Lesson 9: Inference for Two Samples

Lesson 9: Inference for Two Samples

Objectives

Upon successful completion of this lesson, you should be able to:

  • Identify situations in which the z or t distribution may be used to approximate a sampling distribution
  • Construct a confidence interval to estimate the difference in two population proportions and two population means using Minitab Express given summary or raw data
  • Conduct a hypothesis test for two proportions and two means using Minitab Express given summary or raw data

Note: This lesson corresponds to Chapter 6: Sections 3 and 4 in the Lock5 textbook.

The general form of confidence intervals and test statistics will be the same for all of the procedures covered in this lesson:

General Form of a Confidence Interval
\(sample\ statistic\pm(multiplier)\ (standard\ error)\)
General Form of a Test Statistic
\(test\;statistic=\frac{sample\;statistic-null\;parameter}{standard\;error}\)
We will be using a five step hypothesis testing procedure again in this lesson:
  1. Check assumptions and write hypotheses
    The assumptions will vary depending on the test. The null and alternative hypotheses will also be written in terms of population parameters; the null hypothesis will always contain the equality (i.e., \(=\)).
  2. Calculate the test statistic
    This will vary depending on the test, but it will typically be the difference observed between the sample and population divided by a standard error. In this lesson we will see z and t test statistics. Minitab Express will compute the test statistic. 
  3. Determine the \(p\) value
    This can be found using Minitab Express.
  4. Make a decision
    If \(p \leq \alpha\) reject the null hypothesis. If \(p>\alpha\) fail to reject the null hypothesis.
  5. State a "real world" conclusion
    Based on your decision in step 4, write a conclusion in terms of the original research question.

9.1 - Two Independent Proportions

9.1 - Two Independent Proportions

Two independent proportions tests are used to compare the proportions in two unrelated groups. In StatKey these were known as "Difference in Proportions" tests. 

Given that \(n_1 p_1 \ge 10\), \(n_1(1-p_1) \ge 10\), \(n_2 p_2 \ge 10\), and \(n_2(1-p_2) \ge 10\), where the subscript 1 represents one groups and the subscript 2 represents the second group, the sampling distribution will be approximately normal with a standard deviation (i.e., standard error) of \( \sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}\).  Given that the population proportions are not known, they are estimated using the sample proportions. In other words, if there are at least 10 "successes" and at least 10 "failures" in both groups, then the sampling distribution for the difference in proportions will be approximately normal.

If the assumption above is met, the normal approximation method is typically preferred. The normal approximation method uses the z distribution to approximate the sampling distribution, similar to the procedures we used in Lesson 7.

When this assumption is not met, Fisher's exact method, bootstrapping, or randomization methods may be used.


9.1.1 - Confidence Intervals

9.1.1 - Confidence Intervals

Given that \(np \ge 10\) and \(n(1-p) \ge 10\) for both groups, in other words at least 10 "successes" and at least 10 "failures" in each group, the sampling distribution can be approximated using the normal distribution with a mean of \(\widehat p_1 - \widehat p_2\) and a standard error of \(\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}\).

Recall from the general form of a confidence interval:

General Form of a Confidence Interval
\(sample\ statistic\pm(multiplier)\ (standard\ error)\)

Here, the sample statistic is the difference between the two proportions (\(\widehat p_1 - \widehat p_2\)) and the standard error is computed using the formula \(\sqrt{\frac{p_1(1-p_1)}{n_1}+\frac{p_2(1-p_2)}{n_2}}\). Putting this information together, we can derive the formula for a confidence interval for the difference between two proportions.

Confidence Interval for the Difference Between Two Proportions
\((\widehat{p}_1-\widehat{p}_2) \pm z^\ast {\sqrt{\frac{\widehat{p}_1 (1-\widehat{p}_1)}{n_1}+\frac{\widehat{p}_2 (1-\widehat{p}_2)}{n_2}}}\)

Example: Confidence Interval for the Difference Between the Proportion of Females and Males in Favor of Legal Same Sex Marriage

A survey was given to a sample of college students. They were asked whether they think same sex marriage should be legal. Of the 251 females in the sample, 185 said "yes." Of the 199 males in the sample, 107 said "yes." Let’s construct a 95% confidence interval for the difference of the proportion of females and males who responded “yes.” We can apply the 95% Rule and use a multiplier of \(z^\ast\) = 2

  • For females: \(np=185\) and \(n(1-p) = 251-185=66\)
  • For males: \(np=107\) and \(n(1-p)=199-107=92\)

These counts are all at least 10 so the sampling distribution can be approximated using a normal distribution. 

