Question

Acidity of mouthwash. Acid is a primary cause of dental caries (cavities). It is theorized that oral mouthwashes contribute to caries development due to the antiseptic agent oxidizing into acid over time. This theory was tested in the Journal of Dentistry, Oral Medicine, and Dental Education (Vol. 3, 2009). Three bottles of mouthwash, each of a different brand, were randomly selected from a drugstore. Each bottle's pH level (where lower pH levels indicate higher acidity) was measured on the date of purchase and after 30 days. The data are shown in the table on the next page. Conduct an analysis to determine if mouthwash's mean initial pH level differs significantly from the mean pH level after 30 days. Use a = .05 as your level of significance.

Source: Based on K. L. Chunhye and B. C. Schmitz, "Determination of pH, Total Acid, and Total Ethanol in Oral Health Products: Oxidation of Ethanol and Recommendations to Mitigate Its Association with Dental Caries," Journal of Dentistry, Oral Medicine and Dental Education, Vol. 3, No. 1, 2009 (Table 1)

Short answer

$$\begin{gathered} \left(a\right)0.1106 \\ \left(b\right)t=2.92 \end{gathered}$$

Step-by-step solution

Given data

$d=x-4$

To test the hypothesis:

$$\begin{gathered} H_0:{\mu }_d=0 \\ \text{v/s} \\ {H}_1:{\mu }_d>0 \end{gathered}$$

To find the value, " prepare the table as follows:

$$\begin{gathered} \overline{d}\text{=}\frac{\text{Ed}}{\text{n}} \\ =\frac{0.56}{3} \\ =0.1867 \\ \text{where n = 3} \end{gathered}$$

$$\begin{gathered} Sd=\sqrt{\sum {\left({\overline{d}}_i-\overline{d}\right)}^2} \\ =\frac{\sqrt{0.0245}}{2} \\ =0.1106 \end{gathered}$$

Test statistics

$$\begin{gathered} t=\frac{\overline{d}}{Sd/\sqrt{n}} \\ t=\frac{0.1867}{0.1106/\sqrt{3}} \\ t=2.9229 \end{gathered}$$

Critical value: 5% Significance level $t_{n-1},\lambda t_{3-1},0.05=2.92$

$H_0$reject at a 5% significance level.

There is enough data to establish that the average acidity level in mouthwash rose throughout 30 days.

The fundamental premise is that PH levels should be regularly distributed, and observations should be presented as matched pairs.