A laundry detergent company wants to determine if a new formula of detergent, [tex]$A$[/tex], cleans better than the original formula, [tex]$B$[/tex]. Researchers randomly assign 500 pieces of similarly soiled clothes to the two detergents, putting 250 pieces in each group. After washing the clothes, independent reviewers determine the cleanliness of the clothes on a scale of 1-10, with 10 being the cleanest. The researchers calculate the proportion of clothes in each group that receive a rating of 7 or higher. For detergent [tex]$A$[/tex], 228 pieces of clothing received a 7 or higher. For detergent [tex]$B$[/tex], 210 pieces of clothing received a rating of 7 or higher.

Let [tex]$p_A$[/tex] be the true proportion of clothes receiving a rating of 7 or higher for detergent [tex]$A$[/tex] and [tex]$p_B$[/tex] be the true proportion of clothes receiving a rating of 7 or higher for detergent [tex]$B$[/tex]. Which of the following are the correct hypotheses to test the company's claim?

A. [tex]$H_0: \rho_A - \rho_B = 0 ; H_2: \rho_A - \rho_B \ \textgreater \ 0$[/tex]

B. [tex]$H_B: \rho_A - \rho_B = 0 ; H_2: \rho_A - \rho_B \ \textless \ 0$[/tex]

C. [tex]$H_0: \rho_A - \rho_B = 0 ; H_2: \rho_A - \rho_B \neq 0$[/tex]

D. [tex]$H_B: \rho_A - \rho_B \ \textgreater \ 0 ; H_2: \rho_A - \rho_B \ \textless \ 0$[/tex]

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]