Quick Summary
This guide will equip you to construct and interpret a confidence interval for the difference between two population means. You will learn to identify the correct inference procedure, verify the necessary conditions for two independent samples, calculate the interval using the standard formula and your calculator, and communicate your findings using the precise language required for the AP exam.
Key Concepts
When we want to estimate the true difference between the means of two distinct populations (e.g., the difference in average test scores between two teaching methods), we use a two-sample t-interval for a difference of means.
The Core Idea
Parameter of Interest: Our goal is to estimate μ₁ - μ₂, the true difference between the mean of population 1 and the mean of population 2.
Point Estimate: The best guess for μ₁ - μ₂ is the difference between our sample means, x̄₁ - x̄₂.
The Formula: The structure of the confidence interval is the same as always: Point Estimate ± (Critical Value) * (Standard Error).
For a difference of two means, this translates to:
(x̄₁ - x̄₂) ± t * √((s₁^2/n₁) + (s₂^2/n₂))*
(x̄₁ - x̄₂): The point estimate, or the difference in sample means.
t:* The critical value from a t-distribution. It depends on the confidence level and the degrees of freedom.
√((s₁^2/n₁) + (s₂^2/n₂)): The standard error of the difference between two sample means. This formula combines the variability from both samples. Notice we add the variances (s^2) inside the square root.
[Image: A diagram showing two separate normal distributions for Population 1 and Population 2, with means μ₁ and μ₂. Below it, a third normal distribution representing the sampling distribution of (x̄₁ - x̄₂), centered at μ₁ - μ₂.]
Conditions for Inference
Before we can construct a valid confidence interval, we must check three conditions. Crucially, these conditions must be verified for both samples independently.
Random Condition: The data must come from two independent random samples or two groups in a randomized experiment.
Why? This ensures the samples are representative of their respective populations and helps prevent bias.
How to check: Look for the words "random sample" or "randomly assigned" in the problem description.
10% Condition (Independence of Observations): When sampling without replacement, the sample size (n) should be no more than 10% of the population size (N). This is checked for each sample: n₁ \le 0.10N₁ and n₂ \le 0.10N₂.
Why? This allows us to treat the observations as independent, which is a requirement for calculating the standard error.
How to check: If the population size is known, verify the math. If it's not, state that it's reasonable to assume the population is at least 10 times the sample size (e.g., "It is reasonable to assume there are more than 10 * 30 = 300 male students at this large university.").
Normal/Large Sample Condition: The sampling distribution of x̄₁ - x̄₂ must be approximately normal. This is met if one of the following is true for both groups:
The populations are stated to be normally distributed.
The sample sizes are large (n₁ \ge 30 and n₂ \ge 30). This is due to the Central Limit Theorem (CLT).
Graphs of the sample data show no strong skewness or outliers. If n < 30, you must be given the raw data to sketch a quick boxplot or dotplot and comment on its shape.
Degrees of Freedom (df)
The t-distribution requires degrees of freedom. For a two-sample t-interval, the calculation is complex. You have two options:
Technology (Recommended): Your calculator will compute a precise degrees of freedom value using the Welch-Satterthwaite formula. This is the preferred method.
Conservative Method (By Hand): Use the smaller of the two degrees of freedom: df = min(n₁ - 1, n₂ - 1). This will result in a slightly wider (more conservative) interval but is acceptable if you don't have a calculator.
Key Vocabulary
Two-Sample t-Interval for a Difference of Means: A confidence interval used to estimate the true difference, μ₁ - μ₂, between the means of two independent populations.
Point Estimate (for a difference of means): The difference between the two sample means, x̄₁ - x̄₂, which serves as our best guess for the true difference, μ₁ - μ₂.
Standard Error of the Difference: An estimate of the standard deviation of the sampling distribution of x̄₁ - x̄₂. It measures the typical distance between the sample difference (x̄₁ - x̄₂) and the true population difference (μ₁ - μ₂).
Independent Samples: Samples where the selection of individuals for one sample has no bearing on the selection of individuals for the other sample.
Degrees of Freedom (df): A parameter of the t-distribution that determines its specific shape. For a two-sample t-test, it is calculated using a complex formula or a conservative approximation.
Calculator Tech (TI-84)
The primary function for this topic is .
