Question

A variable contains five categories. It is expected that data are uniformly distributed across these five categories. To test this, a sample of observed data is gathered on this variable resulting in frequencies of 27, 30 29, 21, and 24. Using a = .01, the appropriate decision is - O reject the null hypothesis that the observed distribution is uniform reject the null hypothesis that the observed distribution is not uniform O do not reject the null hypothesis that the observed distribution is uniform O do not reject the null hypothesis that the observed distribution is not uniform O do nothing

Answer 1

n here we are testing Ho : data are uniformly distributed among the five categories. H1 : data are not uniformly distributed among the five categories. test statistic is x = (fi – ei)?/e; ; i = 1, 2, 3, ..., n=5, fi is observed frequency and e; is expected frequency. Σfi ej = 5 i=1

2 ei 26 : 2 li 6: 6 : 2\\ 26 . 24 30 29 21 24 26 2 26 2 252 &S ! 1 6" 291 2 1 6 2 1 0 184 1 20 09 15 3 2 1
$\ \chi^2=\sum_{i=1}^{n}(f_i-e_i)^2/e_i $ (follows chi square distribution with n-1=5-1=4 df )$\;;i=1,2,3,...,n\\ n=5,f_i $ is observed frequency and $ e_i $ is expected frequency.$\\ e_i=\frac{\sum f_i}{5}=131/5=26.2\\ \chi^2=2.0915\\ $critical value is $\chi^2_{4}(0.01)=13.27 $ from chi square table$\\ $we reject null hypothesis if $\chi^2>\chi^2_{4}(0.01)$ otherwise don't$\\ $so from the data calculated value is less than tabulated value and hence we don't have enough evidence to reject.$$DONOT REJECT THE NULL HYPOTHESIS THAT THE OBSERVED DISTRIBUTION IS UNIFORM.