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In: Statistics and Probability

A person’s educational attainment and age group was collected by the U.S. Census Bureau in 1984...

A person’s educational attainment and age group was collected by the U.S. Census Bureau in 1984 to see if age group and educational attainment are related. The counts in thousands are in the table below ("Education by age," 2013). Do the data show that educational attainment and age are independent? Test at the 5% level. Table 5: Educational Attainment and Age Group Education Age Group Row Total 25-34 35-44 45-54 55-64 >64 Did not complete HS 5416 5030 5777 7606 13746 37575 Competed HS 16431 1855 9435 8795 7558 44074 College 1-3 years 8555 5576 3124 2524 2503 22282 College 4 or more years 9771 7596 3904 3109 2483 26863 Column Total 40173 20057 22240 22034 26290 130794

Solutions

Expert Solution

Ho: educational attainment and age are independent

H1: educational attainment and age are not independent

Chi-Square Test of independence
Observed Frequencies
0
0 25-34 35-44 45-54 55-64 >64 Total
Did not complete HS 5416 5030 5777 7606 13746 37575
Competed HS 16431 1855 9435 8795 7558 44074
College 1-3 years 8555 5576 3124 2524 2503 22282
College 4 or more years 9771 7596 3904 3109 2483 26863
Total 40173 20057 22240 22034 26290 130794
Expected frequency of a cell = sum of row*sum of column / total sum
Expected Frequencies
25-34 35-44 45-54 55-64 >64 Total
Did not complete HS 40173*37575/130794=11541.053 20057*37575/130794=5762.052 22240*37575/130794=6389.192 22034*37575/130794=6330.012 26290*37575/130794=7552.692 37575
Competed HS 40173*44074/130794=13537.202 20057*44074/130794=6758.66 22240*44074/130794=7494.272 22034*44074/130794=7424.855 26290*44074/130794=8859.011 44074
College 1-3 years 40173*22282/130794=6843.852 20057*22282/130794=3416.9 22240*22282/130794=3788.795 22034*22282/130794=3753.701 26290*22282/130794=4478.751 22282
College 4 or more years 40173*26863/130794=8250.893 20057*26863/130794=4119.388 22240*26863/130794=4567.741 22034*26863/130794=4525.432 26290*26863/130794=5399.546 26863
Total 40173 20057 22240 22034 26290 130794
(fo-fe)^2/fe
Did not complete HS 3250.6803 93.0050 58.658 257.211 5078.5959
Competed HS 618.5966 3557.7886 502.574 252.839 191.0630
College 1-3 years 427.8332 1364.3099 116.6473 402.8464 871.5806
College 4 or more years 280.0576 2934.1335 96.4486 443.3343 1575.3625

Chi-Square Test Statistic,χ² = Σ(fo-fe)^2/fe =   22373.566  
      
Level of Significance =   0.05  
Number of Rows =   4  
Number of Columns =   5  
Degrees of Freedom=(#row - 1)(#column -1) = (4- 1 ) * ( 5- 1 ) =   12  
      
p-Value =   0.0000000   [Excel function: =CHISQ.DIST.RT(χ²,df) ]
Decision:    p-value < α , Reject Ho  

conclusion: educational attainment and age are not independent

orchestra answered 1 year ago
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