Suppose the sample in Exercise 7.64 has produced \(\hat p = .83\) and we wish to test \({H_0}:P = 0.9\) against the alternative \({H_a}:p < .9\)

a. Calculate the value of the z-statistic for this test.

The number of sample size is 100.

The hypothesis are given by

\(\begin{aligned}{H_0}:p = 0.9\\{H_a}:p < 0.9\end{aligned}\)

When the variations are known as well as the sampling size is high, a z-test is used to assess if two population means vary. The Z test is a statistical test performed on data that roughly follows a normally distributed. For hypothesis testing, the z test can be used to one sample, samples collected, as well as percentages.

The z-statistic is computed as

\(\begin{aligned}z &= \frac{{\hat p - {p_0}}}{{\sqrt {\frac{{{p_0}{q_0}}}{n}} }}\\ &= \frac{{0.83 - 0.9}}{{\sqrt {\frac{{0.9 \times 0.1}}{{100}}} }}\\ &= \frac{{ - 0.07}}{{0.03}}\\ &= - 2.333\end{aligned}\)

Therefore, the z-statistic is -2.333.