AP Statistics 4.7 Constructing a Confidence Interval for the Difference Between Two Population Means Study Notes - New Syllabus
AP Statistics 4.7 Two-Sample t-Intervals for the Difference Between Population Means Study Notes – New Syllabus
AP Statistics 4.7 Two-Sample t-Intervals for the Difference Between Population Means Study Notes – As per latest AP Statistics Syllabus.
LEARNING OBJECTIVES
- 4.7.A Identify an appropriate confidence interval procedure including the parameter for the difference between two population means.
- 4.7.B Justify the appropriateness of constructing a confidence interval for the difference between two population means by verifying conditions.
- 4.7.C Calculate an appropriate confidence interval for the difference between two population means.
- 4.7.D Calculate the standard error and margin of error for estimating the difference between two population means.
ESSENTIAL KNOWLEDGE:
- 4.7.A.1 Based on the sample data, a confidence interval can be calculated to estimate the difference between two population means. The appropriate confidence interval procedure for two independent samples is a two-sample t-interval for the difference between population means.
- 4.7.A.2 The parameter for a confidence interval for a two-sample t-interval for the difference between population means should reference the difference in the means, the response variable, and the populations in context.
- 4.7.B.1 A two-sample t-interval for a difference between population means requires that three conditions be met:
- 4.7.B.1.i The randomization condition—the data should be collected using two independent random samples or a randomized experiment.
- 4.7.B.1.ii The 10% condition—when sampling without replacement, the size of each sample should be less than or equal to 10% of the respective population size: \( n_1\le10\%N_1 \) and \( n_2\le10\%N_2 \), where \( N_1 \) is the size of population 1 and \( N_2 \) is the size of population 2. (This condition is unnecessary when the data are from a randomized experiment.)
- 4.7.B.1.iii The sample data condition—both samples should have sample sizes greater than or equal to 30 or it is indicated that both population distributions are approximately normal. If either sample size is less than 30, both sample data distributions should be free from strong skewness and outliers.
- 4.7.C.1 A point estimate for the difference between two population means is the difference in sample means, \( \bar{x}_1-\bar{x}_2 \).
- 4.7.C.2 For the difference between population means when the population standard deviations are unknown, the confidence interval can be constructed as
Point estimate ± (margin of error)
The confidence interval for the difference between population means is
\( (\bar{x}_1-\bar{x}_2)\pm t^*\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}} \)
where \( t^* \) is the critical value for the central C% of a t-distribution with appropriate degrees of freedom that can be found using technology. The degrees of freedom fall between \( n_1+n_2-2 \) and the smaller of \( n_1-1 \) and \( n_2-1 \). - 4.7.D.1 The standard error for the difference between two sample means is
\( SE_{\bar{x}_1-\bar{x}_2}=\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}} \),
where \( s_1 \) and \( s_2 \) are the sample standard deviations. - 4.7.D.2 For the difference between two sample means, the margin of error is the critical value (\( t^* \)) times the standard error (SE) of the difference between two sample means, which equals
\( t^*\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}} \).
4.7.A.1 Two-Sample t-Interval for the Difference Between Two Population Means
When comparing the means of two independent populations, it is often necessary to estimate the true difference between the population means.
A confidence interval provides a range of plausible values for this unknown difference based on information obtained from two independent random samples.
When the population standard deviations are unknown, the appropriate confidence interval procedure is a two-sample t-interval for the difference between population means.
This procedure estimates the parameter
\( \mu_1-\mu_2 \)
where
- \( \mu_1 \) = Population mean of Population 1
- \( \mu_2 \) = Population mean of Population 2
When to Use a Two-Sample t-Interval
- There are two independent random samples.
- The response variable is quantitative.
- The goal is to estimate the difference between two population means.
- The population standard deviations are unknown.
Parameter of Interest
\( \mu_1-\mu_2 \)
The parameter represents the true difference between the two population means.
The order of subtraction is important and should remain consistent throughout the problem.
For example,
Population 1 − Population 2
or
School A − School B
| Characteristic | Two-Sample t-Interval |
|---|---|
| Number of Samples | Two independent samples |
| Response Variable | Quantitative |
| Population Standard Deviations | Unknown |
| Parameter | \( \mu_1-\mu_2 \) |
| Procedure | Two-sample t-interval for the difference between population means |
How to Identify This Procedure on the AP Exam
- The problem compares two independent groups.
- The response variable is quantitative.
- The question asks you to estimate the difference between two population means.
- The population standard deviations are not known.
Important AP Exam Notes
- Use a two-sample t-interval only for independent samples.
- Do not use this procedure for matched pairs data.
- Always define the order of subtraction before interpreting the interval.
- The parameter is the difference between two population means, not the difference between sample means.
- Maintain the same order of subtraction throughout the entire solution.
