The problem provides a contingency table showing the distribution of 2000 individuals based on age and preferred sport. The questions ask us to: 1. Calculate the Chi-square statistic.

1. Problem Description

The problem provides a contingency table showing the distribution of 2000 individuals based on age and preferred sport. The questions ask us to:

1. Calculate the Chi-square statistic.

2. Calculate Cramer's V and Tschuprow's T coefficients.

3. Determine if age influences the choice of sport.

2. Solution Steps

First, let's create the contingency table with observed values. The rows represent age groups, and the columns represent sports.

| Age | Equation | Football | Golf | Natation | Tennis | Row Total |

|-----------|----------|----------|------|----------|--------|-----------|

| <20 | 50 | 140 | 20 | 140 | 150 | 500 |

| [20,30[ | 80 | 150 | 50 | 170 | 250 | 700 |

| [30,40[ | 80 | 50 | 70 | 100 | 200 | 500 |

| 40+ | 30 | 20 | 60 | 90 | 100 | 300 |

| Col Total | 240 | 360 | 200 | 500 | 700 | 2000 |

Now we calculate the expected values for each cell. The expected value for cell (i,j) is calculated as:

$E_{ij} = \frac{(\text{Row Total}_i) \times (\text{Column Total}_j)}{\text{Grand Total}}$

For example, the expected value for the first cell (age < 20, Equation) is:

$E_{11} = \frac{500 \times 240}{2000} = 60$

We calculate the expected values for all cells:

| Age | Equation | Football | Golf | Natation | Tennis |

|-----------|----------|----------|------|----------|--------|

| <20 | 60 | 90 | 50 | 125 | 175 |

| [20,30[ | 84 | 126 | 70 | 175 | 245 |

| [30,40[ | 60 | 90 | 50 | 125 | 175 |

| 40+ | 36 | 54 | 30 | 75 | 105 |

Next, we calculate the Chi-square statistic. The formula is:

$\chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}$

where $O_{ij}$ is the observed value and $E_{ij}$ is the expected value.

$\chi^2 = \frac{(50-60)^2}{60} + \frac{(140-90)^2}{90} + \frac{(20-50)^2}{50} + \frac{(140-125)^2}{125} + \frac{(150-175)^2}{175}$

$+ \frac{(80-84)^2}{84} + \frac{(150-126)^2}{126} + \frac{(50-70)^2}{70} + \frac{(170-175)^2}{175} + \frac{(250-245)^2}{245}$

$+ \frac{(80-60)^2}{60} + \frac{(50-90)^2}{90} + \frac{(70-50)^2}{50} + \frac{(100-125)^2}{125} + \frac{(200-175)^2}{175}$

$+ \frac{(30-36)^2}{36} + \frac{(20-54)^2}{54} + \frac{(60-30)^2}{30} + \frac{(90-75)^2}{75} + \frac{(100-105)^2}{105}$

$\chi^2 = \frac{100}{60} + \frac{2500}{90} + \frac{900}{50} + \frac{225}{125} + \frac{625}{175} + \frac{16}{84} + \frac{576}{126} + \frac{400}{70} + \frac{25}{175} + \frac{25}{245} + \frac{400}{60} + \frac{1600}{90} + \frac{400}{50} + \frac{625}{125} + \frac{625}{175} + \frac{36}{36} + \frac{1156}{54} + \frac{900}{30} + \frac{225}{75} + \frac{25}{105}$

$\chi^2 \approx 1.667 + 27.778 + 18 + 1.8 + 3.571 + 0.190 + 4.571 + 5.714 + 0.143 + 0.102 + 6.667 + 17.778 + 8 + 5 + 3.571 + 1 + 21.407 + 30 + 3 + 0.238$

$\chi^2 \approx 159.176$

Now we need to calculate Cramer's V. The formula is:

$V = \sqrt{\frac{\chi^2}{n \times \min(c-1, r-1)}}$

where $n$ is the grand total (2000), $c$ is the number of columns (5), and $r$ is the number of rows (4).

$V = \sqrt{\frac{159.176}{2000 \times \min(5-1, 4-1)}} = \sqrt{\frac{159.176}{2000 \times 3}} = \sqrt{\frac{159.176}{6000}} \approx \sqrt{0.0265} \approx 0.1628$

Now we calculate Tschuprow's T. The formula is:

$T = \sqrt{\frac{\chi^2}{n \sqrt{(r-1)(c-1)}}}$

$T = \sqrt{\frac{159.176}{2000 \sqrt{(4-1)(5-1)}}} = \sqrt{\frac{159.176}{2000 \sqrt{3 \times 4}}} = \sqrt{\frac{159.176}{2000 \sqrt{12}}} \approx \sqrt{\frac{159.176}{2000 \times 3.464}} \approx \sqrt{\frac{159.176}{6928.2}} \approx \sqrt{0.023} \approx 0.1517$

To determine if age influences the choice of sport, we can look at the Chi-square value. A higher Chi-square value suggests a stronger association. Additionally, Cramer's V and Tschuprow's T give us a measure of the strength of the association. The value is quite low. However, we need to find the degrees of freedom (df) to compare the $\chi^2$ statistic to a critical value. The degrees of freedom are:

$df = (r-1)(c-1) = (4-1)(5-1) = 3 \times 4 = 12$

With $df = 12$ and $\alpha = 0.05$ (a common significance level), the critical value is $\chi^2_{critical} = 21.026$. Since our calculated $\chi^2$ (159.176) is much greater than 21.026, we reject the null hypothesis that age and sport choice are independent. Therefore, we conclude that age likely influences the choice of sport.

3. Final Answer

1. Chi-square statistic: $\chi^2 \approx 159.176$

2. Cramer's V: $V \approx 0.1628$, Tschuprow's T: $T \approx 0.1517$

3. Yes, age influences the choice of sport.