Mathematics, 06.04.2020 23:10 cristinavalencia03

Every few years, the National Assessment of Educational Progress asks a national sample of eighth-graders to perform the same math tasks. The goal is to get an honest picture of progress in math. Suppose these are the last few national mean scores, on a scale of 0 to 500.
Year 1990 1992 1996 2000 2003 2005 2008
Score 263 268 271 272 276 277 279
(a) Find the regression line of mean score on time step-by-step. First calculate the mean and standard deviation of each variable and their correlation (use a calculator with these functions). Then find the equation of the least-squares line from these
(b) What percent of the year-to-year variation in scores is explained by the linear trend?

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Answer from: giany

a)X: 0 2 6 10 13 15 18

Y:263 268 271 272 276 277 279

X represent the number of years since 1990

n=7 $\sum x = 64, \sum y = 1906, \sum xy= 17647, \sum x^2 =858, \sum y^2 =519164$

And in order to calculate the correlation coefficient we can use this formula:

$r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^2 -(\sum x)^2][n\sum y^2 -(\sum y)^2]}}$

$r=\frac{7(17647)-(64)(1906)}{\sqrt{[7(858) -(64)^2][7(519164) -(1906)^2]}}=0.97599$

$\bar X = \frac{\sum_{i=1^n X_i}}{n} = 9.14286$

$\bar Y = \frac{\sum_{i=1^n Y_i}}{n} = 272.286$

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=858-\frac{64^2}{7}=272.857$

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}=17647-\frac{64*1906}{7}=220.714$

And the slope would be:

$m=\frac{220.714}{272.857}=0.809$

Now we can find the means for x and y like this:

And we can find the intercept using this:

$b=\bar y -m \bar x=272.286-(0.809*9.143)=264.889$

So the line would be given by:

y=0.809 x +264.889

b) For this case the percent of variation in scores is explained by the linear trend is given by the determination coefficient r^2 and we got:

r^2 =0.976^2 = 0.9526

So then we can say that the percent of variation explained is approximately 95.26%

Step-by-step explanation:

Pearson correlation coefficient(r), "measures a linear dependence between two variables (x and y). Its a parametric correlation test because it depends to the distribution of the data. And other assumption is that the variables x and y needs to follow a normal distribution".

Solution to the problem

Part a

We assume the following data:

X: 0 2 6 10 13 15 18

Y:263 268 271 272 276 277 279

X represent the number of years since 1990

n=7 $\sum x = 64, \sum y = 1906, \sum xy= 17647, \sum x^2 =858, \sum y^2 =519164$

And in order to calculate the correlation coefficient we can use this formula:

$r=\frac{n(\sum xy)-(\sum x)(\sum y)}{\sqrt{[n\sum x^2 -(\sum x)^2][n\sum y^2 -(\sum y)^2]}}$

$r=\frac{7(17647)-(64)(1906)}{\sqrt{[7(858) -(64)^2][7(519164) -(1906)^2]}}=0.97599$

So then the correlation coefficient would be r =0.976

The mean for X on this case is given by:

$\bar X = \frac{\sum_{i=1^n X_i}}{n} = 9.14286$

$\bar Y = \frac{\sum_{i=1^n Y_i}}{n} = 272.286$

For this case we need to calculate the slope with the following formula:

$m=\frac{S_{xy}}{S_{xx}}$

Where:

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}{n}$

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}$

So we can find the sums like this:

$\sum_{i=1}^n x_i = 64$

$\sum_{i=1}^n y_i =1906$

$\sum_{i=1}^n x^2_i =858$

$\sum_{i=1}^n y^2_i =519164$

$\sum_{i=1}^n x_i y_i =17647$

With these we can find the sums:

$S_{xx}=\sum_{i=1}^n x^2_i -\frac{(\sum_{i=1}^n x_i)^2}{n}=858-\frac{64^2}{7}=272.857$

$S_{xy}=\sum_{i=1}^n x_i y_i -\frac{(\sum_{i=1}^n x_i)(\sum_{i=1}^n y_i)}=17647-\frac{64*1906}{7}=220.714$

And the slope would be:

$m=\frac{220.714}{272.857}=0.809$

Now we can find the means for x and y like this:

And we can find the intercept using this:

$b=\bar y -m \bar x=272.286-(0.809*9.143)=264.889$

So the line would be given by:

y=0.809 x +264.889

Part b

For this case the percent of variation in scores is explained by the linear trend is given by the determination coefficient r^2 and we got:

r^2 =0.976^2 = 0.9526

So then we can say that the percent of variation explained is approximately 95.26%

Answer from: Quest

∠lno ≅ ∠lnm

∠oln ≅ ∠mln

step-by-step explanation:

given that triangle lno is congruent to triangle lnm.

we have to fill up the blanks in the given question.

according to above order of congruency kept in mind we can say that

∠lno ≅ ∠lnm

∠oln ≅ ∠mln

ln ≅ ln due to common in both.

hence, by sas congruency,

the side angle side rule states that if two sides and the included angle of one triangle are congruent to another triangle, then these two triangles are congruent.

Answer from: Quest

lol 2

step-by-step explanation:

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