Many gamers have reported experiencing motion sickness from using virtual reality (VR) glasses. A new brand of VR glasses is more expensive, but the makers claim the new brand is less likely to make users motion sick. The makers say that the proportion of gamers who will experience motion sickness from using their glasses is $26 \%$ or less. You work for the leading competitor and are doubtful of the claim. To test it, you decide to perform a hypothesis test. To do so, you choose 180 gamers at random and have each of them use the glasses. Of these gamers, 63 said that using the glasses made them motion sick.

You confirm that it is appropriate to perform a Z-test.
Why is a $Z$-test appropriate?

Find $z$, the value of the test statistic for your $Z$-test. Round your answer to three or more decimal places.
\[
z=\square
\]

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 \(\boxed{2.753}\)