I used excel to display this table.
Other free alternatives (such as OpenOffice spreadsheet) will be able to do the same.
Take note of the row and column labels along the left and top border.
They are very handy to quickly locate a given cell.
For example, cell C5 has the value 19 in it.
This is the cell that is in the intersection of the "C" column and the row 5.
It is in the intersection of the "widowed" row and "18 to 24" column.
This indicates that there are 19 widowed women who are between age 18 and age 24.
If you aren't familiar with this labeling method, then take a few moments to practice reading it.
You'll most likely will use a spreadsheet sometime later in the future.
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Part (A)
Find the number of women who never married.
This is the value in cell F4, which is 19309
It is in the "never married" row and "total" column.
Basically it sums up everything in the "never married" row (ignoring the total column when you sum).
Divide the value in cell F4 by the value in cell F7, which is the grand total of all of the women surveyed
P(never married) = probability of never married
P(never married) = (number of women never married)/(number of women total)
P(never married) = (value in cell F4)/(value in cell F7)
P(never married) = (19309)/(99585)
P(never married) = 0.19389466285083
P(never married) = 0.193895
The result is approximately 0.193895 which is accurate to 6 decimal places
Answer: 0.193895
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Part (B)
We'll follow the same basic steps as part (A).
There are 48116 women who are married and their age is between 25 and 64 (see cell D3)
There are 99585 women total (see cell F7)
P(married; age 25 to 64) = (number of married women between ages 25 to 64)/(number of women total)
P(married; age 25 to 64) = (value in cell D3)/(value in cell F7)
P(married; age 25 to 64) = 48116/99585
P(married; age 25 to 64) = 0.48316513531154
P(married; age 25 to 64) = 0.483165
Like in part (A), this value is approximate and accurate to 6 decimal places.
Answer: 0.483165
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Part (C)
There are 58929 married women (cell F3). Call this value M, so M = 58929
There are 68709 women aged 25 to 64 (cell D7). Call this value N, so N = 68709
There are 48116 women who are both married and in the age range of 25 to 64 (cell D3). Call this value P, so P = 48116
Let's compute the number of women who are married OR they are between 25 and 64. This "or" I'm using is an inclusive "or". We can pick one, or the other or both.
To do this, we calculate the following
R = M+N - P
R = 58929+68709 - 48116
R = 79522
There are 79522 women who are either married only, between 25 and 64 only, or they are both married and in the age range 25 to 64.
You have to subtract off the number of women who are in both categories to avoid double counting these women.
Divide this value over the total number of women, which is 99585 (cell F7)
P(married OR age range is 25 to 64) = (number of women who are married OR they are in the age range of 25 to 64)/(number of women total)
P(married OR age range is 25 to 64) = ( R )/(value in cell F7)
P(married OR age range is 25 to 64) = 79522/99585
P(married OR age range is 25 to 64) = 0.79853391575037
P(married OR age range is 25 to 64) = 0.798534
Answer: 0.798534
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Part (D)
We're told that the woman chosen is between 18 and 24 years of age. We only need to focus on the "18 to 24" column. Ignore everything else.
There are 260 divorced women in this column (cell C6) out of 12614 women total (cell C7). We do NOT use the grand total because we are ignoring the other columns. We only use the total from the "18 to 24" column.
P(divorced; given woman is between 18 and 24) = (number of divorced women aged 18 to 24)/(number of women total)
P(divorced; given woman is between 18 and 24) = (value in cell C6)/(value in cell C7)
P(divorced; given woman is between 18 and 24) = 260/12614
P(divorced; given woman is between 18 and 24) = 0.02061201839227
P(divorced; given woman is between 18 and 24) = 0.020612
Answer: 0.020612
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Part (E)
Let's define three events
A = event that the woman is between 18 and 24
B = event that the woman is divorced
C = event that the woman is both in the age range of 18 to 24, and the woman is divorced
Note how event C is events A,B combined with the keyword "and".
The aim is to see if events A and B are independent or not. If they aren't independent, then we can conclude age and marital status overall are dependent on one another in some way.
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Using the table, we can find
P(A) = (value in cell C7)/(value in cell F7)
P(A) = (12614)/(99585)
P(A) = 0.12666566249938
P(B) = (value in cell F6)/(value in cell F7)
P(B) = (10267)/(99585)
P(B) = 0.10309785610282
P(C) = (value in cell C6)/(value in cell F7)
P(C) = (260)/(99585)
P(C) = 0.0026108349651
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Now compute P(A)*P(B) to get
P(A)*P(B) = 0.12666566249938*0.10309785610282 = 0.01305895824552
This result (0.01305895824552) is NOT the same as P(C) shown above.
Because of this mismatch in values, this means P(A)*P(B) = P(C) is false leading to P(A)*P(B) = P(A and B) being false.
P(A and B) = P(A)*P(B) is only true if A and B are independent events.
The same can be said in reverse: if A and B are independent events, then P(A and B) = P(A)*P(B) is true.
Therefore, "age 18 to 24" and "divorced" are dependent events.
Overall, age and marital status are dependent.
Answer: No, they are not independent.