Question

$x_1=10,n_1=20,x_2=18,n_2=30;$

left-tailed test, $\alpha =0.10;80\%$confidence interval

a. Determine the sample proportions.

b. Decide whether using the two-proportions z-procedures is appropriate. If so, also do parts (c) and (d).

c. Use she two-proportions z-test to conduct the required hypothesis test.

d. Use the two-proportions z-interval procedure to find the specified confidence interval.

Short answer

(a) The sample proportions are$0.5$and $0.6$.

(b) The two-proportion z-Procedure is appropriate

(c) The data does not provide sufficient evidence to reject the null hypothesis at the$10\%$level of significance.

(d) The specified confidence interval is $-0.284$to $0.084$.

Step-by-step solution

Part (a) Step 1: Given information

Given in the question that,

$x_1=10,n_1=20,x_2=18,n_2=30;$

left-tailed test, $\alpha =0.10;80\%$confidence interval

we need to determine the sample proportions.

Part(a) Step 2: Explanation

The given values are,$x_1=10,n_1=20,x_2=18,n_2=30,\alpha =0.10$, and $80\%$confidence interval.

The formula for ${\tilde{p}}_1$is given by,

${\tilde{p}}_1=\frac{x_1}{n_1}$

Substitute $x_1=10,n_1=20$

${\tilde{p}}_1=\frac{10}{20}$

$=0.5$

The formula for ${\tilde{p}}_2$is given by,

${\tilde{p}}_2=\frac{x_2}{n_2}$

${\tilde{p}}_2=\frac{18}{30}$

role="math" localid="1651479624421" $=0.6$

As a result the sample proportions are $0.5$and $0.6$.

Part (b) Step 1: Given information

Given in the question that,

$x_1=10,n_1=20,x_2=18,n_2=30$;

left-tailed test, $\alpha =0.10;80\%$confidence interval

we need to decide that whether using the two-proportions z-procedures is appropriate. If so, also do parts (c) and (d).

Part(b) Step 2: Explanation

The given values are, $x_1=10,n_1=20,x_2=18,n_2=30,\alpha =0.10$, and $80\%$confidence interval.

To begin, calculate $n_1-x_1$and $n_2-x_2$ . After that, compare the outcome to 5. The two-proportion z-procedure technique is appropriate if it is more than or equal to 5.

The value of $n_1-x_1$is calculated as,

$n_1-x_1=20-10$

$=10$

The value of $n_2-x_2$is calculated as,

$n_2-x_2=30-18$

$=12$

The two-proportion z-procedure technique is appropriate because the values are more than $5$. As a result, the two-proportion z-Procedure is appropriate.

Part (c) Step 1: Given information

Given in the question that,

$x_1=10,n_1=20,x_2=18,n_2=30$

left-tailed test, $\alpha =0.10;80\%$confidence interval

we need to use the two-proportions z-test to conduct the required hypothesis test.

Part (c) Step 2: Explanation

The given values are, $x_1=10,n_1=20,x_2=18,n_2=30,\alpha =0.10$, and $80\%$confidence interval.

The formula for $z$is given by,

$z=\frac{{\tilde{p}}_1-{\tilde{p}}_2}{\sqrt{{\tilde{p}}_p\left(1-{\tilde{p}}_p\right)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$

The formula for ${\tilde{p}}_p$is given by,

${\tilde{p}}_p=\frac{x_1+x_2}{n_1+n_2}$

Substitute $x_1=10,n_1=20,x_2=18,n_2=30$

$=\frac{10+18}{20+30}$

$=\frac{28}{50}$

$=0.56$

Part (c) Step 3: Value of z

The value of$z$is calculated as,

$z=\frac{{\tilde{p}}_1-{\tilde{p}}_2}{\sqrt{{\tilde{p}}_p\left(1-{\tilde{p}}_p\right)}\sqrt{\frac{1}{n_1}+\frac{1}{n_2}}}$

$=\frac{0.5-0.6}{\sqrt{0.56(1-0.56)}\left(\sqrt{\frac{1}{20}+\frac{1}{30}}\right)}$

$=\frac{-0.1}{0.143}$

$=-0.698$

Perform the test at $10\%$level of significance that is $\alpha =0.1$from table-IV (at the bottom) the value of

$z_{\alpha }=1.282$

$z_{0.1}=1.282$

$z-1.282$is the rejected region. As a result, the test static does not fall into the reject zone. As a result, the hypothesis $H_o$is rejected, and the test findings at the $10\%$level are not statistically significant.

As a result, the data does not provide sufficient evidence to reject the null hypothesis at the$10\%$level of significance.

Part (d) Step 1: Given information

Given in the question that,

$x_1=10,n_1=20,x_2=18,n_2=30$;

left-tailed test, $\alpha =0.10;80\%$confidence interval

we need to find the specified confidence interval by using the two-proportions z-interval procedure

Part (d) Step 2: Explanation

The given values are, $x_1=10,n_1=20,x_2=18,n_2=30,\alpha =0.10$, and $80\%$confidence interval.

For confidence level of $(1-\alpha )$the confidence interval for $p_1-p_2$are

$\left({\hat{p}}_1-{\hat{p}}_2\right)\pm z_{1/2}\times \sqrt{{\hat{p}}_1\left(1-{\hat{p}}_1\right)/n_1+{\hat{p}}_2\left(1-{\hat{p}}_2\right)/n_2}$

Calculate the value of $\alpha$,

$80=100(1-\alpha )$

$\alpha =0.2$

The value of $z$at $\alpha /2$from the $z$-score table is $1.282$.

For the difference between the two-population proportion, the needed confidence interval is determined as,

$\left({\tilde{p}}_1-{\tilde{p}}_2\right)\pm z_{\alpha /2}·\sqrt{\frac{{\tilde{p}}_1\left(1-{\tilde{p}}_1\right)}{n_1+2}+\frac{{\tilde{p}}_2\left(1-{\tilde{p}}_2\right)}{n_2+2}}=(0.5-0.6)\pm 1.282$

$.\sqrt{\frac{0.5(1-0.5)}{20}+\frac{0.6(1-0.6)}{30}}$

$=-0.1\pm 0.184$

$=-0.284$to $0.084$

As a result, the difference in adult-American percentages is $-0.284$ to $0.084$.