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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

7.8 Setting Up a Test for the Difference of Two Population Means

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

image courtesy of: pixabay.com

Have you ever looked at two different populations (maybe classes, brands of food, types of materials) and wondered, "Hmm..are these two things REALLY different?" This is exactly where a test for the difference of two population means is useful. As long as your data is quantitative, in other words we are testing means, this is the type of test we will use. If you are using some form of technology, you will select Two Sample T Test. In making note of what test you are performing on the AP Statistics exam, it is best to be more specific and state, "Two Sample T Test for Difference in Two Population Means." 🚂

A two-sample t-test is used to determine whether the means of two independent groups are significantly different from each other. It is a parametric test, meaning that it assumes that the data follows a normal distribution and that the variances of the two groups are equal. 

The test calculates the difference between the two group means and compares it to the standard error of the difference, and then uses this information to calculate a t-statistic. The t-statistic is then used to determine the p-value, which indicates the probability that the difference between the two group means occurred by chance.

Hypotheses

As with any statistical test, the first step necessary to perform the significance test is to write our hypotheses. We always have a null hypothesis, which is the hypothesis that the two populations are not different. Then we have our alternate hypothesis, which states that they are different in some way (either less than, greater than or simply not equal to).

When writing out your hypotheses, you should state them as follows: 📝

Ho:  𝞵1 = 𝞵2

Ha:  𝞵1 ≠ 𝞵2, 𝞵1 < 𝞵2, or 𝞵1 > 𝞵2

Another way of writing them using differences is as follows:

Ho:  𝞵1 - 𝞵2 = 0

Ha:  𝞵1 - 𝞵2 > 0, 𝞵1 - 𝞵2 < 0, or 𝞵1 - 𝞵2 ≠ 0

The first option is more in line with the technology used in actually computing the test statistic and p value that we will cover in Unit 7.9.

Conditions

Also, as with any significance test, there are conditions for inference that we must check to assure that our test is accurately going to be able to draw conclusions about a said population.

(1) Random

When drawing our two samples to perform our t-test, it is absolutely imperative that our samples are random from the given populations. If they are not random, we cannot generalize our results to the two given populations, which renders our tests useless and there is no way to fix sampling bias with the numbers. If the test involves an experimental study, it is important to note that the treatments were randomly assigned. This allows us to make a causation conclusion. ☑️

(2) Independent

Since we are normally sampling without replacement, it is also important to make sure our sample is independently chosen. This can be assumed under the 10% condition which states that as long as our population is 10x our sample, we can assume independence. If dealing with an experimental study, independence is not necessary since we are randomizing treatments. ☑️

(3) Normal

When calculating our p-value, we are going to make use of the t curve to see what the probability of obtaining our samples are. In order to ensure that we can use the t curve for our sampling distribution, we must check to be sure that either: ☑️

  1. Each of our samples are greater than or equal to 30. (Central Limit Theorem)
  2. Our populations are normally distributed. (Given in question prompt)
  3. A boxplot of our sample data shows no obvious skewness or apparent outliers. (Last resort)

Example

Mr. Fleck runs a green bean farm. He has two fields that he normally picks from. Every day, he goes out and picks green beans from both fields and has found that the two fields appear to be yielding different amount of crops. In order to test his theory, he randomly selects 120 days to pick from both fields. Field A yields an average of 580 beans with a standard deviation of 25, while Field B yields an average of 550 with a standard deviation of 12. Do the data give convincing evidence that the two fields yield different amount of beans? 🌽

Hypotheses

Our null hypothesis is that the two fields yield equal amounts of beans, so our Ho: 𝞵A = 𝞵B, where 𝞵A is the true mean number of beans that comes from field A everyday and 𝞵B is the true mean number of beans that comes from Field B everyday.

Our alternate hypothesis is that these two are different, so our Ha: 𝞵A ≠ 𝞵B. 🫘

Conditions

  • Random: both samples are randomly selected by the day chosen
  • Independent: It is reasonable to believe that there are 1200 days that he could pick from the fields.
  • Normal: 120>30, so our sampling distribution of the difference of the two means will be approximately normal

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

Key Terms to Review (19)

