Confidence interval for two proportions examples

## Confidence interval for two proportions examples

In this tutorial we will discuss some examples on confidence interval for difference between two population proportions.

## Example 1

A random sample of 85 parts manufactured by machine A yields 10 defective and a random sample of 110 parts manufactured by machine B shows 28 defective.

Compute 95% confidence interval for difference between the proportions of defectives?

Are the two machine differ significantly with respect to the proportion of defectives?

#### Solution

Given information

. | Machine A | Machine B |
---|---|---|

Sample size | $n_1=85$ | $n_2=110$ |

Observed no. of defectives | $X_1=10$ | $X_2=28$ |

We wish to determine $95$% confidence interval estimate of the difference between two population proportions $(p_1-p_2)$.

#### Step 1 Specify the confidence level

Confidence level is $1-\alpha = 0.95$. Thus, the level of significance is $\alpha = 0.05$.

#### Step 2 Given information

Given that $X_1 = 10$, $X_2 = 28$, $n_1 = 85$, $n_2 = 110$.

The estimate of the population proportions $p_1$ is $\hat{p}_1 =\frac{X_1}{n_1} =\frac{10}{85}=0.1176$ and

the estimate of the population proportion $p_2$ is $\hat{p}_2 =\frac{X_2}{n_2} =\frac{28}{110}=0.2545$.

#### Step 3 Specify the formula

$100(1-\alpha)$% confidence interval estimate of the difference between two population proportions $(p_1-p_2)$ is

` $$ \begin{aligned} (\hat{p}_1-\hat{p}_2) - E \leq (p_1 -p_2) \leq (\hat{p}_1 -\hat{p}_2)+ E. \end{aligned} $$ `

where `$E = Z_{\alpha/2} \sqrt{\frac{\hat{p}_1*(1-\hat{p}_1)}{n_1}+\frac{\hat{p}_2*(1-\hat{p}_2)}{n_2}}$`

and `$Z_{\alpha/2}$`

is the $Z$ value providing an area of $\alpha/2$ in the upper tail of the standard normal probability distribution.

#### Step 4 Determine the critical value

The critical value of $Z$ for given level of significance is $Z_{\alpha/2}$.

Thus `$Z_{\alpha/2} = Z_{0.025} = 1.96$`

.

#### Step 5 Compute the margin of error

The margin of error for the difference $(p_1-p_2)$ is

` $$ \begin{aligned} E & = Z_{\alpha/2} \sqrt{\frac{\hat{p}_1*(1-\hat{p}_1)}{n_1}+\frac{\hat{p}_2*(1-\hat{p}_2)}{n_2}}\\ & = 1.96 \sqrt{\frac{0.1176*(1-0.1176)}{85}+\frac{0.2545*(1-0.2545)}{110}}\\ &= 0.1064. \end{aligned} $$ `

#### Step 6 Determine the confidence interval

$95$% confidence interval estimate of the difference between two population proportions $(p_1-p_2)$ is

` $$ \begin{aligned} (\hat{p}_1-\hat{p}_2) - E &\leq (p_1-p_2) \leq (\hat{p}_1-\hat{p}_2) + E\\ (0.1176-0.2545) - 0.1064 & \leq (p_1-p_2) \leq (0.1176-0.2545) + 0.1064\\ -0.2433 & \leq (p_1-p_2) \leq -0.0305 \end{aligned} $$ `

Thus, $95$% confidence interval estimate of the difference between two population proportions $(p_1-p_2)$ is $(-0.2433,-0.0305)$.

#### Interpretation

We can be $95$% confident that the difference between two population proportions $(p_1-p_2)$ is between $-0.2433$ and $-0.0305$.

Because the $95$% confidence interval does not include the point zero, we conclude that at $0.05$ level of significance the two machine do not differ significantly with respect to the proportion of defectives.

## Example 2

In a recent survey of randomly selected adults 65 or older, 411 of 1012 men and 525 of 1062 women say they suffer from some form of arthritis.

a. Construct a 98% confidence interval for the difference between senior men and women who suffer from arthritis.

b. Does there appear to be a difference between senior men and women as far as suffering from arthritis?

#### Solution

Given information

. | Men | Women |
---|---|---|

Sample size | $n_1=1012$ | $n_2=1062$ |

Suffering from arthritis | $X_1=411$ | $X_2=525$ |

We want to determine $98$% confidence interval estimate of the difference between two population proportions $(p_1-p_2)$.

#### Step 1 Specify the confidence level

Confidence level is $1-\alpha = 0.98$. Thus, the level of significance is $\alpha = 0.02$.

#### Step 2 Given information

Given that $X_1 = 411$, $X_2 = 525$, $n_1 = 1012$, $n_2 = 1062$.

The estimate of the population proportions $p_1$ is $\hat{p}_1 =\frac{X_1}{n_1} =\frac{411}{1012}=0.4061$ and

the estimate of the population proportion $p_2$ is $\hat{p}_2 =\frac{X_2}{n_2} =\frac{525}{1062}=0.4944$.

#### Step 3 Specify the formula

$100(1-\alpha)$% confidence interval estimate of the difference between two population proportions $(p_1-p_2)$ is

` $$ \begin{aligned} (\hat{p}_1-\hat{p}_2) - E \leq (p_1 -p_2) \leq (\hat{p}_1 -\hat{p}_2)+ E. \end{aligned} $$ `

where `$E = Z_{\alpha/2} \sqrt{\frac{\hat{p}_1*(1-\hat{p}_1)}{n_1}+\frac{\hat{p}_2*(1-\hat{p}_2)}{n_2}}$`

and `$Z_{\alpha/2}$`

is the $Z$ value providing an area of $\alpha/2$ in the upper tail of the standard normal probability distribution.

