12.3.101 two proportion z-test

The numbers of successes and the sample sizes for independent simple random samples from two populations are provided for a​ left-tailed test and an​ 80% confidence interval. Complete parts​ (a) through​ (d). \(x_1 = 20, n_1 = 50,x_2 = 23, n_2 = 50, \alpha =.10\)

(a). Determine the sample proportion. We use the formula \(\hat{p}= \frac{x}{n}\)

x1 = 20
n1 = 50
p1 = x1/n1
p1

## [1] 0.4

\(\hat{p_1} = .4\)

x2 = 23
n2 = 50
p2 = x2/n2
p2

## [1] 0.46

round(p2,3)

## [1] 0.46

\(\hat{p_2} = .46\)

p = (x1 + x2)/(n1+n2)
p

## [1] 0.43

round(p,3)

## [1] 0.43

\(\hat{p} = .43\)

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(b) Decide whether using the​ two-proportions z-procedures is appropriate.

There are two conditions for two-proportions z test:

1. simple random sample

2. \(x_1,n_1-x_1,x_2,n_2-x_2,\) are 5 are greater

The first condition is passed since the question specifies a simple random sample

To check the second condition we can run

x1 >= 5

## [1] TRUE

n1 - x1 >= 5

## [1] TRUE

x2 >= 5

## [1] TRUE

n2 - x2 >= 5

## [1] TRUE

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(c).If​ appropriate, use the​ two-proportions z-test to conduct the required hypothesis test.What are the hypotheses for this​ test?. Since this is a left-tailted test, we have

\(H_0:p_1=p_2, H_a:p_1<p_2\)

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To compute test statistic z we use the formula \(z=\frac{\hat{p_1}-\hat{p_2}}{\sqrt{\hat{p_p}(1-\hat{p_p})}.\sqrt{(\frac{1}{n_1})+(\frac{1}{n_2})}}\)

z  = (p1 -p2)/(sqrt(p*(1-p)) * sqrt(1/n1+1/n2))
z

## [1] -0.6059679

Round to two decimal places

round(z,2)

## [1] -0.61

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Identify the critical​ value(s), if appropriate. Select the correct choice below​ and, if​ necessary, fill in the answer box to complete your answer.

It is a left-tailed test with \(\alpha = .1\), so the area to the left of critical value = .1

To find negative critical value, we run

round(qnorm(.1),2)

## [1] -1.28

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Since \(-z_{\alpha}=-1.28< z = -.61\) the test statistic does not lie in rejected region. We do not have enough evidence to reject hypothesis.

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(d) If​ appropriate, use the​ two-proportions z-interval procedure to find the specified confidence interval. Select the correct choice below​ and, if​ necessary, fill in the answer boxes to complete your answer.

The 80% confidence interval is from… to…

First approach we calculate confidence interval using margin of error

To calculate margin of error we use formula \(E=z_{\alpha/2}.\sqrt{\frac{\hat{p_1}(1-\hat{p1})}{n_1}+\frac{\hat{p_2}(1-\hat{p2})}{n_2}}\)

Since the confidence level is .8, we have \(\alpha = .2\) and \(\alpha/2=.1\)

alpha = .2
zalpha2 = abs(qnorm(alpha/2))
E = zalpha2 * sqrt(p1*(1-p1)/n1 + p2*(1-p2)/n2)
E

## [1] 0.1266598

To calculate the confidence interval, we use the formula \((\hat{p_1}-\hat{p_2}) \pm E\)

(p1-p2) + E

## [1] 0.06665983

(p1-p2) - E

## [1] -0.1866598

Round to three decimal places

round((p1-p2) + E, 3)

## [1] 0.067

round((p1-p2) - E, 3)

## [1] -0.187

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Second approach: find confidence interval directly without using margin of error E

To calculate the confidence interval, we use the formula \((\hat{p_1}-\hat{p_2}) \pm z_{\alpha/2}.\sqrt{\frac{\hat{p_1}(1-\hat{p_1})}{n_1}+\frac{\hat{p_2}(1-\hat{p_2})}{n_2}}\)

alpha = .2
zalpha2 = abs(qnorm(alpha/2))
(p1-p2) + zalpha2 * sqrt(p1*(1-p1)/n1 + p2*(1-p2)/n2)

## [1] 0.06665983

(p1-p2) - zalpha2 * sqrt(p1*(1-p1)/n1 + p2*(1-p2)/n2)

## [1] -0.1866598

Round to three decimal places

round((p1-p2) + zalpha2 * sqrt(p1*(1-p1)/n1 + p2*(1-p2)/n2),3)

## [1] 0.067

round((p1-p2) - zalpha2 * sqrt(p1*(1-p1)/n1 + p2*(1-p2)/n2),3)

## [1] -0.187

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Notes: confidence interval is a range of data that contain unknown population and critical value is where we split the data to rejected and non-rejected region

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Hope that helps!