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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:

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