Respuesta :
The missing values a, b, c, d and e are included in the given two-way table. The correct values from the options are: a =15, b=18,e=31
How to form two-way table?
Suppose two dimensions are there, viz X and Y. Some values of X are there as [tex]X_1, X_2, ... , X_n[/tex] and some values of Y are there as [tex]Y_1, Y_2, ... , Y_n[/tex]
List them in title of the rows and left to the columns. There will be [tex]n \times k[/tex] table of values will be formed(excluding titles and totals), such that:
Value(ith row, jth column) = Frequency for intersection of [tex]X_i[/tex] and [tex]Y_j[/tex] (assuming X values are going in rows, and Y values are listed in columns).
Then totals for rows, columns, and whole table are written on bottom and right margin of the final table.
For n = 2, and k = 2, the table would look like:
[tex]\begin{array}{cccc}&Y_1&Y_2&\rm Total\\X_1&n(X_1 \cap Y_1)&n(X_1\cap Y_2)&n(X_1)\\X_2&n(X_2 \cap Y_1)&n(X_2 \cap Y_2)&n(X_2)\\\rm Total & n(Y_1) & n(Y_2) & S \end{array}[/tex]
where S denotes total of totals, also called total frequency.
n is showing the frequency of the bracketed quantity, and intersection sign in between is showing occurrence of both the categories together.
For this case, the table given is:
[tex]\begin{array}{cccc}\rm &\rm \text{Likes swimming}&\rm \text{Doesn't like swimming}&\rm Senior\\\rm \text{Likes baseball}&a&b&33\\\rm \text{Doesn't likes baseball}&c&13&d\\\rm Total & 39& e& 70 \end{array}[/tex]
From this table, we can make 5 equations as:
[tex]a+b = 33\\a+c=39\\b+39=e\\33+d = 70\\39+e=70[/tex]
From the last two equations, we get the values of d and e as:
[tex]d = 70-33 = 37\\e = 70-39 = 31[/tex]
Putting this value of 'e' in last third equation, we get:
[tex]b + 13 = 31\\b = 31-13= 18[/tex]
From this, and first equation, we get:
[tex]a +18 = 33\\a=33-18=15[/tex]
And from this value of a, and second equation, we get:15+c = 39
[tex]c = 39-15 = 24[/tex]
Thus, the correct values are: [tex]a =15, b=18,e=31[/tex]
Learn more about two-way table here:
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