A particular report included the following table classifying 713 fatal bicycle a
ID: 3263977 • Letter: A
Question
A particular report included the following table classifying 713 fatal bicycle accidents according to time of day the accident occurred.
(a) Assume it is reasonable to regard the 713 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 2 value to two decimal places, and round your P-value to three decimal places.)
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 = 0.05. Round your 2 value to two decimal places, and round your P-value to three decimal places.)
What can you conclude?
There is sufficient evidence to reject H0.There is insufficient evidence to reject H0.
Time of Day Number of Accidents Midnight to 3 a.m. 38 3 a.m. to 6 a.m. 29 6 a.m. to 9 a.m. 67 9 a.m. to Noon 77 Noon to 3 p.m. 97 3 p.m. to 6 p.m. 126 6 p.m. to 9 p.m. 164 9 p.m. to Midnight 115Explanation / Answer
a)
Time of Day
Observed
Expected
(E-O)2 / E
Midnight to 3 a.m.
38
89.125
29.32696
3 a.m. to 6 a.m.
29
89.125
40.56119
6 a.m. to 9 a.m.
67
89.125
5.492461
9 a.m. to Noon
77
89.125
1.649544
Noon to 3 p.m.
97
89.125
0.695827
3 p.m. to 6 p.m.
126
89.125
15.25684
6 p.m. to 9 p.m.
164
89.125
62.9034
9 p.m. to Midnight
115
89.125
7.512097
Chi square statistic
163.3983
Here, Chi- square statistic = (Expected – Observed)2 / Expected ~ 2n-1 , under H0
n = 8
Critical value = 20.05;7 = 14.06714
Since chi-square statistic > critical value, we reject the null hypothesis and conclude that fatal bicycle accidents are not equally likely to occur in each of the 3-hour time periods.
p-value = PH0 [2 > 163.3983]
= PH0 [27 > 163.3983]
= 6.177403e-32
< 0.05.
There is sufficient evidence to reject H0.
b)
Time of Day
Observed
Obs.Prop
Exp.Prop
(E-O)2 / E
Midnight to noon
211
0.295932679
0.333333
0.004196
Noon to Midnight
502
0.704067321
0.666667
0.002098
Chi square statistic
0.006295
Here, n = 2
Critical value = 20.05;1 = 3.841459
Since chi-square statistic < critical value, we accept the null hypothesis H0: p1 = 1/3, p2 = 2/3.
p-value = PH0 [2 >0.006295]
= PH0 [21 > 0.006295]
= 0.9367614
> 0.05.
There is insufficient evidence to reject H0.
Time of Day
Observed
Expected
(E-O)2 / E
Midnight to 3 a.m.
38
89.125
29.32696
3 a.m. to 6 a.m.
29
89.125
40.56119
6 a.m. to 9 a.m.
67
89.125
5.492461
9 a.m. to Noon
77
89.125
1.649544
Noon to 3 p.m.
97
89.125
0.695827
3 p.m. to 6 p.m.
126
89.125
15.25684
6 p.m. to 9 p.m.
164
89.125
62.9034
9 p.m. to Midnight
115
89.125
7.512097
Chi square statistic
163.3983
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