Question

A particular report included the following table classifying 712 fatal bicycle accidents according to time of day the accident occurred.

Time of Day	Number of Accidents
Midnight to 3 a.m.	38
3 a.m. to 6 a.m.	27
6 a.m. to 9 a.m.	66
9 a.m. to Noon	77
Noon to 3 p.m.	98
3 p.m. to 6 p.m.	127
6 p.m. to 9 p.m.	164
9 p.m. to Midnight	115

(a) Assume it is reasonable to regard the 712 bicycle accidents summarized in the table as a random sample of fatal bicycle accidents in that year. 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 .05. (Round your χ² value to two decimal places, and round your P-value to three decimal places.)

χ2	=
P-value	=

What can you conclude?

There is sufficient evidence to reject H₀.

There is insufficient evidence to reject H₀.    

(b) Suppose a safety office proposes that bicycle fatalities are twice as likely to occur between noon and midnight as during midnight to noon and suggests the following hypothesis: H₀: p₁ = 1/3, p₂ = 2/3, where p₁ is the proportion of accidents occurring between midnight and noon and p₂ is the proportion occurring between noon and midnight. Do the given data provide evidence against this hypothesis, or are the data consistent with it? Justify your answer with an appropriate test. (Hint: Use the data to construct a one-way table with just two time categories. Use α = 0.05. Round your χ² value to two decimal places, and round your P-value to three decimal places.)

χ2	=
P-value	=

What can you conclude?

There is sufficient evidence to reject H₀.

There is insufficient evidence to reject H₀.    

Answer 1

a) H0: fatal bicycle accidents are equally likely to occur in each of the 3-hour time periods used to construct the table

H1: fatal bicycle accidents are not equally likely to occur in each of the 3-hour time periods used to construct the table

From the given data

	Observed	Expected
S.NO.	Freq (Oi)	Freq Ei	(Oi-Ei)^2 /Ei
1	38	89	29.2247191
2	27	89	43.1910112
3	66	89	5.94382022
4	77	89	1.61797753
5	98	89	0.91011236
6	127	89	16.2247191
7	164	89	63.2022472
8	115	89	7.59550562
Total:	712	712	167.910112

The expected frequencies are calculated as follows i.e. Ei = 712 / 8 = 89 for all i, i = 1,2,3,...8

Test Statistic, X^2: 167.9101
P-Value: 0.0000

since P-value < alpha 0.05 so we reject H0

Answer: There is sufficient evidence to reject H₀.