Chapter 9, Problem 5PT

To determine

To find:The observed value of the test statistic for testing the null hypothesis $H_0:p=0.5$ versus the alternative hypothesis $H_a:p>0.5$

Answer to Problem 5PT

The answer is b) $\frac{0.59-0.5}{\sqrt{\frac{0.5(0.5)}{100}}}$

Explanation of Solution

Given information:

Total number of random samples of voters =100

Number of voters in favour of Candidate $A$ =59

Let $x$ denote the number of voters in favour of candidate A.

Given $x=59$

Let $\widehat{p}$ denote the proportion of voters in favour of candidate A.

  $\begin{array}{cc}\widehat{p}& =\frac{x}{n}\\ & =\frac{59}{100}\\ & =0.59\end{array}$

Test statistic,

  $\begin{array}{cc}z& =\frac{\widehat{p}-p_0}{\sqrt{\frac{p_0\left(1-p_0\right)}{n}}}\\ & =\frac{0.59-0.5}{\sqrt{\frac{0.5(1-0.5)}{100}}}\\ & =\frac{0.59-0.5}{\sqrt{\frac{0.5(0.5)}{100}}}\end{array}$

So the answer is b) $\frac{0.59-0.5}{\sqrt{\frac{0.5(0.5)}{100}}}$