QUES: Von Neumann made fundamental contributions to mathematical statistics. In 1941, he derived the exact distribution of the ratio of the mean square of successive differences to the sample variance for independent and identically normally distributed variables. This ratio was applied to the residuals from regression models and is commonly known as the Durbin–Watson statistic for testing the null hypothesis that the errors are serially independent against the alternative that they follow a stationary first order autoregression.
what does null hypothesis test against?

ANS: alternative that they follow a stationary first order autoregression

QUES: Football is the most popular sport in Portugal. There are several football competitions ranging from local amateur to world-class professional level. The legendary Eusébio is still a major symbol of Portuguese football history. FIFA World Player of the Year winners Luís Figo and Cristiano Ronaldo who won the FIFA Ballon d'Or for 2013 and 2014, are among the numerous examples of other world-class football players born in Portugal and noted worldwide. Portuguese football managers are also noteworthy, with José Mourinho, André Villas-Boas, Fernando Santos, Carlos Queiroz and Manuel José among the most renowned.
Who are some of the most renowned football managers in Portugal?

ANS: José Mourinho, André Villas-Boas, Fernando Santos, Carlos Queiroz and Manuel José

QUES: At a minimum, HDTV has twice the linear resolution of standard-definition television (SDTV), thus showing greater detail than either analog television or regular DVD. The technical standards for broadcasting HDTV also handle the 16:9 aspect ratio images without using letterboxing or anamorphic stretching, thus increasing the effective image resolution.
What does SDTV stand for?

ANS: standard-definition television

QUES: Lancashire is smaller than its historical extent following a major reform of local government. In 1889, the administrative county of Lancashire was created, covering the historical county except for the county boroughs such as Blackburn, Burnley, Barrow-in-Furness, Preston, Wigan, Liverpool and Manchester. The area served by the Lord-Lieutenant (termed now a ceremonial county) covered the entirety of the administrative county and the county boroughs, and was expanded whenever boroughs annexed areas in neighbouring counties such as Wythenshawe in Manchester south of the River Mersey and historically in Cheshire, and southern Warrington. It did not cover the western part of Todmorden, where the ancient border between Lancashire and Yorkshire passes through the middle of the town.
In what year was Cheshire made a county?

ANS:
unanswerable