Question: Part D:
Jeremy Phobe hopes to determine the effectiveness of three different types of soaps on reducing the number of bacteria on a person’s hands. He wants to know if one of the soaps kills bacteria at a higher rate than the others. Phobe invites 24 people to an amusement park for the day. After five hours of fun, Phobe divides the participants into three groups of eight. Each group tries a different soap (Mr. Meyer™ vs. Branch Basic™ vs. Aesop™). Phobe then measures the number of bacteria on each person’s hands (data below). At $\alpha = .05$, test the claim that there is no difference among the means.
Number of Bacteria per cm²
Group 1
Group 2
Group 3
Mr. Meyer
Basic Branch
Aesop
Person
$X_1$
Person
$X_2$
Person
$X_3$
1
2157
9
1602
17
1996
2
1548
10
1890
18
2042
3
1591
11
2031
19
2082
4
1017
12
1042
20
1871
5
1031
13
1794
21
2168
6
1212
14
2178
22
1738
7
1223
15
1349
23
2015
8
1031
16
2124
24
1230
Step 1: In terms of the problem, state the Null Hypothesis & Alternative Hypothesis
1. **State the problem:**
Jeremy Phobe wants to test if there is a difference in the effectiveness of three soaps (Mr. Meyer, Branch Basic, Aesop) in reducing bacteria on hands by comparing the mean bacteria counts after use.
2. **Hypotheses:**
- Null Hypothesis ($H_0$): The mean number of bacteria after using each soap is the same.
$$H_0: \mu_1 = \mu_2 = \mu_3$$
- Alternative Hypothesis ($H_a$): At least one soap has a different mean bacteria count.
$$H_a: \text{At least one } \mu_i \neq \mu_j \text{ for some } i \neq j$$
3. **Significance level:**
$$\alpha = 0.05$$
This means we will reject $H_0$ if the evidence shows a statistically significant difference among the means at the 5% level.
4. **Test to use:**
One-way ANOVA (Analysis of Variance) since we compare means of three independent groups.
Final answer for Step 1:
- $H_0: \mu_1 = \mu_2 = \mu_3$
- $H_a: \text{At least one } \mu_i \neq \mu_j$