Question

In: Statistics and Probability

A door-to-door vacuum salesman purports that the number of knick-knacks sitting on the mantle of any...

A door-to-door vacuum salesman purports that the number of knick-knacks sitting on the mantle of any given home is related to the cleanliness of the home rated on a 0-10 scale (10 being most clean). The salesman randomly samples nine clients’ homes. The raw data has been summarized below. X represents the number of knick-knacks and Y is the rating of cleanliness.

Set alpha= 0.05, two tailed.

What are your hypotheses in symbols?

What is the critical value? DO NOT ROUND

ΣX=115 ΣX2=2,153 ΣY= 42 ΣY2=272 ΣXY=326

What is the obtained value?

What is your decision?

Reject the Null

Fail to Reject the Null

What is your conclusion (in words)?

What proportion of variability in home cleanliness can be explained by the variability of the number of knick-knacks on the mantle? keep in decimal format

Solutions

Expert Solution

Let the linear relation between X and Y be

The appropriate hypotheses are,

Degree of freedom = n-2 = 9-2 = 7

Critical value of t at df = 7 and alpha= 0.05 is 2.365

SSxx = ΣX2 - (ΣX)^2 / n = 2153 - 115^2 / 9 = 683.5556

SSyy = ΣY2 - (ΣY)^2 / n = 272 - 42^2 / 9 = 76

SSxy = ΣXY - (ΣX ΣY)/ n = 326 - (115 * 42)/9 = -210.6667

Slope Coefficient, = SSxy / SSxx = -210.6667 / 683.5556 = -0.3081925

Sum of Squared error, SSE = SSyy - SS^2xy / SSxx = 76 - (-210.6667)^2 / 683.5556 = 11.07411

Standard error of regression, se = = 1.257782

Standard error of slope coefficient, SE() = se / = 1.257782 / = 0.04810813

obtained value (t statistic) = / SE() = -0.3081925 / 0.04810813 = -6.406246

Since the obtained t statistic is less than -2.365, it falls in the rejection region and we Reject the Null.

There is sufficient evidence at 0.05 significance level of significant relation between the number of knick-knacks and the rating of cleanliness.

Sum of Squares Regression, SSR = SS^2xy / SSxx = (-210.6667)^2 / 683.5556 = 64.92589

Sum of Squares Total, SST = SSyy = 76

Proportion of variability in home cleanliness can be explained by the variability of the number of knick-knacks on the mantle

= SSR / SST

= 64.92589 / 76

= 0.854288

orchestra answered 1 year ago

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