Example
A researcher wants to estimate the difference in the average mathematics test scores of students from School A and School B.
Independent random samples are selected from both schools, and the population standard deviations are unknown.
Identify the appropriate confidence interval procedure and the parameter of interest.
▶️ Answer / Explanation
Step 1: Identify the study design.
The problem compares two independent samples of a quantitative variable.
Step 2: Identify the procedure.
The appropriate procedure is a
Two-sample t-interval for the difference between population means.
Step 3: State the parameter.
\( \mu_A-\mu_B \)
where
\( \mu_A \) = the true mean mathematics test score for all students at School A
and
\( \mu_B \) = the true mean mathematics test score for all students at School B.
4.7.A.2 Identifying the Parameter for a Two-Sample t-Interval
Before constructing a confidence interval, the parameter of interest must be clearly identified.
For a two-sample t-interval, the parameter is the difference between two population means.
The parameter should always include:
- The difference in the population means.
- The response variable.
- The two populations being compared.
- The order of subtraction.
Parameter
\( \mu_1-\mu_2 \)
where
- \( \mu_1 \) = True population mean for Population 1
- \( \mu_2 \) = True population mean for Population 2
Example Parameter Statement
\( \mu_A-\mu_B \) = the true difference in the mean mathematics test scores (School A − School B) for all students in the two schools.
| Component | What to Include |
|---|---|
| Population Parameter | \( \mu_1-\mu_2 \) |
| Response Variable | State what is being measured. |
| Populations | Clearly identify both populations. |
| Order of Subtraction | Keep the same order throughout the problem. |
Important AP Exam Notes
- The parameter must always refer to the population means, not the sample means.
- Always define the order of subtraction before interpreting the interval.
- The order of subtraction affects the sign of the estimated difference.
- Write the parameter completely in the context of the problem.
Example
A researcher compares the average daily screen time of students from School X and School Y.
State the parameter for a confidence interval estimating the difference between the two population means.
▶️ Answer / Explanation
Define the order of subtraction:
School X − School Y
The parameter is
\( \mu_X-\mu_Y \)
where
\( \mu_X-\mu_Y \)
represents the true difference in the mean daily screen time (School X − School Y) for all students in the two schools.
4.7.B.1 Conditions for a Two-Sample t-Interval for the Difference Between Two Population Means
Before constructing a two-sample t-interval for the difference between two population means, statisticians must verify that the required conditions are satisfied.
These conditions ensure that the confidence interval provides a valid estimate of the true difference between the two population means.
A two-sample t-interval requires the following three conditions:
- Randomization Condition
- 10% Condition
- Sample Data Condition
4.7.B.1.i Randomization Condition
The data should be collected using two independent random samples or from a randomized experiment.
If the study is observational, each sample should be selected randomly and independently.
If the study is an experiment, subjects should be randomly assigned to the treatment groups.
How to Verify
- Two independent simple random samples (SRS) are selected from the two populations.
- Or, subjects are randomly assigned to treatments in a randomized experiment.
4.7.B.1.ii 10% Condition
When sampling is conducted without replacement, each sample size must be no more than 10% of its respective population size.
This condition ensures that observations within each sample are approximately independent.
Formulas
\( N_1 \ge 10n_1 \)
\( N_2 \ge 10n_2 \)
or equivalently
\( n_1 \le 0.10N_1 \)
\( n_2 \le 0.10N_2 \)
Where:
- \(N_1\) = Population size of Population 1
- \(N_2\) = Population size of Population 2
- \(n_1\) = Sample size from Population 1
- \(n_2\) = Sample size from Population 2
Note: If the data come from a randomized experiment, the 10% Condition is not required.
4.7.B.1.iii Sample Data Condition
The sampling distribution of the difference between the sample means should be approximately normal.
This condition is satisfied if either of the following is true:
- Both population distributions are approximately normal.
- Both sample sizes are at least 30.
If either sample size is less than 30, then both sample data distributions should be free from strong skewness and outliers.
| Situation | Condition Satisfied? |
|---|---|
| Both populations are approximately normal. | ✔ Yes |
| \( n_1\ge30 \) and \( n_2\ge30 \) | ✔ Yes (Central Limit Theorem) |
| One or both sample sizes less than 30, but both sample distributions have no strong skewness or outliers. | ✔ Yes |
| One or both sample sizes less than 30 and either sample distribution has strong skewness or outliers. | ✘ No |
Summary of the Conditions
| Condition | Requirement |
|---|---|
| Randomization Condition | Two independent random samples or a randomized experiment. |
| 10% Condition | \(N_1\ge10n_1\) and \(N_2\ge10n_2\) when sampling without replacement. |
| Sample Data Condition | Both populations are approximately normal, or both sample sizes are at least 30, or if either sample size is less than 30, both sample distributions are free from strong skewness and outliers. |
Important AP Exam Notes
- Always verify all three conditions before constructing a two-sample t-interval.
