To start with, let's list the data provided:
- Number of pieces of clothing tested with detergent [tex]$A$[/tex]: \( 250 \)
- Number of pieces of clothing tested with detergent [tex]$B$[/tex]: \( 250 \)
- Number of pieces of clothing receiving a rating of 7 or higher with detergent [tex]$A$[/tex]: \( 228 \)
- Number of pieces of clothing receiving a rating of 7 or higher with detergent [tex]$B$[/tex]: \( 210 \)
Next, we will calculate the sample proportions:
- Sample proportion of clothes receiving a rating of 7 or higher with detergent [tex]$A$[/tex] (denoted as \( p_A \)):
[tex]\[
p_A = \frac{228}{250} = 0.912
\][/tex]
- Sample proportion of clothes receiving a rating of 7 or higher with detergent [tex]$B$[/tex] (denoted as \( p_B \)):
[tex]\[
p_B = \frac{210}{250} = 0.84
\][/tex]
Now, we need to determine the correct hypotheses to test the company's claim that the new formula (detergent [tex]$A$[/tex]) cleans better than the original formula (detergent [tex]$B$[/tex]):
1. Null Hypothesis (\(H_0\)): This is typically a statement of no effect or no difference. In this context, the null hypothesis is that there is no difference in the proportions of clothes receiving a rating of 7 or higher between the two detergents. Mathematically, we express this as:
[tex]\[
H_0: p_A - p_B = 0
\][/tex]
2. Alternative Hypothesis (\(H_a\)): This is what we are trying to find evidence for. In this context, the alternative hypothesis is that the proportion of clothes receiving a rating of 7 or higher is greater for detergent [tex]$A$[/tex] than for detergent [tex]$B$[/tex]. Mathematically, we express this as:
[tex]\[
H_a: p_A - p_B > 0
\][/tex]
Comparing this to the given options:
1. \( H_0: p_A - p_B = 0 \) and \( H_a: p_A - p_B > 0 \)
2. \( H_0: p_A - p_B = 0 \) and \( H_a: p_A - p_B < 0 \)
3. \( H_0: p_A - p_B = 0 \) and \( H_a: p_A - p_B \neq 0 \)
4. \( H_0: p_A - p_B > 0 \) and \( H_a: p_A - p_B < 0 \)
The correct hypotheses according to the context of this problem are:
[tex]\[
H_0: p_A - p_B = 0, \quad H_a: p_A - p_B > 0
\][/tex]
Thus, the correct answer is the first option:
[tex]\[ H_0: p_A - p_B = 0, \quad H_a: p_A - p_B > 0 \][/tex]
