The College Board SAT college entrance exam consists of three parts: math, writing and critical reading (The World Almanac 2012). Sample data showing the math and writing scores for a sample of twelve students who took the SAT follow.

Student Math Writing Student Math Writing
1 540 474 7 480 430
2 432 380 8 499 459
3 528 463 9 610 615
4 574 612 10 572 541
5 448 420 11 390 335
6 502 526 12 593 613

Use α= 0.05 level of significance and test for a difference between the population mean for the math scores and the population mean for the writing scores. Enter negative values as negative numbers. What is the test statistic?

Yes, there is a difference between the population mean for the math scores and the population mean for the writing scores.

Test Statistics =   follows .

Step-by-step explanation:

We are provided with the sample data showing the math and writing scores for a sample of twelve students who took the SAT ;

Let A = Math Scores ,B = Writing Scores  and D = difference between both

So, = Population mean for the math scores

        = Population mean for the writing scores

 Let = Difference between the population mean for the math scores and the population mean for the writing scores.

             Null Hypothesis, :     or   = 0

     Alternate Hypothesis, :      or   0

Hence, Test Statistics used here will be;

            follows    where, Dbar = Bbar - Abar

                                                                =

                                                               n = 12

Student        Math scores (A)          Writing scores (B)         D = B - A

     1                      540                            474                                   -66

     2                      432                           380                                    -52  

     3                      528                           463                                    -65

     4                       574                          612                                      38

     5                       448                          420                                    -28

     6                       502                          526                                    24

     7                       480                           430                                     -50

     8                       499                           459                                   -40

     9                       610                            615                                       5

     10                      572                           541                                      -31

     11                       390                           335                                     -55

     12                      593                           613                                       20  

Now Dbar = Bbar - Abar = 489 - 514 = -25

 Bbar = =  = 489

 Abar =   = = 514

 ∑ = 22600     and   = = = 37.05

So, Test statistics =   follows

                            = follows   = -2.34

Now at 5% level of significance our t table is giving critical values of -2.201 and 2.201 for two tail test. Since our test statistics doesn't fall between these two values as it is less than -2.201 so we have sufficient evidence to reject null hypothesis as our test statistics fall in the rejection region .

Therefore, we conclude that there is a difference between the population mean for the math scores and the population mean for the writing scores.

answer: a video game has three difficulty levels: easy, medium, and expert. the game's scoring works in such a way that a player has to score a fixed number of points to advance to the next level, and then the scoring starts over. the total score needed to clear all three difficulty levels is 7,500. two times the number of points needed to clear the easy level is 500 more than the number of points needed to clear the medium level. twice the number of points needed to clear the medium difficulty level is 1,500 more than the number of points needed to clear the expert level. if x, y, and z represent the easy, medium, and expert levels, respectively, and a is the coefficient matrix of the system of equations modeling this situation, identify the inverse matrix, a-1, and the solution matrix, x, of the system of equations.

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