Computer software generated 500 random numbers that should look like they are fr

Question

Computer software generated 500 random numbers that should look like they are from the standard normal distribution. They are categorized into five groups: (1) less than or equal to -0.8, (2) greater than -0.8 and less than or equal to -0.2, (3) greater than -0.2 and less than or equal to 0.2, (4) greater than 0.2 and less than or equal to 0.8, and (5) greater than 0.8. The counts in the five groups are 98, 112, 79, 111, and 100, respectively. Find the probabilities for these five intervals using Table A. Then compute the expected number for each interval for a sample of 500. Finally, perform the goodness of fit test and summarize your results.

Explanation / Answer

probability of each

98/500 = 0.196

112/500 = 0.224

79/500 = 0.158

111/500 = 0.222

100/500 = 0.2

observed frequncies are Oi

98 112 79 111 100   

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expected frequencies are Ei

we have five possibilities = 500/5 = 100

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and claiming hypothesis is

null, Ho: no association exists b/w them OR observations are independent

alternative, H1: exists association b/w them OR observations are dependnet

level of significance, = 0.05

from standard normal table, chi square value at right tailed, ^2 /2 =5.9915

since our test is right tailed,reject Ho when ^2 o > 5.9915

we use test statistic ^2 o = (Oi-Ei)^2/Ei

from the table take sum of (Oi-Ei)^2/Ei, we get ^2 o = 7.1

critical value

the value of |^2 | at los 0.05 with d.f, n - 1 = 3 - 1 = 2 is 5.9915

we got | ^2| =7.1 & | ^2 | =5.9915

make decision

hence value of | ^2 o | > | ^2 | and here we reject Ho

^2 p_value =0.0287

ANSWERS

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null, Ho: no association exists b/w them OR observations are independent

alternative, H1: exists association b/w them OR observations are dependnet

test statistic: 7.1

critical value: 5.9915

p-value:0.0287

decision: reject Ho

observations are dependnent to each other

Observed (Oi ) Expected ( Ei) Oi-Ei (Oi-Ei)^2 (Oi-Ei)^2/Ei 0.196 1011.4 -1011 1022534 1011.008 0.224 172.374 -172 29635.6 171.9263 0.158 87.225 -87.1 7580.66 86.9093 920 936.6 -16.6 275.56 0.2942 174 159.626 14.37 206.612 1.2943 83 80.775 2.225 4.9506 0.0613

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