Question

A​ case-control (or​ retrospective) study was conducted to investigate a relationship between the colors of helmets worn by motorcycle drivers and whether they are injured or killed in a crash. Results are given in the accompanying table. Using a 0.05 significance​ level, test the claim that injuries are independent of helmet color. Black White Yellow Red Blue Controls (not injured) 503 373 35 160 56 Cases (injured or killed) 215 122 10 66 28 Identify the null and alternative hypotheses. Choose the correct answer below. A. Upper H 0H0​: Whether a crash occurs and helmet color are independent Upper H 1H1​: Whether a crash occurs and helmet color are dependent B. Upper H 0H0​: Injuries and helmet color are independent Upper H 1H1​: Injuries and helmet color are dependent C. Upper H 0H0​: Whether a crash occurs and helmet color are dependent Upper H 1H1​: Whether a crash occurs and helmet color are independent D. Upper H 0H0​: Injuries and helmet color are dependent Upper H 1H1​: Injuries and helmet color are independent Compute the test statistic. nothing ​(Round to three decimal places as​ needed.) Find the critical​ value(s). nothing ​(Round to three decimal places as needed. Use a comma to separate answers as​ needed.) What is the conclusion based on the hypothesis​ test? ▼ Reject Fail to reject Upper H 0H0. There ▼ is not is sufficient evidence to warrant rejection of the claim that injuries are independent of helmet color. Click to select your answer(s).

Answer 1

Chi square test for independence:

B) H₀ : Injuries and helmet color are independent.

H_a : Injuries and helmet color are dependent.

Observed frequency table :

We need to find expected frequency.

Expected frequency =

For example : expected frequency for control and black = = 516.0625

Expected frequency table :

Chi square statistic:

χ²    =         ; O is observed and E is expected.

χ²     = 6.171

Critical value:

We are given level of significance α = 0.05

Degrees of freedom (df) = (r-1)(c-1)           r is type of color and c is categories of injuries.

r = 5 and c = 2

df = (5-1)*(2-1) = 4

Therefore critical value χ²_(0.05,4) = 9.488 -------- ( From chi square table )

Decision : As χ² = 6.171 is less than χ²_(0.05,4) = 9.488

Fail to reject H₀ ,There is not sufficient evidence to warrant rejection of the claim that injuries are independent of helmet color.