\(\widehat{p}_{f}=\frac{185}{251}=0.737\)

\(\widehat{p}_{m}=\frac{107}{199}=0.538\)

\((0.737-0.538) \pm 2 {\sqrt{\frac{0.737 (1-0.737)}{251}+\frac{0.538 (1-0.538)}{199}}}\)

\(0.199 \pm 2 ( 0.045)\)

\(.199 \pm .090=[.110, .288]\)

We are 95% confident that in the population the difference between the proportion of females and males who are in favor of same sex marriage legalization is between 0.110 and 0.288.

This confidence interval does not contain 0. Therefore it is not likely that the difference between females and males is 0. We can conclude that there is a difference between the proportion of males and females in the population who would respond “yes" to this question.


9.1.1.1 - Minitab Express: Confidence Interval for 2 Proportions

9.1.1.1 - Minitab Express: Confidence Interval for 2 Proportions

Minitab Express can be used to construct a confidence interval for the difference between two proportions using the normal approximation method. Note that the confidence intervals given in the Minitab Express output assume that \(np \ge 10\) and \(n(1-p) \ge 10\) for both groups. If this assumption is not true, you should use bootstrapping methods in StatKey.

MinitabExpress  – Constructing a Confidence Interval with Raw Data

Let's estimate the difference between the proportion of females who have tried weed and the proportion of males who have tried weed. 

  1. Open Minitab Express file:
  2. On a PC: In the menu bar select STATISTICS > Two Samples > Proportions
  3. On a Mac: In the menu bar select Statistics > 2-Sample Inference > Proportions
  4. Double click the variable Try Weed in the box on the left to insert the variable into the Samples box
  5. Double click the variable Biological Sex in the box on the left to insert the variable into the Sample IDs box
  6. Keep the default Options
  7. Click OK

This should result in the following output:

2-Sample Proportions: Try Weed by Biological Sex
Method
Event: Try Weed = Yes
\(p_1\): proportion where Try Weed = Yes and Biological Sex = Female
\(p_2\): proportion where Try Weed = Yes and Biological Sex = Male
Difference: \(p_1-p_2\)
Descriptive Statistics: Try Weed
Biological Sex N Event Sample p
Female 127 56 0.440945
Male 99 62 0.626263
Estimation for Difference
Difference 95% CI for Difference
-0.185318 (-0.313920, -0.056716)
Test
Null hypothesis \(H_0\): \(p_1-p_2=0\)
Alternative hypothesis \(H_1\): \(p_1-p_2\neq0\)
Method Z-Value P-Value
Fisher's exact   0.0072
Normal approximation -2.82 0.0047

95% CI for the Difference

Video Walkthrough

Select your operating system below to see a step-by-step guide for this example.

MinitabExpress  – Constructing a Confidence Interval with Summarized Data

Let's estimate the difference between the proportion of Penn State World Campus graduate students who have children to the proportion of Penn State University Park graduate students who have children. In our representative sample there were 120 World Campus graduate students; 92 had children. There were 160 University Park graduate students; 23 had children.

  1. Open Minitab Express without data
  2. On a PC: In the menu bar select STATISTICS > Two Samples > Proportions
  3. On a Mac: In the menu bar select Statistics > 2-Sample Inference > Proportions
  4. Change Both samples are in one column to Summarized data
  5. For Sample 1 next to Number of events enter 92 and next to Number of trials enter 120
  6. For Sample 2 next to Number of events enter 23 and next to Number of trials enter 160
  7. Keep the default Options
  8. Click OK

This should result in the following output:

2-Sample Proportions
Method
\(p_1\): proportion where Sample 1 = Event
\(p_2\): proportion where Sample 2 = Event
Difference: \(p_1-p_2\)
Descriptive Statistics
Sample N Event Sample p
Sample 1 120 92 0.766667
Sample 2 160 23 0.143750
Estimation for Difference
Difference 95% CI for Difference
0.622917 (0.529740, 0.716093)
Test
Null hypothesis \(H_0\): \(p_1-p_2=0\)
Alternative hypothesis \(H_1\): \(p_1-p_2\neq0\)
Method Z-Value P-Value
Fisher's exact   <0.0001
Normal approximation 13.10 <0.0001

95% CI for the Difference

Video Walkthrough

Select your operating system below to see a step-by-step guide for this example.


9.1.2 - Hypothesis Testing

9.1.2 - Hypothesis Testing

Here we will walk through the five-step hypothesis testing procedure for comparing the proportions from two independent groups. In order to use the normal approximation method there must be at least 10 "successes" and at least 10 "failures" in each group. In other words, \(n p \geq 10\) and \(n (1-p) \geq 10\) for both groups.

