu of your calculator. Write this down! 🖊️
  • Then write out your answer from your calculator:- Confidence Intervals: Just the interval is sufficient- Significance Test: Critical Value, p-value, and df (if necessary)

(4) Conclusion

This is where you follow templates given throughout the unit. 🤏

  • For confidence intervals: "I am _% confident that the true _____ of __________ is between (, __).
  • For significance tests:  "Since the p(</>) alpha, I (fail to reject/reject) the Ho. There (is/is not) convincing evidence of Ha (in context of problem)."

🎥 Watch: AP Stats - Review of Inference: z and t Procedures

Key Terms to Review (19)

Alternative Hypothesis (Ha): The Alternative Hypothesis (Ha) is a statement that suggests there is a significant effect or difference in a given statistical test. It is the hypothesis that researchers aim to support, contrasting with the null hypothesis (H0), which asserts that there is no effect or difference. The Alternative Hypothesis is crucial for determining the direction of a test, whether it be one-tailed or two-tailed, and helps researchers identify significant relationships in categorical data when conducting tests like Chi-Square.
Categorical Data: Categorical data refers to variables that can be divided into distinct groups or categories. This type of data is often used to represent characteristics such as gender, color, or type of vehicle, and it is essential for analyzing relationships between different groups within a dataset. Understanding categorical data is crucial for conducting various statistical tests, visualizations, and interpretations, particularly when it comes to comparing proportions and frequencies across categories.
Chi-Square Test: A Chi-Square Test is a statistical method used to determine if there is a significant association between categorical variables or to assess how well an observed distribution fits an expected distribution. This test helps in understanding relationships and differences within data, making it essential for analyzing survey results, experimental data, and more.
Confidence Interval: A confidence interval is a range of values derived from sample statistics that is likely to contain the true value of an unknown population parameter, with a specified level of confidence. This concept connects statistical inference to the estimation of parameters, allowing researchers to make informed claims about populations based on sample data.
Contingency Table: A contingency table is a statistical tool used to display the frequency distribution of variables, typically categorical ones, to understand the relationship between them. It allows researchers to summarize data and visualize potential associations, helping to identify patterns or correlations that may exist within the data set.
Critical Value: A critical value is a point on the scale of the test statistic that is compared to the test statistic to determine whether to reject the null hypothesis. It serves as a threshold that defines the boundaries of the rejection region, which helps in assessing whether the observed data falls significantly outside what would be expected under the null hypothesis.
Dependent samples t-test: A dependent samples t-test is a statistical method used to determine if there is a significant difference between the means of two related groups. This test is appropriate when the same subjects are measured under different conditions or at different times, allowing for the analysis of paired data. It helps assess whether changes in the mean scores can be attributed to the treatment or intervention rather than random variation.
Hypothesis Test: A hypothesis test is a statistical method used to make inferences or draw conclusions about a population based on sample data. This process involves formulating a null hypothesis and an alternative hypothesis, followed by analyzing the data to determine whether to reject the null hypothesis in favor of the alternative. The validity and reliability of these tests are crucial for assessing claims about population parameters and making data-driven decisions.
Independent Events: Independent events are two or more events that do not influence each other's occurrence. This means the outcome of one event has no effect on the outcome of another event, which is crucial in various statistical methods and calculations.
Inferential Procedures: Inferential procedures are statistical methods used to make inferences or generalizations about a population based on sample data. These procedures involve estimating population parameters, testing hypotheses, and determining the likelihood that observed data occurred by chance. By utilizing inferential statistics, researchers can draw conclusions that extend beyond the immediate data at hand, allowing for broader insights and predictions.
Matched pairs t-test: A matched pairs t-test is a statistical method used to compare the means of two related groups to determine if there is a significant difference between them. This test is particularly useful when each subject in one group is paired with a corresponding subject in the other group, such as before-and-after measurements on the same individuals or matched subjects based on certain characteristics. It focuses on the differences within each pair, allowing for a more sensitive analysis of the data.
Normal Distribution: Normal distribution is a continuous probability distribution characterized by a symmetric, bell-shaped curve, where most of the observations cluster around the central peak and probabilities for values farther away from the mean taper off equally in both directions. This concept is foundational in statistics, as many statistical tests and methods, including confidence intervals and hypothesis tests, rely on the assumption that the underlying data follows a normal distribution.
Null Hypothesis (H0): The Null Hypothesis (H0) is a statement that asserts there is no significant effect or difference in a given context, serving as a baseline for statistical testing. It acts as the starting point for analysis, allowing researchers to determine if any observed effects in data are statistically significant compared to the status quo, guiding the acceptance or rejection of potential alternatives.
P-value: A P-value is a measure used in hypothesis testing to determine the strength of evidence against the null hypothesis. It quantifies the probability of observing test results at least as extreme as the ones obtained, assuming that the null hypothesis is true. A smaller P-value indicates stronger evidence against the null hypothesis, which is crucial for decision-making in various statistical tests.
Random Sampling: Random sampling is a method of selecting individuals from a population in such a way that every member has an equal chance of being chosen. This technique ensures that the sample is representative of the population, minimizing bias and allowing for generalizations to be made about the whole group.
Significance Level: The significance level, often denoted as alpha (\(\alpha\)), is the threshold used to determine whether to reject the null hypothesis in statistical hypothesis testing. It represents the probability of making a Type I error, which occurs when a true null hypothesis is incorrectly rejected. Understanding the significance level is crucial for interpreting results and making informed decisions based on statistical tests.
T-test: A t-test is a statistical test used to determine if there is a significant difference between the means of two groups or to assess the significance of a regression slope. This test is especially useful when the sample sizes are small and the population standard deviation is unknown, making it a key tool in hypothesis testing for both population means and regression analysis.
Two-sample t-test: A two-sample t-test is a statistical method used to determine whether the means of two independent groups are significantly different from each other. This test assumes that the data from both groups are normally distributed and have similar variances, allowing researchers to make inferences about the populations based on sample data.
Z-test: A z-test is a statistical method used to determine if there is a significant difference between the means of two groups, or between a sample mean and a known population mean, assuming that the data follows a normal distribution. It uses the standard normal distribution to calculate the z-score, which indicates how many standard deviations an element is from the mean. This method is commonly used when the sample size is large or the population standard deviation is known.
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7.11 Multiple Choice Questions