Path:STAT -> TESTS -> 0: 2-SampTInt...
You will be presented with two input options: or .
Case 1: You have summary statistics (Stats)
Select and press
ENTER.Enter the summary statistics for both samples:
: Mean of sample 1
: Standard deviation of sample 1
: Sample size of sample 1
: Mean of sample 2
: Standard deviation of sample 2
: Sample size of sample 2
: Enter the confidence level as a decimal (e.g., 0.95 for 95%).
: ALWAYS select "No". The AP curriculum does not require or use pooled procedures, which assume the two populations have equal variances.
Highlight and press
ENTER. The calculator will display the confidence interval, degrees of freedom (df), and the sample statistics.
Case 2: You have raw data (Data)
Enter the data for the first group into list
L1(STAT->EDIT).Enter the data for the second group into list
L2.Go to
STAT->TESTS->0: 2-SampTInt...Select and press
ENTER.:
L1:
L2: 1 (unless you have a frequency list)
: 1
: Enter the confidence level (e.g., 0.99).
: ALWAYS select "No".
Highlight and press
ENTER.
How to Show Work on the FRQ
Use the four-step State-Plan-Do-Conclude process to earn full credit on inference questions.
State:
Define the parameter of interest in context. "We want to estimate μ₁ - μ₂, the true difference in the mean [context] between [population 1] and [population 2]..."
State the confidence level. "...at a [C]% confidence level."
Plan:
Name the procedure. "We will construct a Two-Sample t-Interval for a Difference of Means."
Check the conditions for inference.
Random: "The data come from two independent random samples of [context]..." OR "...two groups in a randomized experiment."
10% Condition: "It is reasonable to assume the number of [population 1 context] is at least 10 * n₁ = [10n₁] and the number of [population 2 context] is at least 10 * n₂ = [10n₂]."
Normal/Large Sample: "Since n₁ = [size] \ge 30 and n₂ = [size] \ge 30, the Central Limit Theorem applies and the sampling distribution of x̄₁ - x̄₂ is approximately normal." (If n < 30, refer to the problem stating the population is normal or comment on a provided graph of sample data).
Do:
Provide the formula: (x̄₁ - x̄₂) ± t* * √((s₁^2/n₁) + (s₂^2/n₂))
Plug in the values from the problem: Show the point estimate, the standard error calculation, and the t* critical value (you can get the df from your calculator).
State the final interval from your calculator. It's good practice to show the calculator function used ().
Example: (2.5 - 1.9) ± 2.045 * √((0.8^2/30) + (0.6^2/32)) = (0.24, 0.96)
Calculator Output: Interval: (0.24, 0.96), df = 55.8
Conclude:
Interpret the interval in context. "We are [C]% confident that the interval from [lower bound] to [upper bound] captures the true difference in the mean [context] between [population 1] and [population 2]."
Crucial Interpretation: If the interval contains 0, there is no convincing evidence of a difference between the two population means. If the interval is entirely positive or entirely negative, there is convincing evidence of a difference.
If 0 is NOT in the interval (e.g., (2.1, 5.3)): "Because 0 is not in this interval, we have convincing evidence that there is a true difference in the mean [context] between [population 1] and [population 2]."
If 0 IS in the interval (e.g., (-1.5, 4.2)): "Because 0 is in this interval, we do not have convincing evidence of a true difference in the mean [context] between [population 1] and [population 2]."
Practice Problems
Problem 1:
A guidance counselor wants to know if there is a difference in the mean number of hours senior boys and senior girls at their high school spend on homework per week. They take a random sample of 45 senior girls and 40 senior boys. The girls reported a mean of 15.6 hours with a standard deviation of 4.1 hours. The boys reported a mean of 13.9 hours with a standard deviation of 5.2 hours. Construct and interpret a 95% confidence interval for the difference in mean homework hours.
Solution:
State: We want to estimate μ₁ - μ₂, the true difference in the mean number of hours spent on homework per week between senior girls (μ₁) and senior boys (μ₂) at this high school, at a 95% confidence level.
Plan: We will construct a Two-Sample t-Interval for a Difference of Means.
Random: The data come from independent random samples of 45 senior girls and 40 senior boys.
