Math / Data & Statistics

Question\begin{enumerate} \item[(iii)] After examining the bar graph in part B. (ii), do you believe that finishing place is independent of event for Quebecois alpine skiers? Explain. \item[(iv)] Test at the $5\%$ significance level the claim that finishing place is independent of event. (Hint: Use the test of independence.) Do Quebecois alpine skiers appear to be better at certain events than other events? Explain. \end{enumerate}

Studdy Solution

STEP 1

1. We are analyzing whether the finishing place of Quebecois alpine skiers is independent of the event they participate in.
2. We will use a chi-square test of independence to statistically test this claim at a 5% significance level.
3. The bar graph provides a visual representation of the distribution of finishing places across different events.

STEP 2

1. Analyze the bar graph to form an initial opinion on independence.
2. Formulate the null and alternative hypotheses for the chi-square test.
3. Calculate the expected frequencies for each category.
4. Compute the chi-square statistic.
5. Determine the critical value and compare it to the chi-square statistic.
6. Draw a conclusion from the statistical test and explain the results.

STEP 3

Examine the bar graph to assess whether the distribution of finishing places appears similar across the different events. Look for patterns or differences in the proportions of skiers finishing in each category (Medal, Places 4 to 10, Remaining Places).

STEP 4

Formulate the hypotheses: - Null Hypothesis ( $H_0$ ): Finishing place is independent of the event. - Alternative Hypothesis ( $H_a$ ): Finishing place is dependent on the event.

STEP 5

Calculate the expected frequencies for each category using the formula:
$E_{ij} = \frac{(Row\ Total_i) \times (Column\ Total_j)}{Grand\ Total}$
where $E_{ij}$ is the expected frequency for the $i$ -th row and $j$ -th column.

STEP 6

Compute the chi-square statistic using the formula:
$\chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}$
where $O_{ij}$ is the observed frequency.

STEP 7

Determine the critical value from the chi-square distribution table with the appropriate degrees of freedom and compare it to the calculated chi-square statistic.

STEP 8

Draw a conclusion: - If the chi-square statistic is greater than the critical value, reject the null hypothesis. - If the chi-square statistic is less than or equal to the critical value, fail to reject the null hypothesis.
Explain whether the data suggests that Quebecois alpine skiers perform better in certain events.

Was this helpful?

Studdy Solution

STEP 1

STEP 2

STEP 3

Examine the bar graph to assess whether the distribution of finishing places appears similar across the different events. Look for patterns or differences in the proportions of skiers finishing in each category (Medal, Places 4 to 10, Remaining Places).

STEP 4

Formulate the hypotheses: - Null Hypothesis ( $H_0$ ): Finishing place is independent of the event. - Alternative Hypothesis ( $H_a$ ): Finishing place is dependent on the event.

STEP 5

Calculate the expected frequencies for each category using the formula:
$E_{ij} = \frac{(Row\ Total_i) \times (Column\ Total_j)}{Grand\ Total}$
where $E_{ij}$ is the expected frequency for the $i$ -th row and $j$ -th column.

STEP 6

Compute the chi-square statistic using the formula:
$\chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}$
where $O_{ij}$ is the observed frequency.

STEP 7

Determine the critical value from the chi-square distribution table with the appropriate degrees of freedom and compare it to the calculated chi-square statistic.

STEP 8

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Studdy Solution

STEP 1

STEP 2

STEP 3

Examine the bar graph to assess whether the distribution of finishing places appears similar across the different events. Look for patterns or differences in the proportions of skiers finishing in each category (Medal, Places 4 to 10, Remaining Places).

STEP 4

Formulate the hypotheses: - Null Hypothesis (H0H_0H0​): Finishing place is independent of the event. - Alternative Hypothesis (HaH_aHa​): Finishing place is dependent on the event.

STEP 5

STEP 6

Compute the chi-square statistic using the formula:χ2=∑(Oij−Eij)2Eij\chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}χ2=∑Eij​(Oij​−Eij​)2​where OijO_{ij}Oij​ is the observed frequency.

STEP 7

Determine the critical value from the chi-square distribution table with the appropriate degrees of freedom and compare it to the calculated chi-square statistic.

STEP 8

Get Studdy

Studdy solves anything!

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

Formulate the hypotheses: - Null Hypothesis ( $H_0$ ): Finishing place is independent of the event. - Alternative Hypothesis ( $H_a$ ): Finishing place is dependent on the event.

Compute the chi-square statistic using the formula:
$\chi^2 = \sum \frac{(O_{ij} - E_{ij})^2}{E_{ij}}$
where $O_{ij}$ is the observed frequency.