10% Condition: The 10% Condition is a guideline used in statistics to ensure that the sample size taken from a population is small enough relative to the population size, typically indicating that the sample size should be less than 10% of the total population. This condition helps justify the use of certain statistical procedures and calculations, as it ensures that sampling without replacement is approximately equal to sampling with replacement, making results more reliable.
Alternate Hypothesis: The alternate hypothesis is a statement that proposes a change, difference, or effect in a statistical test, serving as a counterpoint to the null hypothesis. It reflects what researchers aim to support, indicating that there is an effect or relationship worth investigating. The alternate hypothesis is crucial for determining the outcome of statistical tests and guides the decision-making process in hypothesis testing.
Boxplot: A boxplot is a graphical representation that displays the distribution of a dataset through its quartiles, highlighting the median, and identifying potential outliers. It provides a visual summary that helps compare different datasets, particularly when analyzing the differences between two population means. The box in the plot represents the interquartile range (IQR), while the lines extending from the box, known as whiskers, show the range of the data.
Central Limit Theorem: The Central Limit Theorem (CLT) states that the sampling distribution of the sample mean approaches a normal distribution as the sample size increases, regardless of the population's distribution, given that the samples are independent and identically distributed. This theorem is crucial because it enables statisticians to make inferences about population parameters even when the population distribution is not normal, thereby connecting to hypothesis testing, confidence intervals, and various types of sampling distributions.
Causation Conclusion: A causation conclusion refers to the determination that one event or variable directly influences another. In the context of comparing two population means, establishing a causation conclusion is essential for understanding whether a change in one mean causes a change in another, rather than simply observing a correlation. This concept underscores the importance of experimental design and control in drawing valid inferences about relationships between variables.
Equal Variances: Equal variances, also known as homoscedasticity, refers to the condition in statistical analyses where two or more populations have the same variance. This concept is crucial when comparing the means of two populations, as many statistical tests assume that the variances are equal to ensure valid results. When variances are unequal, it can affect the reliability of the test results and lead to incorrect conclusions.
Experimental Study: An experimental study is a research method where the investigator manipulates one or more variables to determine their effect on a dependent variable while controlling other factors. This type of study is essential for establishing cause-and-effect relationships, particularly when comparing the differences between two population means. By randomly assigning subjects to treatment and control groups, researchers can minimize bias and ensure that observed effects are due to the manipulation of the independent variable.
Independent Samples: Independent samples are two or more groups that are selected in such a way that the members of one group do not influence or affect the members of the other group. This concept is crucial when comparing different populations, as it ensures that the results of a statistical test reflect true differences rather than biases from overlapping influences. Understanding independent samples is essential for accurate hypothesis testing and for constructing confidence intervals, as it allows researchers to make valid inferences about the populations being studied.
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: The null hypothesis is a statement that assumes there is no effect or no difference in a given situation, serving as the foundation for statistical testing. It provides a baseline against which alternative hypotheses are tested, guiding researchers in determining whether observed data significantly deviates from what is expected under this assumption.
Outliers: Outliers are data points that significantly differ from the rest of the data in a dataset. These values can have a major impact on statistical analyses and can affect the results of regression models, correlation calculations, and various statistical tests. Identifying outliers is crucial, as they can indicate variability in measurements, experimental errors, or novel insights into the data.
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.
Parametric Test: A parametric test is a statistical procedure that makes assumptions about the parameters of the population distribution from which a sample is drawn. These tests generally require data to follow a normal distribution and assume equal variances among groups. This approach allows for more powerful statistical inferences when conditions are met, particularly when comparing the means of two populations.
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.
Sampling Bias: Sampling bias occurs when certain individuals or groups within a population are more likely to be selected for a sample than others, leading to an unrepresentative sample. This can distort the results of statistical analyses, affecting conclusions drawn about the entire population and leading to incorrect generalizations.
Significance Test: A significance test is a statistical method used to determine if there is enough evidence to reject a null hypothesis in favor of an alternative hypothesis. This process involves comparing sample data to what is expected under the null hypothesis and calculating a p-value to evaluate the strength of the evidence against the null hypothesis. It plays a crucial role in comparing population means and proportions, interpreting p-values, and drawing conclusions from data.
Skewness: Skewness measures the asymmetry of a probability distribution, indicating whether data points are concentrated on one side of the mean. A positive skewness means that the tail on the right side of the distribution is longer or fatter than the left, while negative skewness indicates a longer or fatter tail on the left. Understanding skewness is crucial in interpreting data distributions, which plays a significant role in regression analysis and hypothesis testing.
T-statistic: The t-statistic is a standardized value that measures the size of the difference relative to the variation in your sample data. It's used primarily in hypothesis testing to determine if there is a significant difference between two population means or to test the slope of a regression model. This statistic allows researchers to make inferences about population parameters based on sample statistics, especially when sample sizes are small and the population standard deviation is unknown.
Two Sample T Test: A Two Sample T Test is a statistical method used to compare the means of two independent groups to determine if there is a significant difference between them. It assesses whether the observed differences in sample means are due to random chance or if they reflect true differences in the population means. This test is essential for analyzing data from experiments or surveys where two separate groups are evaluated under different conditions.
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7.9 Carrying Out a Test for the Difference of Two Population Means