- The two samples must be independent.
- The 10% Condition applies only when sampling is performed without replacement.
- The 10% Condition is not required for a randomized experiment.
- If either sample size is less than 30, both sample distributions must be checked for strong skewness and outliers.
- On the AP Exam, justify each condition separately before stating that a two-sample t-interval is appropriate.
Example
A researcher compares the average mathematics test scores of students from two schools.
An independent random sample of 35 students is selected from School A, which has 900 students.
An independent random sample of 40 students is selected from School B, which has 1,200 students.
Neither population is normally distributed.
Determine whether it is appropriate to construct a two-sample t-interval for the difference between the population means.
▶️ Answer / Explanation
Step 1: Randomization Condition
The problem states that two independent random samples were selected.
✔ The Randomization Condition is satisfied.
Step 2: 10% Condition
School A:
\(10n_1=10(35)=350\)
\(900\ge350\)
✔ Condition satisfied.
School B:
\(10n_2=10(40)=400\)
\(1200\ge400\)
✔ Condition satisfied.
Step 3: Sample Data Condition
The population distributions are not normal, but
\(n_1=35\ge30\)
\(n_2=40\ge30\)
Therefore, by the Central Limit Theorem, the sampling distribution of
\( \bar{x}_1-\bar{x}_2 \)
is approximately normal.
✔ The Sample Data Condition is satisfied.
Conclusion
Since all three conditions are satisfied, it is appropriate to construct a two-sample t-interval for the difference between the population means.
4.7.C.1 Point Estimate for the Difference Between Two Population Means
Before constructing a confidence interval, we first calculate a point estimate for the difference between two population means.
A point estimate is a single numerical value calculated from sample data that is used to estimate an unknown population parameter.
For comparing two population means, the best point estimate is the difference between the two sample means.
Point Estimate Formula
\( \bar{x}_1-\bar{x}_2 \)
Where:
- \( \bar{x}_1 \) = Sample mean from Population 1
- \( \bar{x}_2 \) = Sample mean from Population 2
The order of subtraction must remain the same throughout the entire problem.
Interpretation
The value
\( \bar{x}_1-\bar{x}_2 \)
is the best estimate of the true difference between the two population means,
\( \mu_1-\mu_2 \).
| Population Parameter | Point Estimate |
|---|---|
| \( \mu_1-\mu_2 \) | \( \bar{x}_1-\bar{x}_2 \) |
Important AP Exam Notes
- The point estimate is always the difference between the two sample means.
- The point estimate estimates the population parameter \( \mu_1-\mu_2 \).
- Always define and maintain the same order of subtraction (Population 1 − Population 2).
- A point estimate is a single value, while a confidence interval provides a range of plausible values.
Example
A researcher compares the average mathematics test scores of students from two schools.
The sample statistics are:
- School A: \( \bar{x}_1=82.4 \)
- School B: \( \bar{x}_2=77.9 \)
Find the point estimate for the difference between the population means (School A − School B).
▶️ Answer / Explanation
Step 1: Use the point estimate formula.
\( \bar{x}_1-\bar{x}_2 \)
Step 2: Substitute the sample means.
\( 82.4-77.9=4.5 \)
Answer
The point estimate for the difference between the two population means is
\( 4.5 \)
This indicates that, based on the sample data, School A’s average mathematics test score is estimated to be 4.5 points higher than School B’s.
4.7.C.2 Two-Sample t-Interval for the Difference Between Two Population Means
After calculating the point estimate, a confidence interval can be constructed to estimate the true difference between two population means.
When the population standard deviations are unknown, the appropriate procedure is a two-sample t-interval.
The confidence interval is calculated as
Point Estimate ± Margin of Error
Confidence Interval Formula
\( (\bar{x}_1-\bar{x}_2)\pm t^*\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}} \)
Where:
- \( \bar{x}_1,\bar{x}_2 \) = Sample means
- \( s_1,s_2 \) = Sample standard deviations
- \( n_1,n_2 \) = Sample sizes
- \( t^* \) = Critical value from the t-distribution
The critical value \(t^*\) is obtained using technology because the degrees of freedom are calculated using statistical software or a graphing calculator.
The degrees of freedom lie between
\( \min(n_1-1,\;n_2-1) \)
and
\( n_1+n_2-2 \).
| Component | Meaning |
|---|---|
| Point Estimate | \( \bar{x}_1-\bar{x}_2 \) |
| Standard Error | \( \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}} \) |
| Margin of Error | \( t^*\times \text{Standard Error} \) |
Important AP Exam Notes
- The confidence interval estimates the parameter \( \mu_1-\mu_2 \).
- Maintain the same order of subtraction throughout the problem.
