To analyze whether college students enjoy playing sports more than watching sports, we can perform a hypothesis test for the difference in means between paired samples.
Step-by-step solution:
### a. Null and Alternative Hypotheses
The hypotheses we will test are:
- [tex]\( H_0: \mu_D = 0 \)[/tex] (There is no difference in mean ratings between playing and watching sports)
- [tex]\( H_1: \mu_D \neq 0 \)[/tex] (There is a difference in mean ratings between playing and watching sports)
Here, [tex]\( \mu_D \)[/tex] represents the mean of the differences between each pair of ratings (Play minus Watch).
### b. Test Statistic
To determine the test statistic, we first calculate the difference [tex]\( D_i \)[/tex] for each student:
[tex]\[
\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|}
\hline
\text{Play} & 2 & 4 & 5 & 5 & 8 & 3 & 2 & 7 & 8 & 10 \\
\hline
\text{Watch} & 1 & 2 & 6 & 5 & 5 & 1 & 1 & 5 & 7 & 8 \\
\hline
\text{Difference} & 1 & 2 & -1 & 0 & 3 & 2 & 1 & 2 & 1 & 2 \\
\hline
\end{array}
\][/tex]
We calculate the mean ([tex]\(\overline{D}\)[/tex]) and standard deviation (s) of these differences.
[tex]\[
\overline{D} = 1.3
\][/tex]
[tex]\[
s = 1.160
\][/tex]
Using these values, we calculate the t-statistic:
[tex]\[
t = \frac{\overline{D}}{s / \sqrt{n}}
\][/tex]
Where [tex]\( n = 10 \)[/tex], the number of differences.
[tex]\[
t = \frac{1.3}{1.160 / \sqrt{10}} = 3.545
\][/tex]
Thus, the test statistic [tex]\( v \)[/tex] is:
[tex]\[
v = 3.545
\][/tex]
### c. P-value
To find the p-value associated with the t-statistic, we use the t-distribution with [tex]\( n-1 = 9 \)[/tex] degrees of freedom.
[tex]\[
p\text{-value} = 0.0063
\][/tex]
### d. Comparison with [tex]\(\alpha\)[/tex]
We compare the p-value to the significance level [tex]\(\alpha = 0.01\)[/tex]:
[tex]\[
0.0063 < 0.01
\][/tex]
### e. Decision
Since the p-value is less than [tex]\(\alpha\)[/tex], we reject the null hypothesis:
[tex]\[
\text{Reject } H_0
\][/tex]
### f. Conclusion
Based on our hypothesis test:
The results are statistically significant at [tex]\(\alpha = 0.01\)[/tex], so there is sufficient evidence to conclude that the population mean rating for playing sports is greater than the population mean rating for watching sports.
