7.10 Skills Focus: Selecting, Implementing, and Communicating Inference Procedures
4 min read•january 5, 2023
Josh Argo
Jed Quiaoit
Josh Argo
Jed Quiaoit
image courtesy of: pixabay.com
Multiple Choice
When given a multiple choice question regarding , it is almost always going to pertain to selecting the correct procedure, interpreting a or drawing a conclusion given a . 🎨
Selecting Correct Procedure
When asked to select a correct procedure, the best way of approaching the problem is to ask yourself 2 questions: 🤔
Am I dealing with means (t-___) or proportions (z-____)?
Do I have one or two samples?
This will help you to determine if you are running a one-/two-sample/proportion- t-//interval. These two questions are the guiding factor in answering what type of test/interval we will run.
Special Cases
A (also known as a ) 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 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 with a !
As for multiple proportions, a 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 . It is often used to test hypotheses about , 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 may be the appropriate statistical test to use.
Interpreting P-Value
When asked to interpret a , 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 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 of 0.11. Interpret this .
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 to a . You can follow the chart below in making your decision:
| p < alpha: Reject the H0 | We have significant evidence of the Ha (in context). |
| p > alpha: Fail to Reject the H0 | We 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 , we usually see one of the following 2 prompts: 💬
Do the data give convincing evidence... (Significance Test)
Construct and interpret a ___% ()
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 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: , , 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, denoted as Ha, is a statement that contradicts or challenges the null hypothesis. It suggests that there is a significant relationship or difference between variables being studied.Categorical Data
: Categorical data refers to data that can be divided into categories or groups based on qualitative characteristics.Chi-Square Test
: A statistical test used to determine if there is a significant association between two categorical variables. It compares the observed frequencies with the expected frequencies under the assumption of independence.Confidence Interval
: A confidence interval is a range of values that is likely to contain the true value of a population parameter. It provides an estimate along with a level of confidence about how accurate the estimate is.Contingency Table
: A contingency table is a table that displays the frequencies or counts of two categorical variables. It shows how the categories of one variable are distributed across the categories of another variable.Critical Value
: A critical value is a specific value that separates the rejection region from the non-rejection region in hypothesis testing. It is compared to the test statistic to determine whether to reject or fail to reject the null hypothesis.Dependent samples t-test
: A dependent samples t-test, also known as a paired-samples or repeated measures t-test, is used to determine whether there is a significant difference between the means of two related groups or conditions.Hypothesis Test
: A hypothesis test is a statistical procedure used to make decisions about whether there is enough evidence to support or reject a claim about a population parameter. It involves formulating null and alternative hypotheses, collecting data, and using statistical tests to evaluate those hypotheses.Independent Events
: Independent events are events that have no influence on each other. The outcome of one event does not affect the outcome of another event.Inferential Procedures
: Inferential procedures are statistical methods used to draw conclusions or make predictions about a population based on sample data. These procedures involve making inferences and generalizations from the sample to the larger population.Matched pairs t-test
: A matched pairs t-test is a statistical test used to determine whether there is a significant difference between paired observations or measurements taken on the same subjects under different conditions.Normal Distribution
: A normal distribution is a symmetric bell-shaped probability distribution characterized by its mean and standard deviation. It follows a specific mathematical formula called Gaussian distribution.Null Hypothesis (H0)
: The null hypothesis is a statement that assumes there is no significant difference or relationship between variables in a statistical analysis.P-value
: The p-value is a probability value that helps determine whether an observed result is statistically significant or occurred by chance. It quantifies how strong or weak evidence against a null hypothesis exists.Random Sampling
: Random sampling is a method of selecting individuals from a population in such a way that every individual has an equal chance of being chosen. It helps to ensure that the sample represents the population accurately.Significance Level
: The significance level, also known as alpha (α), determines how much evidence we need to reject the null hypothesis. It represents the probability of making a Type I error.t-test
: A t-test is a statistical test that compares two sample means to determine if they are significantly different from each other.Two-sample t-test
: A two-sample t-test is a statistical test used to compare the means of two independent groups and determine if they are significantly different from each other.z-test
: A z-test is a statistical test used to determine if there is a significant difference between a sample mean and a population mean when the population standard deviation is known. It compares sample data by calculating a z-value based on differences between means and their variability.7.10 Skills Focus: Selecting, Implementing, and Communicating Inference Procedures
4 min read•january 5, 2023
Josh Argo
Jed Quiaoit
Josh Argo
Jed Quiaoit
image courtesy of: pixabay.com
Multiple Choice
When given a multiple choice question regarding , it is almost always going to pertain to selecting the correct procedure, interpreting a or drawing a conclusion given a . 🎨
Selecting Correct Procedure
When asked to select a correct procedure, the best way of approaching the problem is to ask yourself 2 questions: 🤔
Am I dealing with means (t-___) or proportions (z-____)?
