Question

multiple choice 4 points
determine the test statistic ( chi^{2} ).
a sociologist was interested in determining if there was a relationship between the age of a young adult (18 to 35 years old) and the type of movie preferred. a random
sample of 93 adults revealed the following data. use a chi - square independence test to determine if age and type of movie preferred are independent at the 5% level of
significance.
two - way table genre vs age
provided the assumptions of the test are satisfied, find the test statistic ( chi^{2} ).
1.44
12.234
3.623
2.944
cannot be determined.

Explanation:

Step1: Calculate the expected frequency

The formula for the expected frequency \(E_{ij}=\frac{R_i\times C_j}{N}\), where \(R_i\) is the row total, \(C_j\) is the column total, and \(N\) is the grand total.
For the cell corresponding to 18 - 23 years old and Drama: \(E_{13}=\frac{29\times34}{93}\approx10.67\)
For the cell corresponding to 18 - 23 years old and Science Fiction: \(E_{12}=\frac{29\times30}{93}\approx9.46\)
For the cell corresponding to 18 - 23 years old and Comedy: \(E_{11}=\frac{29\times29}{93}\approx9.07\)
For the cell corresponding to 24 - 29 years old and Drama: \(E_{23}=\frac{33\times34}{93}\approx12.09\)
For the cell corresponding to 24 - 29 years old and Science Fiction: \(E_{22}=\frac{33\times30}{93}\approx10.65\)
For the cell corresponding to 24 - 29 years old and Comedy: \(E_{21}=\frac{33\times29}{93}\approx10.34\)
For the cell corresponding to 30 - 35 years old and Drama: \(E_{33}=\frac{31\times34}{93}\approx11.24\)
For the cell corresponding to 30 - 35 years old and Science Fiction: \(E_{32}=\frac{31\times30}{93}\approx10\)
For the cell corresponding to 30 - 35 years old and Comedy: \(E_{31}=\frac{31\times29}{93}\approx9.59\)

Step2: Calculate the \(\chi^{2}\) statistic

The formula for the \(\chi^{2}\) statistic is \(\chi^{2}=\sum\frac{(O - E)^{2}}{E}\), where \(O\) is the observed frequency and \(E\) is the expected frequency.
For the cell (18 - 23, Drama): \(\frac{(8 - 10.67)^{2}}{10.67}=\frac{(- 2.67)^{2}}{10.67}=\frac{7.1289}{10.67}\approx0.67\)
For the cell (18 - 23, Science Fiction): \(\frac{(12 - 9.46)^{2}}{9.46}=\frac{(2.54)^{2}}{9.46}=\frac{6.4516}{9.46}\approx0.68\)
For the cell (18 - 23, Comedy): \(\frac{(9 - 9.07)^{2}}{9.07}=\frac{(-0.07)^{2}}{9.07}=\frac{0.0049}{9.07}\approx0.0005\)
For the cell (24 - 29, Drama): \(\frac{(15 - 12.09)^{2}}{12.09}=\frac{(2.91)^{2}}{12.09}=\frac{8.4681}{12.09}\approx0.7\)
For the cell (24 - 29, Science Fiction): \(\frac{(10 - 10.65)^{2}}{10.65}=\frac{(-0.65)^{2}}{10.65}=\frac{0.4225}{10.65}\approx0.04\)
For the cell (24 - 29, Comedy): \(\frac{(8 - 10.34)^{2}}{10.34}=\frac{(-2.34)^{2}}{10.34}=\frac{5.4756}{10.34}\approx0.53\)
For the cell (30 - 35, Drama): \(\frac{(11 - 11.24)^{2}}{11.24}=\frac{(-0.24)^{2}}{11.24}=\frac{0.0576}{11.24}\approx0.005\)
For the cell (30 - 35, Science Fiction): \(\frac{(8 - 10)^{2}}{10}=\frac{(-2)^{2}}{10}=\frac{4}{10} = 0.4\)
For the cell (30 - 35, Comedy): \(\frac{(12 - 9.59)^{2}}{9.59}=\frac{(2.41)^{2}}{9.59}=\frac{5.8081}{9.59}\approx0.61\)

Sum up all these values: \(\chi^{2}=0.67 + 0.68+0.0005 + 0.7+0.04+0.53+0.005+0.4+0.61\approx3.623\)

Answer:

3.623