Question

Consumer Reports evaluated and rated 46 brands of toothpaste. One attribute examined in the study was whether or not a toothpastebrand carries an American Dental Association (ADA) seal verifying effective decay prevention. The data for the 46 brands (coded 1 = ADA seal, 0 = no ADA seal) are listed here.

a. Give the null and alternative hypotheses for testing whether the true proportion of toothpaste brands with the ADA seal verifying effective decay prevention is less than .5.

b. Locate the p-value on the Minitab printout below

c. Make the appropriate conclusion using$\mathit{\alpha }\mathbf{=}\mathbf{.}\mathbf{10}$

Short answer

a. The hypotheses are$H_0:p=0.50$and$H_a:p<0.50$.

b.The p-value is 0.188

c.At a 10% significance level, we do not have sufficient evidence to conclude that the true proportion of toothpaste brands with the ADA seal verifying effective decay prevention is less than .5.

Step-by-step solution

Given information

As per the study, out of 46 brands of toothpaste, 20 of the toothpaste brand carry an American Dental Association (ADA) seal verifying effective decay prevention.

That is

The size of the sample$n=46$

The sample proportion is

$$\begin{gathered} \hat{p}=\frac{20}{46} \\ =0.435 \end{gathered}$$

Where$\hat{p}$ is the sample proportion of toothpaste brands with the ADA seal verifying effective decay prevention.

Setting up the hypotheses

We have to test the true proportion of toothpaste brands with the ADA seal.

The null and alternative hypotheses are given as

$H_0:p=0.5$

The true proportion of toothpaste brands with the ADA seal is .5.

And

$H_a:p<0.5$

That is, the true proportion of toothpaste brands with the ADA seal is less than .5.

Calculating the p-value

As per the Minitab printout, the p-value is 0.188

We can see that

$p-value=0.188>0.10$

That is, the p-value is greater than the significance level.

Hence, we failed to reject the null hypothesis.

Conclusion

At a 10% significance level, we do not have sufficient evidence to conclude that the true proportion of toothpaste brands with the ADA seal verifying effective decay prevention is less than .5.