Question

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)
$x_{1}=10,n_{1}=90,x_{2}=25,n_{2}=90,α = 0.10$
click here to view a table of areas under the standard normal curve for negative values of z.
click here to view a table of areas under the standard normal curve for positive values of z.
c. if appropriate, use the two-proportions z-test to conduct the required hypothesis test.
what are the hypotheses for this test?
a. $h_{0}:p_{1}=p_{2},h_{a}:p_{1}>p_{2}$
b. $h_{0}:p_{1}c. $h_{0}:p_{1}=p_{2},h_{a}:p_{1}d. $h_{0}:p_{1}=p_{2},h_{a}:p_{1}≠p_{2}$
e. $h_{0}:p_{1}>p_{2},h_{a}:p_{1}=p_{2}$
f. $h_{0}:p_{1}≠p_{2},h_{a}:p_{1}=p_{2}$
g. using the two-proportions z-procedures is not appropriate
determine the test statistic, if appropriate. select the correct choice below and, if necessary, fill in the answer box to complete your answer.
a. $z=-2.83$ (type an integer or a decimal. round to two decimal places as needed.)
b. using the two-proportions z-procedures is not appropriate
identify the critical value(s) for $α = 0.10$, if appropriate. select the correct choice below and, if necessary, fill in the answer box to complete your answer
(type an integer or a decimal. round to two decimal places as needed.)
a. the critical values are ±
b. the critical value is
c. using the two-proportions z-procedures is not appropriate.

Explanation:

Step1: Hypotheses

For a left - tailed test comparing two proportions \(p_1\) and \(p_2\), the null hypothesis \(H_0\) is \(p_1 = p_2\) and the alternative hypothesis \(H_a\) is \(p_1

Step2: Test statistic

First, calculate the sample proportions:
\(\hat{p}_1=\frac{x_1}{n_1}=\frac{10}{90}\approx0.111\), \(\hat{p}_2=\frac{x_2}{n_2}=\frac{25}{90}\approx0.278\)
The pooled proportion \(\hat{p}=\frac{x_1 + x_2}{n_1 + n_2}=\frac{10 + 25}{90+90}=\frac{35}{180}\approx0.194\)
The test statistic \(z=\frac{\hat{p}_1-\hat{p}_2}{\sqrt{\hat{p}(1 - \hat{p})(\frac{1}{n_1}+\frac{1}{n_2})}}\)
\(z=\frac{0.111 - 0.278}{\sqrt{0.194\times(1 - 0.194)\times(\frac{1}{90}+\frac{1}{90})}}\)
\(z=\frac{- 0.167}{\sqrt{0.194\times0.806\times\frac{2}{90}}}\)
\(z=\frac{-0.167}{\sqrt{\frac{0.194\times0.806\times2}{90}}}\)
\(z=\frac{-0.167}{\sqrt{\frac{0.313}{90}}}\)
\(z=\frac{-0.167}{\sqrt{0.00348}}\)
\(z=\frac{-0.167}{0.059}\approx - 2.83\)

Step3: Critical value

For a left - tailed test with \(\alpha = 0.10\), the critical value \(z_{\alpha}\) is the value such that \(P(Z<z_{\alpha})=\alpha\). From the standard normal table, \(z_{0.10}=-1.28\)

Answer:

• Hypotheses: \(H_0:p_1 = p_2\), \(H_a:p_1
• Test statistic: \(z=-2.83\)
• Critical value: \(z=-1.28\)