Questions: Test the claim below about the mean of the differences for a population of paired data at the level of significance α. Assume the samples are random and dependent, and the populations are normally distributed. Claim: μd ≥ 0 ; α=0.10. Sample statistics: d̄=-2.3, sd=1.3, n=19 Identify the null and alternative hypotheses. Choose the correct answer below. A. H0: μd ≠ 0 B. H0: μd ≥ 0 Ha: μd=0 Ha: μd<0 C. H0: μd>0 D. H0: μd ≤ 0 Ha: μd ≤ 0 Ha: μd>0 E. H0: μd<0 F. H0: μd=0 Ha: μd ≥ 0 Ha · μd ≠ 0 The test statistic is t=-7.71. (Round to two decimal places as needed.) The P-value is 0.0. (Round to three decimal places as needed.) Since the P-value is greater than the level of significance, statistically significant evidence to reject the claim. reject the null hypothesis. There is

Test the claim below about the mean of the differences for a population of paired data at the level of significance α. Assume the samples are random and dependent, and the populations are normally distributed. Claim: μd ≥ 0 ; α=0.10. Sample statistics: d̄=-2.3, sd=1.3, n=19 Identify the null and alternative hypotheses. Choose the correct answer below. A. H0: μd ≠ 0 B. H0: μd ≥ 0 Ha: μd=0 Ha: μd<0 C. H0: μd>0 D. H0: μd ≤ 0 Ha: μd ≤ 0 Ha: μd>0 E. H0: μd<0 F. H0: μd=0 Ha: μd ≥ 0 Ha · μd ≠ 0 The test statistic is t=-7.71. (Round to two decimal places as needed.) The P-value is 0.0. (Round to three decimal places as needed.) Since the P-value is greater than the level of significance, statistically significant evidence to reject the claim. reject the null hypothesis. There is
Transcript text: 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: $\mu_{\mathrm{d}} \geq 0 ; \alpha=0.10$. Sample statistics: $\overline{\mathrm{d}}=-2.3, \mathrm{~s}_{\mathrm{d}}=1.3, \mathrm{n}=19$ Identify the null and alternative hypotheses. Choose the correct answer below. A. $\mathrm{H}_{0}: \mu_{\mathrm{d}} \neq 0$ B. $H_{0}: \mu_{d} \geq 0$ $\mathrm{H}_{\mathrm{a}}: \mu_{\mathrm{d}}=0$ $\mathrm{H}_{\mathrm{a}}: \mu_{\mathrm{d}}<0$ C. $\mathrm{H}_{0}: \mu_{\mathrm{d}}>0$ D. $\mathrm{H}_{0}: \mu_{\mathrm{d}} \leq 0$ $H_{a}: \mu_{d} \leq 0$ $\mathrm{H}_{\mathrm{a}}: \mu_{\mathrm{d}}>0$ E. $\mathrm{H}_{0}: \mu_{\mathrm{d}}<0$ F. $\mathrm{H}_{0}: \mu_{\mathrm{d}}=0$ $\mathrm{H}_{\mathrm{a}}: \mu_{\mathrm{d}} \geq 0$ $\mathrm{H}_{\mathrm{a}} \cdot \mu_{\mathrm{d}} \neq 0$ The test statistic is $\mathrm{t}=-7.71$. (Round to two decimal places as needed.) The P -value is 0.0 . (Round to three decimal places as needed.) Since the $P$-value is $\square$ greater than the level of significance, statistically significant evidence to reject the claim. reject $\square$ the null hypothesis. There is $\square$
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Solution

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Solution Steps

Step 1: Identify the Hypotheses

We are testing the claim about the mean of the differences for a population of paired data. The hypotheses are defined as follows:

  • Null Hypothesis: \[ H_0: \mu_d \geq 0 \]
  • Alternative Hypothesis: \[ H_a: \mu_d < 0 \]
Step 2: Calculate the Standard Error

The standard error (\(SE\)) is calculated using the formula: \[ SE = \frac{s_d}{\sqrt{n}} = \frac{1.3}{\sqrt{19}} \approx 0.2982 \]

Step 3: Calculate the Test Statistic

The test statistic (\(t\)) is calculated using the formula: \[ t = \frac{\bar{d} - \mu_0}{SE} = \frac{-2.3 - 0}{0.2982} \approx -7.7119 \]

Step 4: Calculate the P-value

For a left-tailed test, the p-value is determined as follows: \[ P = T(z) = 0.0 \]

Step 5: Decision Based on P-value

We compare the p-value with the significance level (\(\alpha = 0.10\)):

  • Since \(P = 0.0 < 0.10\), we reject the null hypothesis.
Conclusion

There is statistically significant evidence to support the claim that the mean difference is less than 0.

Final Answer

The answer is \(\boxed{H_0: \mu_d \geq 0 \text{ and } H_a: \mu_d < 0}\).

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