Question

A particular report included the following table classifying 819 fatal bicycle accidents that occurred in a certain year according to the time of day the accident occurred.

Time of Day	Number of Accidents
midnight to 3 a.m.	47
3 a.m. to 6 a.m.	52
6 a.m. to 9 a.m.	87
9 a.m. to noon	72
noon to 3 p.m.	79
3 p.m. to 6 p.m.	157
6 p.m. to 9 p.m.	191
9 p.m. to midnight	134

For purposes of this exercise, assume that these 819 bicycle accidents are a random sample of fatal bicycle accidents. Do these data support the hypothesis that fatal bicycle accidents are not equally likely to occur in each of the 3-hour time periods used to construct the table? Test the relevant hypotheses using a significance level of

α = 0.05.

Let p₁, p₂, p₃, p₄, p₅, p₆, p₇, and p₈ be the proportions of accidents occurring in the eight different time periods.

State the appropriate null and alternative hypotheses.

H₀: p₁ = p₂ = p₃ = p₄ = p₅ = p₆ = p₇ = p₈ = 0.08
H_a: H₀ is not true.H₀: p₁ = p₂ = p₃ = p₄ = p₅ = p₆ = p₇ = p₈ = 0.125
H_a: H₀ is not true.    H₀: p₁ = p₂ = p₃ = p₄ = p₅ = p₆ = p₇ = p₈ = 102.375
H_a: H₀ is not true.H₀: p₁ = p₂ = p₃ = p₄ = p₅ = p₆ = p₇ = p₈ = 0.8
H_a: H₀ is not true.H₀: p₁ = p₂ = p₃ = p₄ = p₅ = p₆ = p₇ = p₈ = 819
H_a: H₀ is not true.

Find the test statistic and P-value. (Use technology. Round your test statistic to three decimal places and your P-value to four decimal places.)

X²=

P-value=

State the conclusion in the problem context.

Fail to reject H₀. There is convincing evidence to conclude that fatal accidents are not equally likely to occur in each of the eight time periods.

Reject H₀. There is not convincing evidence to conclude that fatal accidents are not equally likely to occur in each of the eight time periods.   

Fail to reject H₀. There is not convincing evidence to conclude that fatal accidents are not equally likely to occur in each of the eight time periods.

Reject H₀. There is convincing evidence to conclude that fatal accidents are not equally likely to occur in each of the eight time periods.

Answer 1

Given:

A particular report included the following table classifying 819 fatal bicycle accidents according to time of day the accident occurred.

Time of Day	Number of Accidents
Midnight to 3 a.m.	47
3 a.m. to 6 a.m.	52
6 a.m. to 9 a.m.	87
9 a.m. to Noon	72
Noon to 3 p.m.	79
3 p.m. to 6 p.m.	157
6 p.m. to 9 p.m.	191
9 p.m. to Midnight	134

(a) Let p₁, p₂, p₃, p₄, p₅, p₆, p₇, and p₈ be the proportions of accidents occurring in the eight different time periods.

The null and alternative hypotheses is

H₀: p₁ = p₂ = p₃ = p₄ = p₅ = p₆= p₇ = p₈ = 0.125
H_a: H₀ is not true.

Answer - option B

b) Expected frequecy = Total frequency (n) × p

Chi-square goodness of fit test :

Proportion, P	Observed frequency, O	Expected frequency, E	(O-E)^2/E
1/8	47	819*1/8 = 102.375	29.9525
1/8	52	102.375	24.7877
1/8	87	102.375	2.3091
1/8	72	102.375	9.0121
1/8	79	102.375	5.3371
1/8	157	102.375	29.1467
1/8	191	102.375	76.7218
1/8	134	102.375	9.7694
	n = 819		187.037

Test statistics:

2 = (O-E)^2/E = 187.037

Degree of freedom, f = k-1 = 8-1 = 7

P-value:

P-value corresponding to 2 = 187.037 with df = 7 is 0.0000

P-value = 0.0000 .....(from chi square table)

Since P-value is less than significance level, = 0.05, we reject null hypothesis.

c) Conclusion:

Reject H₀. There is convincing evidence to conclude that fatal accidents are not equally likely to occur in each of the eight time periods.

Answer - option D