Step 1 :The problem is asking for the value of the test statistic for a Z-test. The Z-test is appropriate because we are dealing with proportions and we have a large sample size. The Z-test is used when we have quantitative data and when we know the population standard deviation. In this case, we are dealing with proportions, so we can use the Z-test.
Step 2 :The formula for the Z-test statistic is: \(z = \frac{\hat{p} - p_0}{\sqrt{\frac{p_0(1-p_0)}{n}}}\) where: \(\hat{p}\) is the sample proportion, \(p_0\) is the hypothesized population proportion, and \(n\) is the sample size.
Step 3 :In this case, \(\hat{p} = \frac{63}{180}\), \(p_0 = 0.26\), and \(n = 180\).
Step 4 :Substituting these values into the formula, we get: \(z = \frac{0.35 - 0.26}{\sqrt{\frac{0.26(1-0.26)}{180}}}\)
Step 5 :Solving the above expression, we get \(z = 2.7528099422158756\)
Step 6 :Rounding to three decimal places, the final answer is \(\boxed{2.753}\)