test has always the right critical region Case Problem We have in the survey

Test has always the right critical region case

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? −test has always the right critical region ! ? ?
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Iwona Nowakowska 38 Solution: ? ? uptime for the electronic toy produced in the Company A ? ? uptime for the electronic toy produced in the Company B ? ? uptime for the electronic toy produced in the Company C 𝝁 ? the mean uptime for the electronic toy produced in the Company A 𝝁 ? the mean uptime for the electronic toy produced in the Company B 𝝁 ? the mean uptime for the electronic toy produced in the Company C ? − sample size ? = ? + ? + ? = ?? ? − number of groups ? = ? 𝑯 ? : 𝝁 ? = 𝝁 ? = 𝝁 ? 𝑯 ? : ??? ??? ????? ??? ??? ???? 𝑯 ? : 𝝁 ? = 𝝁 ? = 𝝁 ? 𝑯 ? : ⋁ 𝝁 ? ≠ 𝝁 ? ?, ? ?≠? 𝑯 ? : 𝝁 ? = 𝝁 ? = 𝝁 ? 𝑯 ? : ?? ????? ??? ????? ??? ????????? Test statistics:
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Application of Mathematical Statistics (10) Iwona Nowakowska 39 ? = ???????= ????? − ????? − ?
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Iwona Nowakowska 40 ??? = (? − ??) ? + (?? − ??) ? + (?? − ??) ? + (? − ??) ? + (? − ?) ? + (? − ?) ? + ? ? + (? − ?) ? + (? − ?) ? + (?? − ?) ? + (? − ?) ? + (? − ?) ? = = ? ? + ? ? + ? ? + ? ? + ? ? + ? ? + ? ? + ? ? + ? ? + + ? ? + ? ? + ? ? = = ? + ? + ? + ? + ? + ? + ? + ? + ? + ? = ?? ? = ???? ??? = ???? ? − ? ??? ? − ? = ?? ? − ? ?? ?? − ? = ?? ? ?? ? = ??, ? ?, ?? = ?, ?? ? ∶ ? − ???????? (?−?), (?−?) ? 𝜶 critical value To find critical value ? 𝜶 we use ? − ???????? distribution table for the measure of risk 𝜶 with: ?? = (? − ?) and ?? = (? − ?) degrees of freedom 𝜶 = ?, ?? ?? = (? − ?) = ? − ? = ? ?? = (? − ?) = ?? − ? = ? Table for measure of risk 𝜶 = ? , ?? ?? = ? − ? = ? ?? = ? − ? = ? ? , ?? Numerator Degrees of Freedom Denominator Degrees of Freedom
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Application of Mathematical Statistics (10) Iwona Nowakowska 41 ? 𝜶 = ?, ?? ? − test has always the right critical region ! We reject 𝑯 ? . 𝑯 ? is true. 𝑯 ? : 𝝁 ? = 𝝁 ? = 𝝁 ? 𝑯 ? : ?? ????? ??? ????? ??? ????????? The uptime averages for the electronic toys are statistically different in three companies. You should be careful during choosing the electronic toy company. right critical region critical value ? ? 𝜶 = ?, ?? ? ? = ?, ??
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Application of Mathematical Statistics (10) Iwona Nowakowska 42 Analysis of Variance A N O V A Hypothesis testing for differences between many means 𝝁 ? , 𝝁 ? , 𝝁 ? , , 𝝁 ? / comparing several means / Model: ? ? , ? ? , ? ? , . . . , ? ? Independent samples ? ? ∶ ? ( 𝝁 ? , 𝝈 ? ), ? = ?, … , ? 𝝈 ? ? = 𝝈 ? ? = 𝝈 ? ? = ⋯ = 𝝈 ? ? equal variances ? − number of populations (number of groups) ? − total numer of observations ? = ? ? + ? ? + ? ? + ⋯ + ? ? ???? ?????????? ? = ??? ??? = ??? ? − ? ??? ? − ? ? ? the mean in the ? ?? group ? the general mean ? = ? ? ∑ ? ? ∙ ? ? ??? − variance between groups mean square between groups ??? = ??? ? − ? ??? − sum of squares between groups ??? = ∑ ( ? ? ? ) ? ? ? ? ??? − variance within groups mean square within groups ??? = ??? ? − ? ??? − sum of squares within groups ??? = ( ? ?? ? ? ) ? ? ? ? = ? ??. sample test statistics value (observed from the sample) 𝑯 ? : 𝝁 ? = 𝝁 ? = 𝝁 ? = 𝝁 ? = ⋯ = 𝝁 ? 𝑯 ? : ⋁ 𝝁 ? ? , ? 𝝁 ? 𝑯 ? : the means in all populations are the same 𝑯 ? : not all means are equal 𝑯 ?
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  • Spring '17
  • Accounting, Mathematical statistics, Chi-square distribution, Pearson's chi-square test, Iwona Nowakowska

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