Researchers wanted to compare the effectiveness of a water softener used with a filtering process with a water softener used without filtering. Ninety locations were randomly divided into two groups of equal size. Group A locations used a water softener and the filtering process, while group B used only the water softener. At the end of three months, a water sample was tested at each location for its level of softness. (Water softness was measured on a scale of 1 to 5, with 5 being the softest water.) The results were as follows: Group A (water softener and filtering) x_1 = 2.1 s_1 = 0.7 Group B (water softener only) x_2 = 1.7 s_2 = 0.4 Determine, at the 90% confidence level, whether there is a difference between the two types of treatments.

Researchers wanted to compare the effectiveness of a water softener used with a filtering process with a water softener used without filtering. Ninety locations were randomly divided into two groups of equal size. Group A locations used a water softener and the filtering process, while group B used only the water softener. At the end of three months, a water sample was tested at each location for its level of softness. (Water softness was measured on a scale of 1 to 5, with 5 being the softest water.) The results were as follows: Group A (water softener and filtering) x_1 = 2.1 s_1 = 0.7 Group B (water softener only) x_2 = 1.7 s_2 = 0.4 Determine, at the 90% confidence level, whether there is a difference between the two types of treatments.

Question
Comparing two groups
asked 2020-10-23
Researchers wanted to compare the effectiveness of a water softener used with a filtering process with a water softener used without filtering. Ninety locations were randomly divided into two groups of equal size. Group A locations used a water softener and the filtering process, while group B used only the water softener. At the end of three months, a water sample was tested at each location for its level of softness. (Water softness was measured on a scale of 1 to 5, with 5 being the softest water.) The results were as follows: Group A (water softener and filtering) \(\displaystyle{x}_{{1}}={2.1}\)
\(\displaystyle{s}_{{1}}={0.7}\) Group B (water softener only) \(\displaystyle{x}_{{2}}={1.7}\)
\(\displaystyle{s}_{{2}}={0.4}\) Determine, at the 90% confidence level, whether there is a difference between the two types of treatments.

Answers (1)

2020-10-24

Group A \(\displaystyle{x}_{{1}}={2.1}\)
\(\displaystyle{s}_{{1}}={0.7}\)
\(\displaystyle{n}_{{1}}={45}\) Group B \(\displaystyle{x}_{{2}}={1.7}\)
\(\displaystyle{s}_{{2}}={0.4}\)
\(\displaystyle{n}_{{2}}={45}\) \(\displaystyle{S}.{E}.=\sqrt{{{\mathfrak{{a}}}{a}{c}{\left\lbrace{\left({0.7}\right)}^{{2}}\right\rbrace}{\left\lbrace{45}\right\rbrace}+{\frac{{{\left({0.4}\right)}^{{2}}}}{{{45}}}}}}={0.120}\)
\(\displaystyle{d}={x}_{{1}}-{x}_{{2}}={2.1}-{1.7}={0.4}\)
\(\displaystyle{t}_{{c}}{n}_{{c}}{\left({a}{t}{0.10}{l}{o}{s}{\quad\text{and}\quad}{90}-{2}{d}{f}\right)}={1.65}\)
\(\displaystyle\therefore{C}.{I}.={d}\pm{t}_{{c}}{n}_{{c}}{S}.{E}.\)
\(\displaystyle={0.4}\pm{1.65}{\left({0.120}\right)}\)
\(\displaystyle={0.4}\pm{0.198}\)
\(\displaystyle\therefore{C}.{I}.={\left({0.202},{0.598}\right)}\)

0

Relevant Questions

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