If this assumption is not met you should use Fisher's exact method in Minitab Express or bootstrapping methods in StatKey.


9.1.2.1 - Normal Approximation Method Formulas

9.1.2.1 - Normal Approximation Method Formulas

1. Check any necessary assumptions and write null and alternative hypotheses.

To use the normal approximation method a minimum of 10 successes and 10 failures in each group are necessary (i.e., \(n p \geq 10\) and \(n (1-p) \geq 10\)).

The two groups that are being compared must be unpaired and unrelated (i.e., independent).

Below are the possible null and alternative hypothesis pairs:

Research Question Are the proportions of group 1 and group 2 different? Is the proportion of group 1 greater than the proportion of group 2? Is the proportion of group 1 less than the proportion of group 2?
Null Hypothesis, \(H_{0}\) \(p_1 - p_2=0\) \(p_1 - p_2=0\) \(p_1 - p_2=0\)
Alternative Hypothesis, \(H_{a}\) \(p_1 - p_2 \neq 0\) \(p_1 - p_2> 0\) \(p_1 - p_2<0\)
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional
2. Calculate an appropriate test statistic.

The null hypothesis is that there is not a difference between the two proportions (i.e., \(p_1 = p_2\)). If the null hypothesis is true then the population proportions are equal. When computing the standard error for the difference between the two proportions a pooled proportion is used as opposed to the two proportions separately (i.e., unpooled). This pooled estimate will be symbolized by \(\widehat{p}\). This is similar to a weighted mean, but with two proportions. 

Pooled Estimate of \(p\)
\(\widehat{p}=\frac{\widehat{p}_1n_1+\widehat{p}_2n_2}{n_1+n_2}\)

The standard error for the difference between two proportions is symbolized by \(SE_{0}\). The subscript 0 tells us that this standard error is computed under the null hypothesis (\(H_0: p_1-p_2=0\)).

Standard Error

\(SE_0={\sqrt{\frac{\widehat{p} (1-\widehat{p})}{n_1}+\frac{\widehat{p}(1-\widehat{p})}{n_2}}}=\sqrt{\widehat{p}(1-\widehat{p})\left ( \frac{1}{n_1}+\frac{1}{n_2} \right )}\)

Note that the default in many statistical programs, including Minitab Express, is to estimate the two proportions separately (i.e., unpooled). In order to obtain results using the pooled estimate of the proportion you will need to change the method.

Also note that this standard error is different from the one that you used when constructing a confidence interval for \(p_1-p_2\). While the hypothesis testing procedure is based on the null hypothesis that \(p_1-p_2=0\), the confidence interval approach is not based on this premise. The hypothesis testing approach uses the pooled estimate of \(p\) while the confidence interval approach will use an unpooled method. 

Test Statistic for Two Independent Proportions
\(z=\frac{\widehat{p}_1-\widehat{p}_2}{SE_0}\)
3. Determine the p value associated with the test statistic.

The \(z\) test statistic found in Step 2 is used to determine the \(p\) value. The \(p\) value is the proportion of the \(z\) distribution (normal distribution with a mean of 0 and standard deviation of 1) that is more extreme than the test statistic in the direction of the alternative hypothesis. 

4. Decide between the null and alternative hypotheses.

If \(p \leq \alpha\) reject the null hypothesis. If \(p>\alpha\) fail to reject the null hypothesis.

5. State a "real world" conclusion.

Based on your decision in Step 4, write a conclusion in terms of the original research question.


9.1.2.1.1 – Example: Ice Cream

9.1.2.1.1 – Example: Ice Cream

Example: Ice Cream

The Creamery wants to compare men and women in terms of preference for eating their ice cream out of a cone to determine if there is a gender difference. They take a sample of 500 customers (240 men and 260 women) and ask if they prefer cones over bowls. They found that 124 men preferred cones and 90 women preferred cones.

1. Check any necessary assumptions and write hypotheses.

\(H_0: p_m - p_f = 0\)

\(H_a:p_m - p_f \ne 0\)
 

\(n_m \widehat p_m = 124\)

\(n_m (1-\widehat p_m) = 240 - 124 = 116\)

\(n_f \widehat p_f = 90\)

\(n_f (1-\widehat p_f) = 260-90 = 170\)

These counts are all at least 10 so we can use the normal approximation method.

