17. + -/1 points MIntroStat9 2.E.086. My Notes Ask Your Teacher Every few years,

Question

17. + -/1 points MIntroStat9 2.E.086. My Notes Ask Your Teacher 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 2011 2013 Score 264 266 271 272 276 277 279 284 286 (a) Make a time plot of the mean scores, by hand. This is just a scatterplot of score against year. There is a slow linear increasing trend Score 300 Score 300 290 290 280 280 270 270 260 260 250 1990 250 Year Year 1990 2000 2010 2020 2000 2010 2020 Score 300 Score 300 290 290

Explanation / Answer

Solution

Part (a)

The scatter plot of the given data is correctly depicted in the ‘top-right’ graph. ANSWER

Note that,

i. in top left graph, (1990, 264) is not correctly plotted

ii. given data on scores is strictly increasing over the years, but in ‘bottom-left’ graph, from 2003 to 2004, there is a dip in the score.

iii. in ‘bottom-right’ graph, last two points (i.e., the right most points) have score greater than 290 while in the given data, the maximum score itself is 286.

Part (b)

Equation of least square line: Back-up Theory

Let X = year and Y = score. Then the linear model is: Y = + X + , where is the error term, which is assumed to be Normally distributed with mean 0 and variance 2.

The least square estimates of and are given by:

cap = a = Ybar – b.Xbar and cap = b = r.(Y/X), where

Ybar and Xbar are means and Y and X are standard deviations of Y and X respectively, r being the correlation coefficient between X and Y, based on the sample.

Excel calculations yield: [reckoning x = 2003 as 0]

Ybar = 275; Xbar = 0; Y = sqrt(50); X = sqrt(60/9); r = 0.992.

Thus, least square estimates are: a = 275, b = 2.717 and the equation of the least square line is: [with 2003 as x = 0] ycap = 275 + 2.717x ANSWER1

The square of r gives the percentage of year-to-year variation in scores explained by the regression equation = 0.9922 = 0.9840 or 98.4% ANSWER 2

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