To solve this hypothesis testing problem, let's go through it step-by-step:
### Step 1: State the Hypotheses
Given the problem, we need to test the claim that the mean of the differences in heights (president's height minus opponent's height) is greater than \(0 \text{ cm}\).
The hypotheses are stated as:
- Null Hypothesis (\(H_0\)): \(\mu_d = 0 \text{ cm}\)
- Alternative Hypothesis (\(H_1\)): \(\mu_d > 0 \text{ cm}\)
### Step 2: Identify the Test Statistic
We are using the sample data:
[tex]\[
\text{Heights of Presidents (cm)}: [193, 170, 166, 180, 198, 180]
\][/tex]
[tex]\[
\text{Heights of Main Opponents (cm)}: [176, 180, 181, 183, 193, 178]
\][/tex]
Firstly, we find the differences between the heights of presidents and their main opponents:
[tex]\[
\text{Differences (cm)} = [193-176, 170-180, 166-181, 180-183, 198-193, 180-178]
\][/tex]
[tex]\[
= [17, -10, -15, -3, 5, 2]
\][/tex]
These differences form our sample.
#### Sample Mean of the Differences (\(\bar{d}\))
The sample mean difference is:
[tex]\[
\bar{d} = \frac{17 + (-10) + (-15) + (-3) + 5 + 2}{6} \approx -0.67
\][/tex]
#### Sample Standard Deviation of the Differences (\(s_d\))
The standard deviation of the differences is approximately:
[tex]\[
s_d \approx 11.40
\][/tex]
#### Sample Size (\(n\))
The number of data points is:
[tex]\[
n = 6
\][/tex]
Now we calculate the test statistic using the formula for the t-statistic:
[tex]\[
t = \frac{\bar{d} - \mu_{d_0}}{s_d / \sqrt{n}}
\][/tex]
Where:
- \(\bar{d}\) is the sample mean of the differences,
- \(\mu_{d_0}\) is the hypothesized population mean difference (which is 0),
- \(s_d\) is the sample standard deviation of the differences,
- \(n\) is the sample size.
By substituting the values, we get:
[tex]\[
t = \frac{-0.67 - 0}{11.40 / \sqrt{6}} \approx -0.14
\][/tex]
### Conclusion
So, the test statistic is:
[tex]\[
t \approx -0.14
\][/tex]
Therefore, the completed solution is:
[tex]\[
t = -0.14
\][/tex]
So, in summary, the test statistic for this hypothesis test is approximately [tex]\( t = -0.14 \)[/tex].