2. Calculate an appropriate test statistic.
Pooled Estimate of \(p\)
\(\widehat{p}=\frac{\widehat{p}_1n_1+\widehat{p}_2n_2}{n_1+n_2}\)

\(\widehat{p}=\frac{124+90}{240+260}=\frac{214}{500}=0.428\)

Standard Error of \(\widehat{p}\)
\(SE_{0}={\sqrt{\frac{\widehat{p} (1-\widehat{p})}{n_1}+\frac{\widehat{p}(1-\widehat{p})}{n_2}}}=\sqrt{\widehat{p}(1-\widehat{p})\left ( \frac{1}{n_1}+\frac{1}{n_2} \right )}\)

\(SE_{0}=\sqrt{0.428(1-0.428)\left ( \frac{1}{240}+\frac{1}{260} \right )}=0.04429\)

Test Statistic for Two Independent Proportions
\(z=\frac{\widehat{p}_1-\widehat{p}_2}{SE_0}\)

\(z=\frac{\frac{124}{240}-\frac{90}{260}}{0.04429}=3.850\)

3. Determine the p-value associated with the test statistic.

This is a two-tailed test. Our p-value will be the area of the \(z\) distribution more extreme than \(3.850\).

Standard normal distribution showing the p-value given z=3.850 for a two-tailed test

\(p = 0.0000591 \times 2 = 0.0001182\)

4. Decide between the null and alternative hypotheses.

\(p \le 0.05\)

Reject the null hypothesis

5. State a "real world" conclusion.

There is evidence that the proportion of men who prefer cones is different from the proportion of women who prefer cones in the population.


9.1.2.1.2 – Example: Same Sex Marriage

9.1.2.1.2 – Example: Same Sex Marriage

Example: Same Sex Marriage

A survey was given to a random sample of college students. They were asked whether they think same sex marriage should be legal. We'll compare the proportion of males and females who responded "yes." Of the 251 females in the sample, 185 said "yes." Of the 199 males in the sample, 107 said "yes."

1. Check any necessary assumptions and write null and alternative hypotheses.

For females, there were 185 who said "yes" and 66 who said "no." For males, there were 107 who said "yes" and 92 who said "no." There are at least 10 successes and failures in each group so the normal approximation method can be used.

  • \(\widehat{p}_{f}=\frac{185}{251}\)
  • \(\widehat{p}_{m}=\frac{107}{199}\)

This is a two-tailed test because we are looking for a difference between males and females, we were not given a specific direction. 

  • \( H_{0} : p_{f}- p_{m}=0 \)
  • \( H_{a} : p_{f}- p_{m}\neq 0 \)
2. Calculate an appropriate test statistic.

\(\widehat{p}=\frac{185+107}{251+199}=\frac{292}{450}=0.6489\)

\(SE_0=\sqrt{\frac{292}{450}\left ( 1-\frac{292}{450} \right )\left ( \frac{1}{251}+\frac{1}{199} \right )}=0.0453\)

\(z=\frac{\frac{185}{251}-\frac{107}{199}}{0.0453}=4.400\)

Our test statistic is \(z=4.400\)

3. Determine a p value associated with the test statistic.

Distribution Plot of Density vs X - Normal, Mean=0, StDev=1

\(P(z>4.400)=0.0000054\), this is a two-tailed test, so this value must be multiplied by two: \(0.0000054\times 2= 0.0000108\)

\(p<0.0001\)

4. Decide between the null and alternative hypotheses.

\(p\leq0.05\), therefore we reject the null hypothesis.

5. State a "real world" conclusion.

There is evidence that there is a difference between the proportion of females and males who think that same sex marriage should be legal.


9.1.2.2 - Minitab Express: Difference Between 2 Independent Proportions

9.1.2.2 - Minitab Express: Difference Between 2 Independent Proportions

When conducting a hypothesis test comparing the proportions of two independent proportions in Minitab Express two p-values are provided. If \(np \ge 10\) and \(n(1-p) \ge 10\), use the p-value associated with the normal approximation method. If this assumption is not met, use the p-value associated with Fisher's exact method.

When conducting a hypothesis test for a difference between two independent proportions in Minitab Express you need to remember to change the "test method" to "use the pooled estimate of the proportion." This is because the null hypothesis is that the two proportions are equal.

MinitabExpress  – Testing Two Independent Proportions using Raw Data

Let's compare the proportion of females who have tried weed to the proportion of males who have tried weed. 

