Questions: Assume that the differences are normally distributed. Complete parts (a) through (d) below. Observation 1 2 3 4 5 6 7 8 Xi 41.2 44.0 50.8 48.0 52.0 49.4 51.2 50.0 Yi 45.5 45.1 52.1 53.4 53.4 51.8 53.4 50.6 (a) Determine di = Xi - Yi for each pair of data. Observation 1 2 3 4 5 6 7 8 di -4.3 -1.1 -1.3 - 5.4 -1.4 -2.4 -2.2 -0.6 (Type integers or decimals.) (b) Compute d̄ and sd. d̄ = -2.338 (Round to three decimal places as needed.) sd = 1.68 (Round to three decimal places as needed.) (c) Test if μd < 0 at the α = 0.05 level of significance. What are the correct null and alternative hypotheses? A. H0: μd = 0 B. H0: μd < 0 H1: μd < 0 H1: μd = 0 C. H0: μd > 0 D. H0: μd < 0 H1: μd < 0

Assume that the differences are normally distributed. Complete parts (a) through (d) below.
Observation 1 2 3 4 5 6 7 8
Xi 41.2 44.0 50.8 48.0 52.0 49.4 51.2 50.0
Yi 45.5 45.1 52.1 53.4 53.4 51.8 53.4 50.6
(a) Determine di = Xi - Yi for each pair of data.
Observation 1 2 3 4 5 6 7 8
di -4.3 -1.1 -1.3 - 5.4 -1.4 -2.4 -2.2 -0.6
(Type integers or decimals.)
(b) Compute d̄ and sd.
d̄ = -2.338 (Round to three decimal places as needed.)
sd = 1.68 (Round to three decimal places as needed.)
(c) Test if μd < 0 at the α = 0.05 level of significance.

What are the correct null and alternative hypotheses?
A. H0: μd = 0
B. H0: μd < 0 H1: μd < 0 H1: μd = 0
C. H0: μd > 0
D. H0: μd < 0 H1: μd < 0

Transcript text: Assume that the differences are normally distributed. Complete parts (a) through (d) below. Observation 1 2 3 4 5 6 7 8 $\mathbf{X}_{\mathbf{i}}$ 41.2 44.0 50.8 48.0 52.0 49.4 51.2 50.0 $\mathbf{Y}_{\mathbf{i}}$ 45.5 45.1 52.1 53.4 53.4 51.8 53.4 50.6 (a) Determine $d_{i}=X_{i}-Y_{i}$ for each pair of data. Observation 1 2 3 4 5 6 7 8 $d_{i}$ -4.3 -1.1 -1.3 - 5.4 -1.4 -2.4 -2.2 -0.6 (Type integers or decimals.) (b) Compute $\overline{\mathrm{d}}$ and $\mathrm{s}_{\mathrm{d}}$. $\overline{\mathrm{d}}=-2.338$ (Round to three decimal places as needed.) $\mathrm{s}_{\mathrm{d}}=1.68$ (Round to three decimal places as needed.) (c) Test if $\mu_{d}<0$ at the $\alpha=0.05$ level of significance. What are the correct null and alternative hypotheses? A. $H_{0}: \mu_{d}=0$ B. $H_{0}: \mu_{d}<0$ $H_{1}: \mu_{d}<0$ $H_{1}: \mu_{d}=0$ C. $\mathrm{H}_{0}: \mu_{\mathrm{d}}>0$ D. $H_{0}: \mu_{d}<0$ $H_{1}: \mu_{d}<0$

Solution

Solution Steps

Step 1: Calculate Differences

The differences $ d_i = X_i - Y_i $ for each observation are calculated as follows:

\[ d = [-4.3, -1.1, -1.3, -5.4, -1.4, -2.4, -2.2, -0.6] \]

Step 2: Compute Mean and Standard Deviation

The mean of the differences $ \overline{d} $ is calculated using the formula:

\[ \overline{d} = \frac{\sum_{i=1}^N d_i}{N} = \frac{-18.7}{8} = -2.337 \]

The standard deviation $ s_d $ is calculated using the formula for sample standard deviation:

\[ s_d = \sqrt{\frac{\sum (d_i - \overline{d})^2}{n-1}} = 1.68 \]

Step 3: Hypothesis Testing

We will test the hypothesis $ H_0: \mu_d = 0 $ against the alternative hypothesis $ H_1: \mu_d < 0 $ at the significance level $ \alpha = 0.05 $.

Calculate Standard Error (SE): \[ SE = \frac{s_d}{\sqrt{n}} = \frac{1.68}{\sqrt{8}} \approx 0.594 \]
Calculate Test Statistic: \[ t = \frac{\overline{d} - \mu_0}{SE} = \frac{-2.337 - 0}{0.594} \approx -3.9345 \]
Calculate P-value: For a left-tailed test, the P-value is given by: \[ P = T(z) \approx 0.0028 \]

Final Answer

The calculated test statistic is approximately $-3.9345$ and the P-value is $0.0028$. Since the P-value $0.0028$ is less than the significance level $0.05$, we reject the null hypothesis.

Thus, the answer is: \[ \boxed{H_0: \mu_d = 0} \]

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