submit test
Test the claim below about the mean of the differences for a population of paired data at the level of significance $\alpha$. Assume the samples are random and dependent, and the populations are normally distributed.
Claim: $\mu_{d} \geq 0 ; \alpha=0.10$. Sample statistics: $\bar{d}=-2.2, s_{d}=1.2, n=16$
Identify the null and alternative hypotheses. Choose the correct answer below.
A.
\[
\begin{array}{l}
H_{0}: \mu_{d}=0 \\
H_{a}: \mu_{d} \neq 0
\end{array}
\]
C.
\[
\begin{array}{l}
H_{0}: \mu_{d} \leq 0 \\
H_{a}: \mu_{d}> 0
\end{array}
\]
E.
\[
\begin{array}{l}
H_{0}: \mu_{d} \geq 0 \\
H_{a}: \mu_{d}< 0
\end{array}
\]
B.
\[
\begin{array}{l}
H_{0}: \mu_{d} \neq 0 \\
H_{a}: \mu_{d}=0
\end{array}
\]
D.
\[
\begin{array}{l}
H_{0}: \mu_{d}< 0 \\
H_{a}: \mu_{d} \geq 0
\end{array}
\]
F.
\[
\begin{array}{l}
H_{0}: \mu_{d}> 0 \\
H_{a}: \mu_{d} \leq 0
\end{array}
\]
The test statistic is $\mathrm{t}=\square$.
(Round to two decimal places as needed.)
The P-value is $\square$.
(Round to three decimal places as needed.)
Since the $\mathrm{P}$-value is the level of significance, the null hypothesis. There statistically significant evidence to reject the claim.
Answer
\(\boxed{\text{There is statistically significant evidence to reject the claim that } \mu_{d} \geq 0}\)
Step 1 ：The null hypothesis (H0) is \(\mu_{d} \geq 0\) and the alternative hypothesis (Ha) is \(\mu_{d} < 0\)
Step 2 ：Calculate the test statistic using the formula: t = (\(\bar{d}\) - \(\mu_{d}\)) / (s\(_{d}\) / \(\sqrt{n}\))
Step 3 ：Substitute the given values into the formula: t = (-2.2 - 0) / (1.2 / \(\sqrt{16}\)) = -7.33
Step 4 ：The test statistic t = -7.33
Step 5 ：Calculate the P-value. The P-value is the probability of obtaining a result as extreme as the observed data, assuming the null hypothesis is true
Step 6 ：Since we are dealing with a one-tailed test (because Ha: \(\mu_{d} < 0\)), we look up the P-value in a t-distribution table for t = -7.33 and df = n - 1 = 16 - 1 = 15
Step 7 ：The t-value is so extreme that it's off the charts, and the P-value is less than 0.001
Step 8 ：So, the P-value < 0.001
Step 9 ：Since the P-value is less than the level of significance (0.10), we reject the null hypothesis
Step 10 ：\(\boxed{\text{There is statistically significant evidence to reject the claim that } \mu_{d} \geq 0}\)