  1. Open Minitab Express file:
  2. On a PC: In the menu bar select STATISTICS > Two Samples > Proportions
  3. On a Mac: In the menu bar select Statistics > 2-Sample Inference > Proportions
  4. Double click the variable Try Weed in the box on the left to insert the variable into the Samples box
  5. Double click the variable Biological Sex in the box on the left to insert the variable into the Sample IDs box
  6. Under the Options tab change the Test method to Use the pooled estimate of the proportion
  7. Click OK

This should result in the following output:

2-Sample Proportions: Try Weed by Biological Sex
Method
Event: Try Weed = Yes
\(p_1\): proportion where Try Weed = Yes and Biological Sex = Female
\(p_2\): proportion where Try Weed = Yes and Biological Sex = Male
Difference: \(p_1-p_2\)
Descriptive Statistics: Try Weed
Biological Sex N Event Sample p
Female 127 56 0.440945
Male 99 62 0.626263
Estimation for Difference
Difference 95% CI for Difference
-0.185318 (-0.313920, -0.056716)
Test
Null hypothesis \(H_0\): \(p_1-p_2=0\)
Alternative hypothesis \(H_1\): \(p_1-p_2\neq0\)
Method Z-Value P-Value
Fisher's exact   0.0072
Normal approximation -2.77 0.0057

The pooled estimate of the proportion (0.522124) is used for the tests.

Video Walkthrough

Select your operating system below to see a step-by-step guide for this example.

Five Step Hypothesis Testing Procedure

Step 1:
\(H_0 : p_f - p_m =0\)
\(H_a : p_f - p_m \neq 0\)

\(n_f p_f = 56\)
\(n_f (1-p_f) = 127-56 = 71\)
\(n_m p_m = 62\)
\(n_m (1-p_m) = 99-62 = 37\)

All \(np\) and \(n(1-p)\) are at least ten, therefore we can use the normal approximation method.

Step 2:
From output, \(z=-2.77\)

Step 3:
From output, \(p=0.0057\)

Step 4:
\(p \leq \alpha\), reject the null hypothesis

Step 5:
There is evidence that in the population the proportion of females who have tried weed is different from the proportion of males who have tried weed.

MinitabExpress  – Testing Two Independent Proportions using Summarized Data

Let's compare the proportion of Penn State World Campus graduate students who have children to the proportion of Penn State University Park graduate students who have children.  In a representative sample there were 120 World Campus graduate students; 92 had children.  There were 160 University Park graduate students; 23 had children.

  1. Open Minitab Express without data
  2. On a PC: In the menu bar select STATISTICS > Two Samples > Proportions
  3. On a Mac: In the menu bar select Statistics > 2-Sample Inference > Proportions
  4. Change Both samples are in one column to Summarized data
  5. For Sample 1 next to Number of events enter 92 and next to Number of trials enter 120
  6. For Sample 2 next to Number of events enter 23 and next to Number of trials enter 160
  7. Under the Options tab change the Test method to Use the pooled estimate of the proportion
  8. Click OK

This should result in the following output:

2-Sample Proportions
Method
\(p_1\): proportion where Sample 1 = Event
\(p_2\): proportion where Sample 2 = Event
Difference: \(p_1-p_2\)
Descriptive Statistics
Sample N Event Sample p
Sample 1 120 92 0.766667
Sample 2 160 23 0.143750
Estimation for Difference
Difference 95% CI for Difference
0.622917 (0.529740, 0.716093)
Test
Null hypothesis \(H_0\): \(p_1-p_2=0\)
Alternative hypothesis \(H_1\): \(p_1-p_2\neq0\)
Method Z-Value P-Value
Fisher's exact   <0.0001
Normal approximation 10.49 <0.0001

The pooled estimate of the proportion (0.410714) is used for the tests.

Video Walkthrough

Select your operating system below to see a step-by-step guide for this example.

Five Step Hypothesis Testing Procedure

Step 1:
\(H_0 : p_W - p_U =0\)
\(H_a : p_W - p_U \neq 0\)

\(n_W p_W = 92\)
\(n_W (1-p_W) = 120-92 = 28\)
\(n_U p_U = 23\)
\(n_U (1-p_u) = 160-23 = 137\)

All \(np\) and \(n(1-p)\) are at least ten, therefore we can use the normal approximation method.

Step 2:
From output, \(z=10.49\)

Step 3:
From output, \(p<.0001\)

Step 4:
\(p \leq \alpha\), reject the null hypothesis

Step 5:
There is evidence that in the population the proportion of World Campus students who have children is different from the proportion of University Park students who have children.


9.1.2.2.1 - Example: Dating

9.1.2.2.1 - Example: Dating

Example: Dating

This example uses the course survey dataset:

A random sample of Penn State University Park undergraduate students were asked, "Would you date someone with a great personality even if you did not find them attractive?"  Let's compare the proportion of males and females who responded "yes" to determine if there is evidence of a difference.

