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

The degrees of freedom are given by:

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

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

Step-by-step 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:

represent the sample mean

represent the sample standard deviation

sample size  

represent the value to check

represent the significance level

t would represent the statistic

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:  

Alternative hypothesis:  

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

 (1)  

Replacing the info given we got:

The degrees of freedom are given by:

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

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

the correct option is:

b. a value of the variable that makes the polynomial equal to zero.

the root or roots of a polynomial are the values of the variable that makes the polynomial equal to 0. we can find the roots of a polynomial making the expression equal to 0 and then, isolating the variable (it it has just one variable).

for example, given the following polynomial:

the root or zero of the polynomial, can be found by making the expression equal to 0, and then, isolating the variable, so, finding the root we have:

therefore, we have that the root of the polynomial is equal to 2.

we must remember that the number of roots that a polynomial can have, will depend on the degree of the polynomial, a first degree polynomial have will have one single root, a second degree polynomial (quadratic expression) may have two roots and so.

hence the correct option is:

b. a value of the variable that makes the polynomial equal to zero.

have a nice day!