# Researchers wanted to compare the effectiveness of a water softener used with a filtering process with a water softener used without filtering. Ninety

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.
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stuth1

Group A ${x}_{1}=2.1$
${s}_{1}=0.7$
${n}_{1}=45$ Group B ${x}_{2}=1.7$
${s}_{2}=0.4$
${n}_{2}=45$ $S.E.=\sqrt{\frac{\left(0.7{\right)}^{2}}{45}+\frac{\left(0.4{\right)}^{2}}{45}}=0.120$
$d={x}_{1}-{x}_{2}=2.1-1.7=0.4$

$\therefore C.I.=d±{t}_{c}{n}_{c}S.E.$
$=0.4±1.65\left(0.120\right)$
$=0.4±0.198$
$\therefore C.I.=\left(0.202,0.598\right)$