Math / Data & Statistics

Question(Note if any boxes seem not applicable, leave blank.) A random sample of 130 business executives was classified according to age and the degree of risk aversion as measured by a psychological test.
Degree of Risk Aversion \begin{tabular}{|c|c|c|c|c|} \hline Age & \multicolumn{3}{|c|}{ Low } & Medium \\ High & Total \\ \hline Below 45 & 14 & 22 & 7 & 43 \\ \hline $45-55$ & 16 & 33 & 12 & 61 \\ \hline Over 55 & 4 & 15 & 7 & 26 \\ \hline Total & & & & 130 \\ \hline \end{tabular}
Do these data demonstrate an association between risk aversion and age?
Test Statistic: $\square$ According to your table, the P -value is bounded by: .50 $\square$ Is there sufficient evidence to demonstrate an association between

Studdy Solution

STEP 1

1. We are using a chi-square test for independence to determine if there is an association between age and degree of risk aversion.
2. The data is categorical, and the sample size is sufficiently large for the chi-square approximation to be valid.
3. The null hypothesis ( $H_0$ ) is that there is no association between age and degree of risk aversion.
4. The alternative hypothesis ( $H_a$ ) is that there is an association between age and degree of risk aversion.

STEP 2

1. Construct the observed frequency table.
2. Calculate the expected frequencies.
3. Compute the chi-square test statistic.
4. Determine the degrees of freedom.
5. Find the p-value and make a conclusion.

STEP 3

Construct the observed frequency table from the given data:
$\begin{array}{|c|c|c|c|c|} \hline \text{Age} & \text{Low} & \text{Medium} & \text{High} & \text{Total} \\ \hline \text{Below 45} & 14 & 22 & 7 & 43 \\ \hline 45-55 & 16 & 33 & 12 & 61 \\ \hline \text{Over 55} & 4 & 15 & 7 & 26 \\ \hline \text{Total} & & & & 130 \\ \hline \end{array}$

STEP 4

Calculate the expected frequencies using the formula:
$E_{ij} = \frac{( \text{Row Total}_i \times \text{Column Total}_j )}{\text{Grand Total}}$
Calculate expected frequencies for each cell.

STEP 5

Calculate the expected frequency for the cell (Below 45, Low):
$E_{11} = \frac{(43 \times \text{Low Total})}{130}$
Continue this calculation for each cell in the table.

STEP 6

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

STEP 7

Determine the degrees of freedom for the test:
$\text{Degrees of Freedom} = (\text{Number of Rows} - 1) \times (\text{Number of Columns} - 1)$

STEP 8

Using the chi-square distribution table, find the p-value corresponding to the calculated chi-square statistic and degrees of freedom.

STEP 9

Compare the p-value to a significance level (commonly 0.05) to determine if there is sufficient evidence to reject the null hypothesis.
The test statistic and p-value bounds will be calculated based on the chi-square distribution table and the computed chi-square statistic.

Was this helpful?

Studdy Solution

STEP 1

STEP 2

1. Construct the observed frequency table.
2. Calculate the expected frequencies.
3. Compute the chi-square test statistic.
4. Determine the degrees of freedom.
5. Find the p-value and make a conclusion.

STEP 3

STEP 4

Calculate the expected frequencies using the formula:
$E_{ij} = \frac{( \text{Row Total}_i \times \text{Column Total}_j )}{\text{Grand Total}}$
Calculate expected frequencies for each cell.

STEP 5

Calculate the expected frequency for the cell (Below 45, Low):
$E_{11} = \frac{(43 \times \text{Low Total})}{130}$
Continue this calculation for each cell in the table.

STEP 6

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

STEP 7

Determine the degrees of freedom for the test:
$\text{Degrees of Freedom} = (\text{Number of Rows} - 1) \times (\text{Number of Columns} - 1)$

STEP 8

Using the chi-square distribution table, find the p-value corresponding to the calculated chi-square statistic and degrees of freedom.

STEP 9

Compare the p-value to a significance level (commonly 0.05) to determine if there is sufficient evidence to reject the null hypothesis.
The test statistic and p-value bounds will be calculated based on the chi-square distribution table and the computed chi-square statistic.

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

1. Construct the observed frequency table.2. Calculate the expected frequencies.3. Compute the chi-square test statistic.4. Determine the degrees of freedom.5. Find the p-value and make a conclusion.

STEP 3

STEP 4

Calculate the expected frequencies using the formula:Eij=(Row Totali×Column Totalj)Grand TotalE_{ij} = \frac{( \text{Row Total}_i \times \text{Column Total}_j )}{\text{Grand Total}}Eij​=Grand Total(Row Totali​×Column Totalj​)​Calculate expected frequencies for each cell.

STEP 5

Calculate the expected frequency for the cell (Below 45, Low):E11=(43×Low Total)130E_{11} = \frac{(43 \times \text{Low Total})}{130}E11​=130(43×Low Total)​Continue this calculation for each cell in the table.

STEP 6

Compute the chi-square test 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 and EijE_{ij}Eij​ is the expected frequency.

STEP 7

Determine the degrees of freedom for the test:Degrees of Freedom=(Number of Rows−1)×(Number of Columns−1)\text{Degrees of Freedom} = (\text{Number of Rows} - 1) \times (\text{Number of Columns} - 1)Degrees of Freedom=(Number of Rows−1)×(Number of Columns−1)

STEP 8

Using the chi-square distribution table, find the p-value corresponding to the calculated chi-square statistic and degrees of freedom.

STEP 9

Compare the p-value to a significance level (commonly 0.05) to determine if there is sufficient evidence to reject the null hypothesis. The test statistic and p-value bounds will be calculated based on the chi-square distribution table and the computed chi-square statistic.

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.

1. Construct the observed frequency table.
2. Calculate the expected frequencies.
3. Compute the chi-square test statistic.
4. Determine the degrees of freedom.
5. Find the p-value and make a conclusion.

Calculate the expected frequencies using the formula:
$E_{ij} = \frac{( \text{Row Total}_i \times \text{Column Total}_j )}{\text{Grand Total}}$
Calculate expected frequencies for each cell.

Calculate the expected frequency for the cell (Below 45, Low):
$E_{11} = \frac{(43 \times \text{Low Total})}{130}$
Continue this calculation for each cell in the table.

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

Determine the degrees of freedom for the test:
$\text{Degrees of Freedom} = (\text{Number of Rows} - 1) \times (\text{Number of Columns} - 1)$

Compare the p-value to a significance level (commonly 0.05) to determine if there is sufficient evidence to reject the null hypothesis.
The test statistic and p-value bounds will be calculated based on the chi-square distribution table and the computed chi-square statistic.