- The critical value \(t^*\) is obtained using technology.
- The degrees of freedom are also determined using technology.
- The confidence interval is always written as Point Estimate ± Margin of Error.
Example
The following summary statistics were obtained from two independent random samples.
| Population 1 | Population 2 | |
|---|---|---|
| Sample Mean | 85 | 79 |
| Sample Standard Deviation | 10 | 12 |
| Sample Size | 40 | 35 |
Construct a 95% two-sample t-confidence interval for the difference between the population means using technology.
▶️ Answer / Explanation
Step 1: Enter the summary statistics into the calculator.
STAT → TESTS → 2-SampTInt → Stats
Step 2: Enter
- \( \bar{x}_1=85,\; s_1=10,\; n_1=40 \)
- \( \bar{x}_2=79,\; s_2=12,\; n_2=35 \)
- Confidence Level = 0.95
Step 3: Calculator Output (approximately)
95% CI: \( (0.90,\;11.10) \)
Answer
The 95% confidence interval for the difference between the two population means is
\((0.90,\;11.10)\)
We are 95% confident that the true difference in the population means (Population 1 − Population 2) is between 0.90 and 11.10.
4.7.D.1 Standard Error for the Difference Between Two Population Means
When estimating the difference between two population means, the standard error (SE) measures the typical variability of the difference between two sample means from repeated random sampling.
The standard error estimates how much the sample difference
\( \bar{x}_1-\bar{x}_2 \)
is expected to vary from the true difference between the population means.
Because the population standard deviations are usually unknown, the standard error is calculated using the sample standard deviations.
Standard Error Formula
\( SE_{\bar{x}_1-\bar{x}_2}=\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}} \)
Where:
- \( s_1 \) = Sample standard deviation from Population 1
- \( s_2 \) = Sample standard deviation from Population 2
- \( n_1 \) = Sample size from Population 1
- \( n_2 \) = Sample size from Population 2
The standard error is measured in the same units as the response variable.
| Component | Meaning |
|---|---|
| \( s_1,s_2 \) | Sample standard deviations |
| \( n_1,n_2 \) | Sample sizes |
| \( SE_{\bar{x}_1-\bar{x}_2} \) | Estimated standard deviation of the sampling distribution of \( \bar{x}_1-\bar{x}_2 \) |
Important AP Exam Notes
- The standard error measures the variability of the difference between sample means, not the population means.
- Always use the sample standard deviations (\(s_1\) and \(s_2\)), not the population standard deviations.
- Increasing either sample size decreases the standard error.
- A smaller standard error produces a more precise confidence interval.
Example
Two independent random samples produce the following statistics.
- \( s_1=10,\; n_1=40 \)
- \( s_2=12,\; n_2=35 \)
Calculate the standard error for the difference between the two sample means.
▶️ Answer / Explanation
Step 1: Write the formula.
\( SE_{\bar{x}_1-\bar{x}_2}=\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}} \)
Step 2: Substitute the values.
\(=\sqrt{\frac{10^2}{40}+\frac{12^2}{35}} \)
\(=\sqrt{\frac{100}{40}+\frac{144}{35}} \)
\(=\sqrt{2.50+4.11} \)
\(=\sqrt{6.61}\approx2.57 \)
Answer
The standard error of the difference between the two sample means is
\(2.57\)
4.7.D.2 Margin of Error for a Two-Sample t-Interval
The margin of error (ME) measures the maximum expected difference between the point estimate and the true population parameter for a given confidence level.
For a two-sample t-interval, the margin of error equals the critical value multiplied by the standard error.
Margin of Error Formula
\( ME=t^*\left(\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\right) \)
Where:
- \( t^* \) = Critical value from the t-distribution
- \( SE=\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}} \)
The confidence interval is then written as
\( (\bar{x}_1-\bar{x}_2)\pm ME \)
| Component | Formula |
|---|---|
| Standard Error | \( \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}} \) |
| Margin of Error | \( t^*\times SE \) |
Important AP Exam Notes
- The margin of error depends on both the critical value and the standard error.
- Higher confidence levels produce larger critical values and therefore larger margins of error.
- Larger sample sizes reduce the standard error and decrease the margin of error.
- The confidence interval is always written as Point Estimate ± Margin of Error.
Example
Using the previous example, suppose the standard error is
\( SE=2.57 \)
and the technology gives a critical value of
\( t^*=2.00 \)
Calculate the margin of error.
▶️ Answer / Explanation
Step 1: Write the formula.
\( ME=t^*\times SE \)
Step 2: Substitute the values.
\( ME=2.00(2.57) \)
\( ME=5.14 \)
Answer
The margin of error is
\(5.14\)
If the point estimate were \(6.00\), the confidence interval would be
\(6.00\pm5.14\)
or
\((0.86,\;11.14)\)