Question

A recent study conducted by the state government attempts to determine whether the voting public supports a further increase in cigarette taxes. The opinion poll recently sampled 1500 voting age citizens. 1020 of the sampled citizens were in favor of an increase in cigarette taxes. The state government would like to decide if there is enough evidence to establish whether the proportion of citizens supporting an increase in cigarette taxes is significantly greater than 0.66. At α = 0.10:

Select which null and alternate is correct.
H_0:p≤1200 , H_a:p>1200
H_0:p≤1500 , H_a:p>1500
H_0:p≤0.66 , H_a:p>0.66
H_0:μ<0.66 , H_a:P≥0.66


At α =0.10 what is the "test statistic"?


What is the "critical value"?


What is the conclusion of the hypothesis test and why?

Answer 1

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.

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`

Alternative hypothesis,
`{H_a}:p > 0.66`

So Option c is true.

Part b:

The value of sample proportion is,

`\begin{array}{c}\\\hat p = (x)/(n)\\\\ = \frac{{1020}}{{1500}}\\\\ = 0.68\\\end{array}`

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}`

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.

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.

Hence, there is sufficient evidence to conclude that the proportion of citizens supporting an increase in cigarette taxes is significantly greater than 0.66.

There is sufficient evidence to conclude that the proportion of citizens supporting an increase in cigarette taxes is significantly greater than 0.66.