#### Step 4 Determine the critical value

The critical value of $Z$ for given level of significance is $Z_{\alpha/2}$.

Thus `$Z_{\alpha/2} = Z_{0.01} = 2.33$`

.

#### Step 5 Compute the margin of error

The margin of error for the difference $(p_1-p_2)$ is

` $$ \begin{aligned} E & = Z_{\alpha/2} \sqrt{\frac{\hat{p}_1*(1-\hat{p}_1)}{n_1}+\frac{\hat{p}_2*(1-\hat{p}_2)}{n_2}}\\ & = 2.33 \sqrt{\frac{0.4061*(1-0.4061)}{1012}+\frac{0.4944*(1-0.4944)}{1062}}\\ &= 0.0507. \end{aligned} $$ `

#### Step 6 Determine the confidence interval

$98$% confidence interval estimate of the difference between two population proportions $(p_1-p_2)$ is

` $$ \begin{aligned} (\hat{p}_1-\hat{p}_2) - E &\leq (p_1-p_2) \leq (\hat{p}_1-\hat{p}_2) + E\\ (0.4061-0.4944) - 0.0507 & \leq (p_1-p_2) \leq (0.4061-0.4944) + 0.0507\\ -0.1389 & \leq (p_1-p_2) \leq -0.0375 \end{aligned} $$ `

Thus, $98$% confidence interval estimate of the difference between two population proportions $(p_1-p_2)$ is $(-0.1389,-0.0375)$.

#### Interpretation

We can be $98$% confident that the difference between two population proportions $(p_1-p_2)$ is between $-0.1389$ and $-0.0375$.

Because the $98$% confidence interval does not include the point zero, we conclude that at $0.02$ level of significance there appears to be no significant difference between the proportion of senior men and proportion of women who suffer from arthritis.

## Example 3

Two machines used in the same operation are to be compared. A random sample of 80 parts from the first machine yields 6 non-conforming ones. A random sample of 120 parts from the second machine shows 14 non-conforming ones.

Find a 95% confidence interval for difference in the proportion of non-conforming parts between the two machines.

#### Solution

Given information

. | First Machine | Second Machine |
---|---|---|

Sample size | $n_1=80$ | $n_2=200$ |

no.of non-confirming | $X_1=6$ | $X_2=14$ |

We want to determine $95$% confidence interval estimate of the difference between two population proportions $(p_1-p_2)$.

#### Step 1 Specify the confidence level

Confidence level is $1-\alpha = 0.95$. Thus, the level of significance is $\alpha = 0.05$.

#### Step 2 Given information

Given that $X_1 = 6$, $X_2 = 14$, $n_1 = 80$, $n_2 = 200$.

The estimate of the population proportions $p_1$ is $\hat{p}_1 =\frac{X_1}{n_1} =\frac{6}{80}=0.075$ and the estimate of the population proportion $p_2$ is $\hat{p}_2 =\frac{X_2}{n_2} =\frac{14}{200}=0.07$.

#### Step 3 Specify the formula

$100(1-\alpha)$% confidence interval estimate of the difference between two population proportions $(p_1-p_2)$ is

` $$ \begin{aligned} (\hat{p}_1-\hat{p}_2) - E \leq (p_1 -p_2) \leq (\hat{p}_1 -\hat{p}_2)+ E. \end{aligned} $$ `

where `$E = Z_{\alpha/2} \sqrt{\frac{\hat{p}_1*(1-\hat{p}_1)}{n_1}+\frac{\hat{p}_2*(1-\hat{p}_2)}{n_2}}$`

and `$Z_{\alpha/2}$`

is the $Z$ value providing an area of $\alpha/2$ in the upper tail of the standard normal probability distribution.

#### Step 4 Determine the critical value

The critical value of $Z$ for given level of significance is $Z_{\alpha/2}$.

Thus `$Z_{\alpha/2} = Z_{0.025} = 1.96$`

.

#### Step 5 Compute the margin of error

The margin of error for the difference $(p_1-p_2)$ is

` $$ \begin{aligned} E & = Z_{\alpha/2} \sqrt{\frac{\hat{p}_1*(1-\hat{p}_1)}{n_1}+\frac{\hat{p}_2*(1-\hat{p}_2)}{n_2}}\\ & = 1.96 \sqrt{\frac{0.075*(1-0.075)}{80}+\frac{0.07*(1-0.07)}{200}}\\ &= 0.0677. \end{aligned} $$ `

#### Step 6 Determine the confidence interval

$95$% confidence interval estimate of the difference between two population proportions $(p_1-p_2)$ is

` $$ \begin{aligned} (\hat{p}_1-\hat{p}_2) - E &\leq (p_1-p_2) \leq (\hat{p}_1-\hat{p}_2) + E\\ (0.075-0.07) - 0.0677 & \leq (p_1-p_2) \leq (0.075-0.07) + 0.0677\\ -0.0627 & \leq (p_1-p_2) \leq 0.0727 \end{aligned} $$ `

Thus, $95$% confidence interval estimate of the difference between two population proportions $(p_1-p_2)$ is $(-0.0627,0.0727)$.

#### Interpretation

We can be $95$% confident that the difference between two population proportions $(p_1-p_2)$ is between $-0.0627$ and $0.0727$.

Because the $95$% confidence interval include the point zero, we conclude that at $0.05$ level of significance there is a significant difference between the proportion of non-confirming parts between the two machines.

Do read my step by step tutorial on Confidence interval for two proportions, tutorial will help you to understand Step by step procedure to estimate the confidence interval for difference between two population proportions is as follows:

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