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
\((\widehat{p}_1-\widehat{p}_2) \pm z^\ast {\sqrt{\dfrac{\widehat{p}_1 (1-\widehat{p}_1)}{n_1}+\dfrac{\widehat{p}_2 (1-\widehat{p}_2)}{n_2}}}\)

Example: Confidence Interval for the Difference Between the Proportion of Women and Men 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 women in the sample, 185 said "yes." Of the 199 men in the sample, 107 said "yes." Let’s construct a 95% confidence interval for the difference of the proportion of women and men who responded “yes.” We can apply the 95% Rule and use a multiplier of \(z^\ast\) = 2

  • For women: \(np=185\) and \(n(1-p) = 251-185=66\)
  • For men: \(np=107\) 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. 

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

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

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

\(0.199 \pm 2 ( 0.045)\)

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

We are 95% confident that in the population the difference between the proportion of women and men 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 women and men is 0. We can conclude that there is a difference between the proportion of women and men in the population who would respond “yes" to this question.

9.1.1.1 - Minitab: Confidence Interval for 2 Proportions

9.1.1.1 - Minitab: Confidence Interval for 2 Proportions

Minitab 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 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.

Minitab®  – 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 file: class_survey.mpx
  2. Select Stat > Basic Statistics > 2 Proportions
  3. Select Both samples are in one column from the dropdown
  4. Double click the variable Try Weed  in the Samples box
  5. Double click the variable Biological Sex in the Sample IDs box
  6. Keep the default Options
  7. Click OK

This should result in the following output:

Method

Event: Try_Wee = 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)

CI based on normal approximation

Test
Null hypothesis \(H_0\): \(p_1-p_2=0\)
Alternative hypothesis \(H_1\): \(p_1-p_2\neq0\)
Method Z-Value P-Value
     
Normal approximation -2.82 0.005
Fisher's exact   0.007

Minitab®  – 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
  2. Select Stat > Basic Statistics > Two-Sample Proportion
  3. Select Summarized data in the dropdown
  4. For Sample 1 next to Number of events enter 92 and next to Number of trials enter 120
  5. For Sample 2 next to Number of events enter 23 and next to Number of trials enter 160
  6. Keep the default Options
  7. Click OK

This should result in the following output:

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)

CI based on normal approximation

Test
Null hypothesis \(H_0\): \(p_1-p_2=0\)
Alternative hypothesis \(H_1\): \(p_1-p_2\neq0\)
Method Z-Value P-Value
Normal approximation 13.10 0.000
Fisher's exact   0.000
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