(a) identify the claim and state $H_0$ and $H_a$, (b) decide which nonparametric test to use, (c) find the critical value(s), (d) find the test statistic, (e) decide whether to reject or fail to reject the null hypothesis, and (f) interpret the decision in the context of the original claim. The table shows the numbers of emails sent and the numbers of emails received in a week for a random sample of nine people. At $\alpha=0.01,$ can you conclude that there is a significant correlation between the number of emails sent and the number of emails received?

$\begin{matrix} \text{Emails sent} & \text{30} & \text{30} & \text{25} & \text{26} & \text{24} & \text{18} & \text{18} & \text{25} & \text{28}\\ \text{Emails received} & \text{32} & \text{36} & \text{21} & \text{22} & \text{20} & \text{20} & \text{22} & \text{23} & \text{23}\\ \end{matrix}$

Solution

VerifiedGiven:

$\begin{align*} \alpha&=\text{significance level}=0.01 \\ n&=\text{Sample size}=9 \end{align*}$

(a) Given claim: correlation

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 zero.

$\begin{align*} H_0&:\rho_s =0\text{ (There is no correlation.)} \\ H_a&:\rho_s \neq 0\text{ (There is a correlation.)} \end{align*}$

(b) Since we require a test about the correlation, we need to use the $\textbf{Spearman rank correlation coefficient test}$.

(c) The critical value is given in the row $n=9$ and in the column $\alpha=0.01$ of the critical values of Spearman's rank Correlation Coefficient Table in the appendix:

$r=0.833$

Thus the rejection region then contains all values greater than or equal to 0.833 and all values smaller than or equal to $-0.833$.

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