Question

A report classified fatal bicycle accidents according to the month in which the accident occurred, resulting in the accompanying table.

Month	Number of Accidents
January	40
February	30
March	45
April	59
May	76
June	72
July	100
August	87
September	66
October	64
November	40
December	38

(a) Use the given data to test the null hypothesis H₀: p₁ = 1/12, p₂= 1/12, .., p₁₂= 1/12, where p1 is the proportion of fatal bicycle accidents that occur in January, p2 is the proportion for February, and so on. Use a significance level of 0.01.

Calculate the test statistic. (Round your answer to two decimal places.)

χ² =

What is the P-value for the test? (Use a statistical computer package to calculate the P-value. Round your answer to four decimal places.)

P-value =

What can you conclude?
Reject H₀. There is not enough evidence to conclude that fatal bicycle accidents are not equally likely to occur in each of the months.
Do not reject H₀. There is not enough evidence to conclude that fatal bicycle accidents are not equally likely to occur in each of the months.
Do not reject H₀. There is convincing evidence to conclude that fatal bicycle accidents are not equally likely to occur in each of the months.
Reject H₀. There is convincing evidence to conclude that fatal bicycle accidents are not equally likely to occur in each of the months.

(b) The null hypothesis in part (a) specifies that fatal accidents were equally likely to occur in any of the 12 months. But not all months have the same number of days. What null and alternative hypotheses would you test to determine if some months are riskier than others if you wanted to take differing month lengths into account? (Assume this data was collected during a leap year, with 366 days.) (Enter your probabilities as fractions.)

Identify the null hypothesis by specifying the proportions of accidents we expect to occur in each month if the length of the month is taken into account. (Enter your probabilities as fractions.)

p₁= p₂= p₃=   p₄= p₅= p₆=   p₇= p₈= p₉= p₁₀= p₁₁= p₁₂=

Identify the correct alternative hypothesis.

H₀ is true. None of the proportions is not correctly specified under H₀.
H₀ is not true. At least one of the proportions is not correctly specified under H₀.
H₀ is true. At least one of the proportions is not correctly specified under H₀.
H₀ is not true. None of the proportions is correctly specified under H₀.

(c) Test the hypotheses proposed in part (b) using a 0.05 significance level.

Calculate the test statistic. (Round your answer to two decimal places.)

χ² =

What is the P-value for the test? (Use a statistical computer package to calculate the P-value. Round your answer to four decimal places.)

P-value =

What can you conclude?
Reject H₀. There is not enough evidence to conclude that fatal bicycle accidents do not occur in the twelve months in proportion to the lengths of the months.
Do not reject H₀. There is convincing evidence to conclude that fatal bicycle accidents do not occur in the twelve months in proportion to the lengths of the months.
Do not reject H₀. There is not enough evidence to conclude that fatal bicycle accidents do not occur in the twelve months in proportion to the lengths of the months.
Reject H₀. There is convincing evidence to conclude that fatal bicycle accidents do not occur in the twelve months in proportion to the lengths of the months.

Answer 1

1)applying chi square tesT:

	relative	observed	Expected	residual	Chi square
category	frequency(p)	Oi	Ei=total*p	R2i=(Oi-Ei)/√Ei	R2i=(Oi-Ei)2/Ei
1	1/12	40.000	59.750	-2.56	6.528
2	1/12	30.000	59.750	-3.85	14.813
3	1/12	45.000	59.750	-1.91	3.641
4	1/12	59.000	59.750	-0.10	0.009
5	1/12	76.000	59.750	2.10	4.419
6	1/12	72.000	59.750	1.58	2.512
7	1/12	100.000	59.750	5.21	27.114
8	1/12	87.000	59.750	3.53	12.428
9	1/12	66.000	59.750	0.81	0.654
10	1/12	64.000	59.750	0.55	0.302
11	1/12	40.000	59.750	-2.56	6.528
12	1/12	38.000	59.750	-2.81	7.917
total	1.000	717	717		86.8661

X² =86.87

p value =0.0000

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

b)

H₀ is not true. At least one of the proportions is not correctly specified under H₀.

Applying chi square test:

	relative	observed	Expected	residual	Chi square
category	frequency(p)	Oi	Ei=total*p	R2i=(Oi-Ei)/√Ei	R2i=(Oi-Ei)2/Ei
1	31/366	40.000	60.730	-2.66	7.076
2	29/366	30.000	56.811	-3.56	12.653
3	31/366	45.000	60.730	-2.02	4.074
4	5/61	59.000	58.770	0.03	0.001
5	31/366	76.000	60.730	1.96	3.840
6	5/61	72.000	58.770	1.73	2.978
7	31/366	100.000	60.730	5.04	25.394
8	31/366	87.000	60.730	3.37	11.364
9	5/61	66.000	58.770	0.94	0.889
10	31/366	64.000	60.730	0.42	0.176
11	5/61	40.000	58.770	-2.45	5.995
12	31/366	38.000	60.730	-2.92	8.507
total	1.000	717	717		82.9478

X² =82.95

p value =0.0000

Reject H₀. There is convincing evidence to conclude that fatal bicycle accidents do not occur in the twelve months in proportion to the lengths of the months.