A researcher wants to determine if the number of minutes spent watching television per day is independent of gender. A random sample of 315 adults was selected and the results are shown below. Calculate the chi -square statistic $\chi ^ { 2 }$ to determine if there is enough evidence to conclude that the number of minutes spent watching television per day is related to gender. Use $\alpha = 0.05$.
$\text { Gender } $$\quad $$ \text { Minutes spent watching TV per day } $
$\begin{array} { l | c c c c }
\hline & 0 - 30 & 30 - 60 & 60 - 90 & 90 - \text { over } \
\text { Male } & 25 & 35 & 75 & 45 \
\text { Female } & 30 & 45 & 45 & 15
\end{array}$

A) 18.146
B) 19.874
C) 20.912
D) 21.231

Question

A researcher wants to determine if the number of minutes spent watching television per day is independent of gender. A random sample of 315 adults was selected and the results are shown below. Calculate the chi -square statistic $\chi ^ { 2 }$ to determine if there is enough evidence to conclude that the number of minutes spent watching television per day is related to gender. Use $\alpha = 0.05$.
$\text { Gender } $$\quad $$ \text { Minutes spent watching TV per day } $ 
$\begin{array} { l | c c c c } 
\hline & 0 - 30 & 30 - 60 & 60 - 90 & 90 - \text { over } \
\text { Male } & 25 & 35 & 75 & 45 \
\text { Female } & 30 & 45 & 45 & 15
\end{array}$

A) 18.146
B) 19.874
C) 20.912
D) 21.231

Quizplus · Accepted Answer

The chi-square statistic is calculated as:$\chi ^ { 2 } = \sum \frac{(O - E) ^ { 2 }}{E}$where O is the observed frequency and E is the expected frequency.The expected frequency is calculated as:$E = \frac{\text { row total } \times \text { column total }}{\text { grand total }}$The grand total is 315.The row totals are:Male: 25 + 35 + 75 + 45 = 180Female: 30 + 45 + 45 + 15 = 135The column totals are:0 - 30: 25 + 30 = 5530 - 60: 35 + 45 = 8060 - 90: 75 + 45 = 12090 - over: 45 + 15 = 60The expected frequencies are:$E_{11} = \frac{180 \times 55}{315} = 30.25$$E_{12} = \frac{180 \times 80}{315} = 44$$E_{13} = \frac{180 \times 120}{315} = 66$$E_{14} = \frac{180 \times 60}{315} = 33.75$$E_{21} = \frac{135 \times 55}{315} = 24.75$$E_{22} = \frac{135 \times 80}{315} = 36$$E_{23} = \frac{135 \times 120}{315} = 54$$E_{24} = \frac{135 \times 60}{315} = 26.25$The chi-square statistic is:$\chi ^ { 2 } = \frac{(25 - 30.25) ^ { 2 }}{30.25} + \frac{(35 - 44) ^ { 2 }}{44} + \frac{(75 - 66) ^ { 2 }}{66} + \frac{(45 - 33.75) ^ { 2 }}{33.75} + \frac{(30 - 24.75) ^ { 2 }}{24.75} + \frac{(45 - 36) ^ { 2 }}{36} + \frac{(45 - 54) ^ { 2 }}{54} + \frac{(15 - 26.25) ^ { 2 }}{26.25} = 18.146$The degrees of freedom are:$df = (r - 1) \times (c - 1) = (4 - 1) \times (4 - 1) = 9$The critical value of chi-square with 9 degrees of freedom and $\alpha = 0.05$ is 16.919.Since the chi-square statistic (18.146) is greater than the critical value (16.919), we reject the null hypothesis and conclude that there is enough evidence to conclude that the number of minutes spent watching television per day is related to gender.

A Researcher Wants to Determine If the Number of Minutes χ2\chi ^ { 2 }χ2

A Researcher Wants to Determine If the Number of Minutes $\chi ^ { 2 }$