1. Check any necessary assumptions and write hypotheses.

We are looking for a "difference," so this is a two-tailed test.

\(H_{0} : p_1 - p_2 =0\)
\( H_{a} :p_1 - p_2 \neq 0 \)

Method
Event: DatePerly = Yes
\(p_1\): proportion where DatePerly = Yes and Gender = Female
\(p_2\): proportion where DatePerly = Yes and Gender = Male
Difference: \(p_1-p_2\)
Descriptive Statistics: DatePerly
Gender N Event Sample p
Female 571 367 0.642732
Male 433 148 0.341801
Estimation for Difference
Difference 95% CI for Difference
0.300931 (0.241427, 0.360435)
Test
Null hypothesis \(H_0\): \(p_1-p_2=0\)
Alternative hypothesis \(H_1\): \(p_1-p_2\neq0\)
Method Z-Value P-Value
Fisher's exact   <0.0001
Normal approximation 9.45 <0.0001

The pooled estimate of the proportion (0.512948) is used for the tests.

\(n_f p_f = 367\)

\(n_f (1-p_f) = 571 - 367 = 204\)

\(n_m p_m = 148\)

\(n_m (1 - p_m) = 433 - 148 = 285\)

All of these counts are at least 10 so we will use the normal approximation method.

2. Calculate an appropriate test statistic.

From output, \(z=9.45\)

3. Determine the p-value associated with the test statistic.

From output, \(p<0.0001\)

4. Decide between the null and alternative hypotheses.

\(p \leq \alpha\), reject the null hypothesis

5. State a "real world" conclusion.

There is evidence that in the population of all Penn State University Park undergraduate students the proportion of men who would date someone with a great personality even if they did not find them attractive is different from the proportion of women who would date someone with a great personality even if they did not find them attractive.


9.2 - Two Independent Means

9.2 - Two Independent Means

Let's explore how we can compare the means of two independent groups. If the populations are known to be approximately normally distributed, or if both sample sizes are at least 30, then the sampling distribution can be estimated using the \(t\) distribution. If this assumption is not met then simulation methods (i.e., bootstrapping or randomization) may be used.


9.2.1 - Confidence Intervals

9.2.1 - Confidence Intervals

Given that the populations are known to be normally distributed, or if both sample sizes are at least 30, then the sampling distribution can be approximated using the \(t\) distribution, and the formulas below may be used. Here you will be introduced to the formulas to construct a confidence interval using the \(t\) distribution. Minitab Express will do all of these calculations for you, however, it uses a more sophisticated method to compute the degrees of freedom so answers may vary slightly, particularly with smaller sample sizes. 

General Form of a Confidence Interval
\(point \;estimate \pm (multiplier) (standard \;error)\)

Here, the point estimate is the difference between the two mean, \(\overline X _1 - \overline X_2\).

Standard Error
\(\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}\)
Confidence Interval for Two Independent Means
\((\bar{x}_1-\bar{x}_2) \pm t^\ast{ \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\)

The degrees of freedom can be approximated as the smallest sample size minus one.

Estimated Degrees of Freedom

\(df=smallest\;n-1\)

Example: Exam Scores by Learner Type

A STAT 200 instructor wants to know how traditional students and adult learners differ in terms of their final exam scores. She collected the following data from a sample of students:

Descriptive Statistics
  Traditional Students Adult Learners

\(\overline x\)

41.48 40.79
\(s\) 6.03 6.79
\(n\) 239 138

She wants to construct a 95% confidence interval to estimate the mean difference. 

 

The point estimate, or "best estimate," is the difference in sample means: \(\overline x _1 - \overline x_2 = 41.48-40.79=0.69\)

The standard error can be computed next: \(\sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}=\sqrt{\frac{6.03^2}{239}+\frac{6.79^2}{138}}=0.697\)

To find the multiplier, we need to construct a t distribution with \(df=smaller\;n-1=138-1=137\) to find the t scores that separate the middle 95% of the distribution from the outer 5% of the distribution:

Minitab Express was used to construct a t distribution with 137 degrees of freedom to find the t scores that separate the middle 95% from the outer 5%. Those t scores are plus and minus 1.97743

\(t^*=1.97743\)

Now, we can combine all of these values to construct our confidence interval: 

\(point \;estimate \pm (multiplier) (standard \;error)\)

\(0.69 \pm 1.97743 (0.697)\)

\(0.69 \pm 1.379\) The margin of error is 1.379

\([-0.689, 2.069]\) 

We are 95% confident that the mean difference in traditional students' and adult learners' final exam scores is between -0.689 points and +2.069 points. 


