To test the claim that having a pet helps lower stress levels, we will conduct a hypothesis test to compare the means of two independent samples (pet owners and non-owners). We will follow these steps:
1. Define the Hypotheses
- Null Hypothesis ([tex]\(H_0\)[/tex]): There is no difference in stress levels between pet owners and non-owners.
[tex]\[
H_0: \mu_1 = \mu_2
\][/tex]
- Alternative Hypothesis ([tex]\(H_1\)[/tex]): Pet owners have lower stress levels than non-owners.
[tex]\[
H_1: \mu_1 < \mu_2
\][/tex]
2. Given Data
- Mean of pet owners ([tex]\(\bar{X}_1\)[/tex]): 15.25
- Mean of non-owners ([tex]\(\bar{X}_2\)[/tex]): 21.95
- Standard deviation of pet owners ([tex]\(s_1\)[/tex]): 3.00
- Standard deviation of non-owners ([tex]\(s_2\)[/tex]): 4.10
- Sample size of pet owners ([tex]\(n_1\)[/tex]): 27
- Sample size of non-owners ([tex]\(n_2\)[/tex]): 20
- Significance level ([tex]\(\alpha\)[/tex]): 0.05
3. Calculate the Pooled Standard Deviation
The pooled standard deviation [tex]\( s_p \)[/tex] is calculated as follows:
[tex]\[
s_p = \sqrt{\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}}
\][/tex]
Plugging in the values, we get:
[tex]\[
s_p = \sqrt{\frac{3.00^2}{27} + \frac{4.10^2}{20}} \approx 1.083
\][/tex]
4. Calculate the t-Statistic
The t-statistic for two independent samples is calculated using:
[tex]\[
t = \frac{\bar{X}_1 - \bar{X}_2}{s_p}
\][/tex]
Substituting the means and pooled standard deviation:
[tex]\[
t = \frac{15.25 - 21.95}{1.083} \approx -6.184
\][/tex]
5. Calculate the Degrees of Freedom
The degrees of freedom for a t-test with unequal variances (Welch’s t-test) is given by:
[tex]\[
df = \frac{\left(\frac{s_1^2}{n_1} + \frac{s_2^2}{n_2}\right)^2}{\frac{\left(\frac{s_1^2}{n_1}\right)^2}{n_1 - 1} + \frac{\left(\frac{s_2^2}{n_2}\right)^2}{n_2 - 1}}
\][/tex]
Substituting the given values, we get:
[tex]\[
df \approx 33.238
\][/tex]
6. Determine the Critical t-Value
For a one-tailed t-test at the 0.05 significance level with approximately 33 degrees of freedom, the critical t-value can be found using t-distribution tables or software:
[tex]\[
t_{\text{critical}} \approx 1.692
\][/tex]
7. Decision Rule
We reject the null hypothesis if the calculated t-statistic is less than the negative of the critical t-value:
[tex]\[
t < -t_{\text{critical}}
\][/tex]
8. Make a Decision
- Calculated t-statistic: [tex]\( -6.184 \)[/tex]
- Critical t-value: [tex]\( 1.692 \)[/tex]
Since [tex]\( -6.184 < -1.692 \)[/tex], we reject the null hypothesis.
9. Conclusion
At the 0.05 significance level, there is sufficient evidence to support the claim that pet owners have a lower average stress level than non-owners.
