Question

A consumer agency is interested in examining whether there is a difference in two common sealant products used to waterproof residential backyard decks. With cooperation of several builders in the area, they randomly assign 38 newly constructed decks to be treated with Very Clear deck sealant and another 37 newly constructed decks to be treated with Sure Seal deck sealant. After one year of being exposed to similar weather conditions, the decks are rated on a scale of 1 to 100. The mean rating for the decks treated with Very Clear is 89.2 with a standard deviation of 3.1. The mean rating for the decks treated with Sure Seal is 92.4 with a standard deviation of 3.8.
Which of the following represents the 90 percent confidence interval to estimate the difference (Very Clear minus Sure Seal) in mean ratings for the two deck sealants?

a.(89.2−92.4)±1.9603.1238+3.8237−−−−−−−−√
b.(89.2−92.4)±1.6883.1238+3.8237−−−−−−−−√
c.(89.2−92.4)±1.6453.1238+3.8237−−−−−−−−√
d.(89.2−92.4)±1.6883.138+3.837−−−−−−−√
e.(89.2−92.4)±1.645(3.138√+3.837√)

Answer 1

Answer:

$CI = (89.2 - 92.4) \pm 1.6883 \sqrt{\frac{3.1^{2} }{38} +\frac{3.8^{2} }{37} }$

Step-by-step explanation:

Data of newly constructed decks treated with very clear deck sealants:

n₁ = 38, $\bar {x_{1} } = 89.2$,  S₁ = 3.1

Data of newly constructed decks treated with sure seal deck sealants:

n₁ = 37, $\bar {x_{2} } = 92.4$, S₂ = 3.8

First calculate the degree of freedom

$df = \frac{[\frac{s_{1} ^{2} }{n_{1} } + \frac{s_{2} ^{2} }{n_{2} } ]^{2} }{\frac{(\frac{s_{1} ^{2} }{n_{1} } )^{2}}{n_{1}-1 } + \frac{(\frac{s_{2} ^{2} }{n_{2} } )^{2}}{n_{2}-1 }}$

$df = \frac{[\frac{3.1^{2} }{38} + \frac{3.8^{2} }{37}]^2}{\frac{(\frac{3.1^{2} }{38})^{2} }{37} + \frac{(\frac{3.8^{2} }{37})^{2} }{36}}$

df = 69.4

df = 70, α = 1 - 0.9 = 0.1

$t_{\frac{\alpha}{2} } = t_{0.05 }= 1.688$

$CI = \bar{X_{1} } - \bar{X_{2} } + t_{\frac{\alpha}{2} } \sqrt{\frac{S_{1} ^{2} }{n_{1} +} \frac{S_{2} ^{2} }{n_{2} } }$

$CI = (89.2 - 92.4) \pm 1.6883 \sqrt{\frac{3.1^{2} }{38} +\frac{3.8^{2} }{37} }$

Answer 2

You can use formula for finding CI here as shown below.

The needed confidence interval is given by:

$CI = (89.2 - 92.4) \pm1.688\sqrt{\dfrac{3.1^2}{38} + \dfrac{3.8^2}{37}}$

Given that:

• For newly constructed decks treated with very clear deck sealants:  $n_1 = 38, \: \overline{x}_1 = 89.2, \: s_1 = 3.1$
• For newly constructed decks treated with sure seal deck sealant:$n_2 = 37, \: \overline{x}_2 = 92.4, \: s_2 = 3.18$

Finding Confidence Interval:

The formula for Confidence Interval to estimate the difference in mean ratings for the two deck sealants is given by:

$CI = \overline{x}_1 - \overline{x}_2 \pm t_{\frac{\alpha}{2}}\sqrt{\dfrac{s^2_1}{n_1} + \dfrac{s^2_2}{n_2}$

The degree of freedom can be calculated as:

$df = \dfrac{(\dfrac{s_1^2}{n_1} + \dfrac{s_1^2}{n_1})^2}{\dfrac{(\dfrac{s_1^2}{n_1})^2}{n_1 - 1} + \dfrac{(\dfrac{s_2^2}{n_2})^2}{n_2 - 1}}$

df = 69.4

Since 90% confidence interval is there, thus $\alpha = 1 - 0.9 = 0.1$

Thus, $t_{\frac{\alpha}{2}} = t_{0.05} = 1.6883$

Thus, putting values in the formula of Confidence Interval:

$CI = (89.2 - 92.4) \pm1.688\sqrt{\dfrac{3.1^2}{38} + \dfrac{3.8^2}{37}}$

Learn more about confidence interval here:

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