51. Math and gender 2009 Below are mean PISA (Programme for International Student Assessment) math scores for samples of 15-year-old male and female students from a number of randomly selected schools in each of various OECD and other industrialized nations (4000–10 000 per country).

Country Male mean Female mean Israel 451 443 Italy 490 475 Japan 534 524 Korea 548 544 Luxembourg 499 479 Mexico 425 412 Netherlands 534 517 New Zealand 523 515 Norway 500 495 Poland 497 493 Portugal 493 481 Slovak Republic 498 495 Slovenia 502 501 Spain 493 474 Sweden 493 495 Switzerland 544 524 Turkey 451 440 United Kingdom 503 482 United States 497 477 OECD total 496 481 OECD average 501 490 Partners Albania 372 383 Argentina 394 383 Azerbaijan 435 427 Brazil 394 379 Bulgaria 426 430 Colombia 398 366 Croatia 465 454 Dubai (UAE) 454 451 Hong Kong-China 561 547 Indonesia 371 372 Jordan 386 387 Kazakhstan 405 405 Kyrgyzstan 328 334 Latvia 483 481 Liechtenstein 547 523 Lithuania 474 480 Macao-China 531 520 Montenegro 408 396 Panama 362 357 Peru 374 356 Qatar 366 371 Romania 429 425 Russian Federation 469 467 Serbia 448 437 Shanghai-China 599 601 Singapore 565 559 Chinese Taipei 546 541 Thailand 421 417 Trinidad and Tobago 410 418 Tunisia 378 366 Uruguay 433 421

a) Plot male mean score versus female mean score, and superimpose the least-squares regression line on the plot (for prediction of male means from female means).

b) Plot residuals from the regression versus female means or versus predicted (fitted) values. What do you learn?

c) Plot a histogram and Normal probability plot of the residuals. Are they approximately Normally distributed?

d) Plot residuals versus row number. Are the OECD countries less variable than the non-OECD countries?

e) Which two countries have the greatest leverage on the fit? Explain in simple words what this means.

f) The slope does not differ much from 1.0. What would a slope of 1.0 indicate about the nature of the relationship? If we fitted a model with the slope fixed at 1.0, what prediction equation would you expect to get? (Hint: Find the mean scores for males and females.) g) If we examined only those countries with means over 500 for both sexes, what would happen to the correlation? Why?

Country Male mean Female mean OECD Australia 519 509 Austria 506 486 Belgium 526 504. Canada 533 521 Chile 431 410 Czech Republic 495 490 Denmark 511 495 Estonia 516 508 Finland 542 539 France 505 489 Germany 520 505 Greece 473 459 Hungary 496 484 Iceland 508 505 Ireland 491 483