To determine whether there is a significant difference in the amount of airborne bacteria between carpeted and uncarpeted rooms, let's follow these steps:
1. State the hypothesis:
- Null hypothesis ([tex]\(H_0\)[/tex]): There is no difference in the mean bacterial concentration between carpeted and uncarpeted rooms.
- [tex]\(H_0: \mu_{\text{carpeted}} = \mu_{\text{uncarpeted}}\)[/tex]
- Alternative hypothesis ([tex]\(H_1\)[/tex]): There is a difference in the mean bacterial concentration between carpeted and uncarpeted rooms.
- [tex]\(H_1: \mu_{\text{carpeted}} \neq \mu_{\text{uncarpeted}}\)[/tex]
2. Collect the given data:
- Mean concentration in carpeted rooms ([tex]\(\bar{x}_{\text{carpeted}}\)[/tex]): 184
- Standard deviation in carpeted rooms ([tex]\(s_{\text{carpeted}}\)[/tex]): 22.0
- Number of carpeted rooms ([tex]\(n_{\text{carpeted}}\)[/tex]): 7
- Mean concentration in uncarpeted rooms ([tex]\(\bar{x}_{\text{uncarpeted}}\)[/tex]): 175
- Standard deviation in uncarpeted rooms ([tex]\(s_{\text{uncarpeted}}\)[/tex]): 16.9
- Number of uncarpeted rooms ([tex]\(n_{\text{uncarpeted}}\)[/tex]): 7
3. Calculate the standard error for the differences in means:
The formula for the standard error (SE) of the difference between two means is:
[tex]\[
SE_{\text{diff}} = \sqrt{\left( \frac{s_{\text{carpeted}}^2}{n_{\text{carpeted}}} \right) + \left( \frac{s_{\text{uncarpeted}}^2}{n_{\text{uncarpeted}}} \right)}
\][/tex]
- Substituting the given values:
[tex]\[
SE_{\text{diff}} = \sqrt{\left( \frac{22.0^2}{7} \right) + \left( \frac{16.9^2}{7} \right)}
\][/tex]
- Result:
[tex]\[
SE_{\text{diff}} = 10.485432070939456
\][/tex]
4. Calculate the t-statistic for testing the difference in means:
The formula for the t-statistic is:
[tex]\[
t = \frac{\bar{x}_{\text{carpeted}} - \bar{x}_{\text{uncarpeted}}}{SE_{\text{diff}}}
\][/tex]
- Substituting the given values:
[tex]\[
t = \frac{184 - 175}{10.485432070939456}
\][/tex]
- Result:
[tex]\[
t = 0.8583337280819971
\][/tex]
5. Determine the numerical value of the t-statistic:
- The calculated t-statistic is approximately 0.858.
Given the multiple choices provided:
- 0.858
- 1.312
- 3.818
- 0.414
The correct numerical value of the two-sample t statistic for this test, based on our calculations, is:
[tex]\[
\boxed{0.858}
\][/tex]