9.2.1.1 - Minitab Express: Confidence Interval Between 2 Independent Means

9.2.1.1 - Minitab Express: Confidence Interval Between 2 Independent Means

Minitab Express can be used to construct a confidence interval for the difference between two independent means. Note that the confidence intervals given in the Minitab Express output assume that either the populations are normally distributed or that both sample sizes are at least 30.

MinitabExpress  – Confidence Interval Between 2 Independent Means

Let's estimate the difference between the mean weight (in pounds) of females and the mean weight of males. Both sample sizes are at least 30 so the sampling distribution can be approximated using the t distribution. Above, we found that the sample standard deviations are similar, so we will assume equal variances. 

  1. Open the Minitab Express file:
  2. On a PC: In the menu bar select STATISTICS > Two Samples > t
  3. On a Mac: In the menu bar select Statistics > 2-Sample Inference > t
  4. Double click the variable Weight in the box on the left to insert the variable into the Samples box
  5. Double click the variable Gender in the box on the left to insert the variable into the Sample IDs box
  6. Click OK

This should result in the following output:

2-Sample t: Weight by Gender
Method
\(\mu_1\): mean of Weight when Gender = Female
\(\mu_2\): mean of Weight when Gender = Male
Difference: \(\mu_1-\mu_2\)

Equal variances are not assumed for this analysis.

Descriptive Statistics: Weight
Gender N Mean StDev SE Mean
Female 126 136.722 23.362 2.081
Male 99 172.717 27.301 2.744
Estimation for Difference
Difference 95% CI for Difference
-35.995 (-42.787, -29.202)
Video Walkthrough

Select your operating system below to see a step-by-step guide for this example.

I am 95% confident that in the population the mean weight of females is between 29.202 pounds and 42.787 pounds less than the mean weight of males.


9.2.2 - Hypothesis Testing

9.2.2 - Hypothesis Testing

The formula for the test statistic follows the same general format as the others that we have seen this week:

Test Statistic
\(test\; statistic = \dfrac{sample \; statistic - null\;parameter}{standard \;error}\)

Minitab Express will compute the test statistic for you! You will just need to determine if equal variances should be assumed or not. There is one example below walking through these procedures by hand, but you are strongly encouraged to use Minitab Express whenever possible.

1. Check any necessary assumptions and write null and alternative hypotheses.

There are two assumptions: (1) the two samples are independent and (2) both populations are normally distributed or \(n_1 \geq 30\) and \(n_2 \geq 30\). If the second assumption is not met then you can conduct a randomization test.

Below are the possible null and alternative hypothesis pairs:

Research Question Are the means of group 1 and group 2 different? Is the mean of group 1 greater than the mean of group 2? Is the mean of group 1 less than the mean of group 2?
Null Hypothesis, \(H_{0}\) \(\mu_1 = \mu_2\) \(\mu_1 = \mu_2\) \(\mu_1 = \mu_2\)
Alternative Hypothesis, \(H_{a}\) \(\mu_1 \neq \mu_2\) \(\mu_1 > \mu_2\) \(\mu_1 < \mu_2\)
Type of Hypothesis Test Two-tailed, non-directional Right-tailed, directional Left-tailed, directional
2. Calculate an appropriate test statistic.

Standard Error

\(\sqrt{\dfrac{s_1^2}{n_1}+\dfrac{s_2^2}{n_2}}\)

Test Statistic for Independent Means

\(t=\dfrac{\bar{x}_1-\bar{x}_2}{ \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\)

Estimated Degrees of Freedom

\(df=smallest\;n - 1\)

3. Determine the p-value associated with the test statistic.

The \(t\) test statistic found in Step 2 is used to determine the p-value. 

4. Decide between the null and alternative hypotheses.

If \(p \leq \alpha\) reject the null hypothesis. If \(p>\alpha\) fail to reject the null hypothesis.

5. State a "real world" conclusion.

Based on your decision in Step 4, write a conclusion in terms of the original research question.


9.2.2.1 - Minitab Express: Independent Means t Test

9.2.2.1 - Minitab Express: Independent Means t Test

Here we will use Minitab Express to conduct an independent means t test. Note that Minitab Express uses a more complicated formula for computing the degrees of freedom for this test. To read more about the formulas that Minitab Express uses, see Minitab Express Support.

Within Minitab Express, the procedure for obtaining the test statistic and confidence interval for independent means is identical.

MinitabExpress  – Conducting an Independent Means t Test

Let's compare the mean SAT-Math scores of students who have and have not ever cheated. Both sample sizes are at least 30 so the sampling distribution can be approximated using the \(t\) distribution. Above, we found that the sample standard deviations are similar, so we will assume equal variances. 

