Question

A household goods manufacturer wants to increase the absorption capacity of a dish washing sponge. Based on past data, the average sponge could absorb 3.5 ounces. After the redesign, the absorption amounts of a sample of sponges were (in ounces): 4.1, 3.7, 3.3, 3.5, 3.8, 3.9, 3.6, 3.8, 4.0, and 3.9. What is the decision rule at the 0.01 level of significance to test if the new design increased the absorption amount of the sponge

Answer 1

`t=(3.76-3.5)/((0.241)/(√(10)))=3.407`

The degrees of freedom are given by:

`df=n-1=10-1=9`

And the p value using the alternative hypothesis is given by:

`p_v =P(t_((9))>3.407)=0.0039`

Since the p value is lower than the significance level provided of 0.10 we have enough evidence to reject the null hypothesis and we can conclude that the new design increased the absorption amount of the sponge

Explanation:

Information provided

4.1, 3.7, 3.3, 3.5, 3.8, 3.9, 3.6, 3.8, 4.0, and 3.9

We can find the sample mean and deviation with the following formulas:

`\bar X = (\sum_(i=1)^n X_i)/(n)`

`s= \sqrt{(\sum_(i=1)^n (X_i -\bar X)^2)/(n-1)}`

`\bar X=3.76` represent the sample mean

`s=0.241` represent the sample standard deviation

`n=10` sample size

`\mu_o =3.5` represent the value to check

`\alpha=0.01` represent the significance level

t would represent the statistic

`p_v` represent the p value

System of hypothesis

We want to test if the new design increased the absorption amount of the sponge (3.5), the system of hypothesis would be:

Null hypothesis:
`\mu \leq 3.5`

Alternative hypothesis:
`\mu > 3.5`

Since we don't know the deviation the statistic is given by:

`t=(\bar X-\mu_o)/((s)/(√(n)))` (1)

Replacing the info given we got:

`t=(3.76-3.5)/((0.241)/(√(10)))=3.407`

The degrees of freedom are given by:

`df=n-1=10-1=9`

And the p value using the alternative hypothesis is given by:

`p_v =P(t_((9))>3.407)=0.0039`

Since the p value is lower than the significance level provided of 0.10 we have enough evidence to reject the null hypothesis and we can conclude that the new design increased the absorption amount of the sponge