## Related questions with answers

Use a sign test to test the claim by doing the following. (a) Identify the claim and state $H_0$ and $H_a$. (b) Find the critical value. (c) Find the test statistic. (d) Decide whether to reject or fail to reject the null hypothesis. (e) Interpret the decision in the context of the original claim. A store manager claims that the median number of customers per day is no more than 650. The numbers of customers per day for 17 randomly selected days are listed below. At $\alpha=0.01$, can you reject the manager’s claim?

$\begin{matrix} \text{675 665 601 642 554 653 639 650 645}\\ \text{550 677 569 650 660 682 689 590}\\ \end{matrix}$

Solution

VerifiedGiven:

$\begin{align*} n&=\text{Sample size}-\text{Ties}=17-2=15 \\ \alpha&=\text{Significance level}=0.01 \end{align*}$

(a) Given claim: MEDIAN$\leq 650$

The claim is either the null hypothesis or the alternative hypothesis. The null hypothesis and the alternative hypothesis state the opposite of each other. The null hypothesis needs to contain the value mentioned in the claim.

$H_0:MEDIAN\leq 650$

$H_a:MEDIAN>650$

(b) Determine the critical value in the row with $n=15$ and in the table with $\alpha=0.01$ (one-tailed) using table 8 ($n\leq 25$):

$x_c=2$

(c) The number of plus signs is the number of data values larger than the median in the claim, while the number of minus signs is the number of data values smaller than the median in the claim..

$\begin{align*} \# \text{ + signs}&=7 \\ \#\text{ - signs}&=8 \end{align*}$

When the sample size $n$ is at most 25, the the value of the test statistic is the minimum of the number of positive signs and the number of minus signs.

$x=\min(\# \text{ + signs}, \#\text{ - signs})=\min (7,8)=7$

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