10% Condition: It is reasonable to assume there are at least 10 * 45 = 450 senior girls and 10 * 40 = 400 senior boys at this high school.
Normal/Large Sample: Since n₁ = 45 \ge 30 and n₂ = 40 \ge 30, the Central Limit Theorem applies, and the sampling distribution of the difference in sample means is approximately normal.
Do:
Using a calculator ():
x̄₁ = 15.6, s₁ = 4.1, n₁ = 45
x̄₂ = 13.9, s₂ = 5.2, n₂ = 40
C-Level = 0.95, Pooled = No
The calculator gives an interval of (-0.318, 3.718) with df = 75.9.
By hand calculation for reference:
Point Estimate = 15.6 - 13.9 = 1.7
Standard Error = √((4.1^2/45) + (5.2^2/40)) = 1.025
t* with df \approx 75 is approx. 1.992.
Interval = 1.7 ± 1.992 * (1.025) = 1.7 ± 2.042 = (-0.342, 3.742). (Note the slight difference from the more precise calculator df).
Conclude: We are 95% confident that the interval from -0.318 to 3.718 hours captures the true difference in the mean number of hours spent on homework per week between senior girls and senior boys. Because 0 is included in this interval, we do not have convincing evidence of a difference in the mean homework time between girls and boys at this school.
Problem 2:
A researcher is testing a new fertilizer. They randomly assign 10 tomato plants to receive the new fertilizer (Group A) and 10 plants to receive a standard fertilizer (Group B). After one month, they measure the height of each plant in centimeters. The data are below. Construct and interpret a 90% confidence interval for the difference in mean plant height.
Group A (New): 45, 48, 52, 49, 44, 50, 51, 47, 46, 48
Group B (Standard): 42, 44, 39, 41, 45, 40, 43, 44, 38, 41
Solution:
State: We want to estimate μ₁ - μ₂, the true difference in the mean height (in cm) of tomato plants treated with the new fertilizer (μ₁) and plants treated with the standard fertilizer (μ₂), at a 90% confidence level.
Plan: We will construct a Two-Sample t-Interval for a Difference of Means.
Random: The 20 plants were randomly assigned to the two fertilizer groups.
10% Condition: This is a randomized experiment, not sampling from a population, so this condition is not applicable.
Normal/Large Sample: Since n₁ = 10 and n₂ = 10, which are less than 30, we must check the sample data. A quick sketch of dotplots for each group shows no strong skewness or outliers. Therefore, it is reasonable to proceed with a t-procedure.
Do:
First, enter the Group A data into L1 and Group B data into L2.
Using a calculator ( with input):
List1: L1, List2: L2, Freq1: 1, Freq2: 1
C-Level = 0.90, Pooled = No
The calculator gives an interval of (4.07 cm, 7.33 cm) with df = 17.1.
(The calculator also provides the sample stats: x̄₁ = 48, s₁ = 2.58, x̄₂ = 41.7, s₂ = 2.21)
Conclude: We are 90% confident that the interval from 4.07 cm to 7.33 cm captures the true difference in the mean height between plants with the new fertilizer and plants with the standard fertilizer. Because the entire interval is positive (0 is not included), we have convincing evidence that the new fertilizer results in a greater mean plant height than the standard fertilizer.
Common Mistakes to Avoid
Pooling the Variances: On the TI-84 calculator, never select "Yes" for "Pooled." This option assumes the two populations have the same variance, a condition we do not test or assume in AP Statistics. Always use "No."
Checking Conditions for Only One Sample: The Random, 10%, and Normal/Large Sample conditions must be discussed and verified for both groups. Forgetting one group will result in a loss of credit.
Misinterpreting "No Difference": A confidence interval for a difference of means that contains 0 (e.g., -5.2 to 11.3) does not prove that there is no difference. It simply means we do not have convincing statistical evidence of a difference. A plausible value for the true difference is zero.
Using Z instead of T:** When you are working with sample means and do not know the population standard deviations (σ₁ and σ₂), you must use a t-distribution and a t* critical value. Using z* is incorrect and will lead to an interval that is too narrow.
Subtracting Standard Deviations: The standard error formula involves the square root of the sum of the variances (s^2/n). Never subtract the variances or the standard deviations. Variability always adds.