Question

a color code personality test categorizes people into four colors - red (power), blue (intimacy), green (peace), and yellow (fun). in general, 10% of people are red, 20% blue, 30% green, and 40% yellow. a sample of 115 art students is tested at a university and 16 are found to be red, 14 blue, 24 green and 61 yellow. can it be concluded that personality type has an impact on students areas of interest and talents, such as artistic students?

$h_{o}:p_{r}=0.1;p_{b}=0.2;p_{g}=0.3;p_{y}=0.4$

$h_{1}$: at least one proportion is different. (claim)

complete the table. report all answers accurate to 4 decimal places.

what is the chi - square test - statistic for this data?

$\chi^{2}=\square$

what is the p - value?

$p - value=\square$

for significance level alpha 0.05,

what would be the decision and conclusion of this hypothesis test?

Explanation:

Step1: Calculate expected frequencies

Expected frequency = Total sample size × Proportion. Total sample size = 115.
Red: $115×0.1 = 11.0000$
Blue: $115×0.2 = 23.0000$
Green: $115×0.3 = 34.5000$
Yellow: $115×0.4 = 46.0000$

Step2: Compute chi-square statistic

$\chi^2 = \sum \frac{(O-E)^2}{E}$
Red: $\frac{(16-11)^2}{11} ≈ 2.2727$
Blue: $\frac{(14-23)^2}{23} ≈ 3.5217$
Green: $\frac{(24-34.5)^2}{34.5} ≈ 3.2362$
Yellow: $\frac{(61-46)^2}{46} ≈ 4.8913$
Sum: $2.2727 + 3.5217 + 3.2362 + 4.8913 ≈ 13.9219$

Step3: Find degrees of freedom

$df = \text{Categories} - 1 = 4 - 1 = 3$

Step4: Calculate p-value

Using chi-square distribution with $df=3$, p-value for $\chi^2≈13.92$ is $P(\chi^2>13.92)≈0.0078$

Step5: Make decision (α=0.05)

p-value (0.0078) < α (0.05), reject $H_0$.

Answer:

Completed Table:

Category	Observed Frequency	Expected Frequency
Blue	14	23.0000
Green	24	34.5000
Yellow	61	46.0000

Chi-square test-statistic:

13.9219

P-Value:

0.0078

Decision and Conclusion:

Reject the null hypothesis. There is sufficient evidence to support the claim that personality type impacts students' areas of interest (e.g., artistic students) at the 0.05 significance level.