Do I have one or two samples?
This will help you to determine if you are running a one-/two-sample/proportion- t-//interval. These two questions are the guiding factor in answering what type of test/interval we will run.
Special Cases
A (also known as a ) 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 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 with a !
As for multiple proportions, a 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 . It is often used to test hypotheses about , 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 may be the appropriate statistical test to use.
Interpreting P-Value
When asked to interpret a , 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 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 of 0.11. Interpret this .
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 to a . You can follow the chart below in making your decision:
| p < alpha: Reject the H0 | We have significant evidence of the Ha (in context). |
| p > alpha: Fail to Reject the H0 | We 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 , we usually see one of the following 2 prompts: 💬
Do the data give convincing evidence... (Significance Test)
Construct and interpret a ___% ()
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 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: , , 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, denoted as Ha, is a statement that contradicts or challenges the null hypothesis. It suggests that there is a significant relationship or difference between variables being studied.Categorical Data
: Categorical data refers to data that can be divided into categories or groups based on qualitative characteristics.Chi-Square Test
: A statistical test used to determine if there is a significant association between two categorical variables. It compares the observed frequencies with the expected frequencies under the assumption of independence.Confidence Interval
: A confidence interval is a range of values that is likely to contain the true value of a population parameter. It provides an estimate along with a level of confidence about how accurate the estimate is.Contingency Table
: A contingency table is a table that displays the frequencies or counts of two categorical variables. It shows how the categories of one variable are distributed across the categories of another variable.Critical Value
: A critical value is a specific value that separates the rejection region from the non-rejection region in hypothesis testing. It is compared to the test statistic to determine whether to reject or fail to reject the null hypothesis.Dependent samples t-test
: A dependent samples t-test, also known as a paired-samples or repeated measures t-test, is used to determine whether there is a significant difference between the means of two related groups or conditions.Hypothesis Test
: A hypothesis test is a statistical procedure used to make decisions about whether there is enough evidence to support or reject a claim about a population parameter. It involves formulating null and alternative hypotheses, collecting data, and using statistical tests to evaluate those hypotheses.Independent Events
: Independent events are events that have no influence on each other. The outcome of one event does not affect the outcome of another event.Inferential Procedures
: Inferential procedures are statistical methods used to draw conclusions or make predictions about a population based on sample data. These procedures involve making inferences and generalizations from the sample to the larger population.Matched pairs t-test
: A matched pairs t-test is a statistical test used to determine whether there is a significant difference between paired observations or measurements taken on the same subjects under different conditions.Normal Distribution
: A normal distribution is a symmetric bell-shaped probability distribution characterized by its mean and standard deviation. It follows a specific mathematical formula called Gaussian distribution.Null Hypothesis (H0)
: The null hypothesis is a statement that assumes there is no significant difference or relationship between variables in a statistical analysis.P-value
: The p-value is a probability value that helps determine whether an observed result is statistically significant or occurred by chance. It quantifies how strong or weak evidence against a null hypothesis exists.Random Sampling
: Random sampling is a method of selecting individuals from a population in such a way that every individual has an equal chance of being chosen. It helps to ensure that the sample represents the population accurately.Significance Level
: The significance level, also known as alpha (α), determines how much evidence we need to reject the null hypothesis. It represents the probability of making a Type I error.t-test
: A t-test is a statistical test that compares two sample means to determine if they are significantly different from each other.Two-sample t-test
: A two-sample t-test is a statistical test used to compare the means of two independent groups and determine if they are significantly different from each other.z-test
: A z-test is a statistical test used to determine if there is a significant difference between a sample mean and a population mean when the population standard deviation is known. It compares sample data by calculating a z-value based on differences between means and their variability.
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