  1. Open the Minitab Express file:
  2. On a PC: In the menu bar select STATISTICS > Two Samples > t
  3. On a Mac: In the menu bar select Statistics > 2-Sample Inference > t
  4. Double click the variable SATM in the box on the left to insert the variable into the Samples box
  5. Double click the variable Ever Cheat in the box on the left to insert the variable into the Sample IDs box
  6. Click OK

This should result in the following output:

2-Sample t: SATM by Ever Cheat
Method
\(\mu_1\): mean of SATM when Ever Cheat = No
\(\mu_2\): mean of SATM when Ever Cheat = Yes
Difference: \(\mu_1-\mu_2\)

Equal variances are not assumed for this analysis.

Descriptive Statistics: SATM
Ever Cheat N Mean StDev SE Mean
No 163 603.988 86.893 6.806
Yes 53 583.68 79.18 10.88
Estimation for Difference
Difference 95% CI for Difference
20.31 (-5.16, 45.78)
Test
Null hypothesis \(H_0\): \(\mu_1-\mu_2=0\)
Alternative hypothesis \(H_1\): \(\mu_1-\mu_2\neq0\)
T-Value DF P-Value
1.58 95 0.1168
Video Walkthrough

Select your operating system below to see a step-by-step guide for this example.

The result of our two independent means t test is \(t(95) = 1.58, p = 0.1168\). Our p-value is greater than the standard alpha level of 0.05 so we fail to reject the null hypothesis. There is not evidence to state that the mean SAT-Math scores of students who have and have not ever cheated are different. 

Note that we could also interpret the confidence interval in this output. We are 95% confident that the mean difference in the population is between -5.16 and 45.78.

The example above uses a dataset. The following examples show how you can conduct this type of test using summarized data.


9.2.2.1.1 - Video Example: SAT-Verbal Scores

9.2.2.1.1 - Video Example: SAT-Verbal Scores

Research question: Do students who complete test prep program A have higher SAT-Verbal scores than students who complete test prep program B?

The following data were collected from representative samples of students who were randomly assigned to complete either program A or program B.

  Program A Program B
\(\overline X\) 530 510
\(s\) 90 80
\(n\) 35 30

The sampling distribution may be approximated using the t distribution because \(n_A \ge 30\) and \(n_B \ge 30\). 

Let's take a look at how we can go about solving this research question using Minitab Express.


9.2.2.1.2 - Video Example: Comparing Kids' Weight by Gender

9.2.2.1.2 - Video Example: Comparing Kids' Weight by Gender

A school nurse wants to know if the fifth grade boys at his school weigh more than the fifth grade girls on average. This is an example of a one-tailed (directional) test of two independent means using summarized data.


9.2.2.1.3 - Video Example: Hotel Ratings

9.2.2.1.3 - Video Example: Hotel Ratings

A group of hotel management students wants to compare guest satisfaction ratings from two hotels. A random sample guests from each hotel are asked to rate their level of satisfaction on a scale from 0 to 100. This is an example of a two-tailed (non-directional) test for two independent means using summarized data.


9.3 - Lesson 9 Summary

9.3 - Lesson 9 Summary

Objectives

Upon successful completion of this lesson, you should be able to:

  • Identify situations in which the z or t distribution may be used to approximate a sampling distribution
  • Construct a confidence interval to estimate the difference in two population proportions and two population means using Minitab Express given summary or raw data
  • Conduct a hypothesis test for two proportions and two means using Minitab Express given summary or raw data
  Confidence Interval Test Statistic
Two Independent Proportions
At least 10 successes and 10 failures in both samples.
\((\widehat{p}_1-\widehat{p}_2) \pm z^\ast {\sqrt{\frac{\widehat{p}_1 (1-\widehat{p}_1)}{n_1}+\frac{\widehat{p}_2 (1-\widehat{p}_2)}{n_2}}}\) \(z=\frac{\widehat{p}_1-\widehat{p}_2}{\sqrt{\widehat{p}(1-\widehat{p})\left ( \frac{1}{n_1}+\frac{1}{n_2} \right )}}\)
Two Independent Means
Both sample size are at least 30 OR populations are normally distributed.

\((\bar{x}_1-\bar{x}_2) \pm t^\ast{ \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\)

\(Estimated \;df = smallest\; n - 1\)

\(t=\frac{\bar{x}_1-\bar{x}_2}{ \sqrt{\frac{s_1^2}{n_1}+\frac{s_2^2}{n_2}}}\)

\(Estimated \;df = smallest\; n - 1\)


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