Question

A particular report included the following table classifying 716 fatal bicycle accidents according to time of day the accident occurred.

Time of Day	Number of Accidents
Midnight to 3 a.m.	36
3 a.m. to 6 a.m.	28
6 a.m. to 9 a.m.	65
9 a.m. to Noon	78
Noon to 3 p.m.	99
3 p.m. to 6 p.m.	128
6 p.m. to 9 p.m.	166
9 p.m. to Midnight	116

(a) Assume it is reasonable to regard the 716 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	=

(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	=

Answer 1

a)

	relative	observed	Expected	residual	Chi square
category	frequency(p)	Oi	Ei=total*p	R2i=(Oi-Ei)/√Ei	R2i=(Oi-Ei)2/Ei
1	0.125	36.0	89.50	-5.66	31.980
2	0.125	28.0	89.50	-6.50	42.260
3	0.125	65.0	89.50	-2.59	6.707
4	0.125	78.0	89.50	-1.22	1.478
5	0.125	99.0	89.50	1.00	1.008
6	0.125	128.0	89.50	4.07	16.561
7	0.125	166.0	89.50	8.09	65.388
8	0.125	116.0	89.50	2.80	7.846
total	1.000	716	716		173.2291
test statistic X2 =		173.2291
test statistic X2 =		173.23

p value =0.0000

b)

	relative	observed	Expected	residual	Chi square
category	frequency(p)	Oi	Ei=total*p	R2i=(Oi-Ei)/√Ei	R2i=(Oi-Ei)2/Ei
1	0.333	207.0	238.67	-2.05	4.202
2	0.667	509.0	477.33	1.45	2.101
total	1.000	716	716		6.3024
test statistic X2 =		6.30
p value =		0.012