Answer:
Part a: Option c is true.
Part b: The value of test statistic is 1.64.
Part c: The value is critical z is found as 1.28.
Part d: There is sufficient evidence to conclude that the proportion of citizens supporting an increase in cigarette taxes is significantly greater than 0.66.
Step-by-step explanation:
Part a:
From the given data
x=1020
n=1500
Let α be the level of significance = 0.10.
Test the claim that the proportion of citizens supporting an increase in cigarette taxes is significantly greater than 0.66.
The null and alternative hypothesis is as follows:
Null hypothesis, ![{H_0}:p \le 0.66](https://tex.z-dn.net/?f=%7BH_0%7D%3Ap%20%5Cle%200.66)
Alternative hypothesis, ![{H_a}:p > 0.66](https://tex.z-dn.net/?f=%7BH_a%7D%3Ap%20%3E%200.66)
So Option c is true.
Part b:
The value of sample proportion is,
![\begin{array}{c}\\\hat p = \frac{x}{n}\\\\ = \frac{{1020}}{{1500}}\\\\ = 0.68\\\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bc%7D%5C%5C%5Chat%20p%20%3D%20%5Cfrac%7Bx%7D%7Bn%7D%5C%5C%5C%5C%20%3D%20%5Cfrac%7B%7B1020%7D%7D%7B%7B1500%7D%7D%5C%5C%5C%5C%20%3D%200.68%5C%5C%5Cend%7Barray%7D)
The test static value is given as
![\begin{array}{c}\\Z = \frac{{\hat p - p}}{{\sqrt {\frac{{p\left( {1 - p} \right)}}{n}} }}\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\,\\\\ = \frac{{\left( {0.68 - 0.66} \right)}}{{\sqrt {\frac{{0.66\left( {1 - 0.66} \right)}}{{1500}}} }}\\\\ = 1.64\\\end{array}](https://tex.z-dn.net/?f=%5Cbegin%7Barray%7D%7Bc%7D%5C%5CZ%20%3D%20%5Cfrac%7B%7B%5Chat%20p%20-%20p%7D%7D%7B%7B%5Csqrt%20%7B%5Cfrac%7B%7Bp%5Cleft%28%20%7B1%20-%20p%7D%20%5Cright%29%7D%7D%7Bn%7D%7D%20%7D%7D%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%2C%5C%5C%5C%5C%20%3D%20%5Cfrac%7B%7B%5Cleft%28%20%7B0.68%20-%200.66%7D%20%5Cright%29%7D%7D%7B%7B%5Csqrt%20%7B%5Cfrac%7B%7B0.66%5Cleft%28%20%7B1%20-%200.66%7D%20%5Cright%29%7D%7D%7B%7B1500%7D%7D%7D%20%7D%7D%5C%5C%5C%5C%20%3D%201.64%5C%5C%5Cend%7Barray%7D)
So the value of test statistic is 1.64.
Part c:
The critical value for α =0.10 is found from the z tables for the value
1-α =1-0.10 =0.90
So, the value is critical z is found as 1.28.
Part d:
Compare the test statistic value with the critical value.
<em>From the values of test statistic and critical value, the test statistic value is greater than critical value. So reject the null hypothesis at 10% level of significance.</em>
<em>Hence, there is sufficient evidence to conclude that the proportion of citizens supporting an increase in cigarette taxes is significantly greater than 0.66.</em>
There is sufficient evidence to conclude that the proportion of citizens supporting an increase in cigarette taxes is significantly greater than 0.66.