N experiment was conducted to test the effect of a new stain remover. Twenty stains were treated with stain remover A and stain remover B for a week, then the number of stains removed was measured. A significance test was conducted at the α = 0.10 level for the difference in the means of the number of stains removed between stain remover A and stain remover B. The test resulted in t = 1.37 and p = 0.17. If the alternative hypothesis in question is Ha: μA − μB ≠ 0, where μA equals the mean number of stains removed with stain remover A and μB equals the mean number of stains removed with stain remover B, what conclusion can be drawn?

So the p value is a very high value and using the significance level given[tex]\alpha=0.1[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and there is enough evidence to conclude that we have a difference between the two means at 10% of significance.

Step-by-step explanation:

Data given and notation

[tex]\bar X_{A}[/tex] represent the mean for the sample for A

[tex]\bar X_{B}[/tex] represent the mean for the sample B

[tex]s_{A}[/tex] represent the sample standard deviation for the sample A

[tex]s_{B}=[/tex] represent the sample standard deviation for the sample B

[tex]n_{A}=20[/tex] sample size for the group poisoned

[tex]n_{B}=20[/tex] sample size for the group unpoisoned

t would represent the statistic (variable of interest)

Concepts and formulas to use

We need to conduct a hypothesis in order to check if the two means are equal or not , the system of hypothesis would be:

Null hypothesis:[tex]\mu_{A} - \mu_{B} =0[/tex]

Alternative hypothesis:[tex]\mu_{A} - \mu_{B} \neq 0[/tex]

For this case is better apply a t test to compare means, and the statistic is given by:

[tex]t=\frac{\bar X_{A}-\bar X_{B}}{\sqrt{\frac{s^2_{A}}{n_{A}}+\frac{s^2_{B}}{n_{B}}}}[/tex] (1)

t-test: Is used to compare group means. Is one of the most common tests and is used to determine whether the means of two groups are equal to each other.

Calculate the statistic

For this case the statistic calculated is [tex] t= 1.37[/tex]

Statistical decision

The significance level is 0.1 [tex]\alpha=0.1[/tex], but we can calculate the p value for this test. The first step is calculate the degrees of freedom, on this case:

[tex]df=n_{A}+n_{B}-2[/tex]

The p value on this case is given

[tex]p_v =0.17[/tex]

So the p value is a very high value and using the significance level given[tex]\alpha=0,1[/tex] we see that [tex]p_v>\alpha[/tex] so we can conclude that we have enough evidence to FAIL to reject the null hypothesis, and there is enough evidence to conclude that we have a difference between the two means at 10% of significance.