A particular report included the following table classifying 711 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. 29
6 a.m. to 9 a.m. 64
9 a.m. to Noon 78
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 113
(a) Assume it is reasonable to regard the 711 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 =

To answer part (a) of the question, we need to perform a chi-square test to determine if the fatal bicycle accidents are equally likely to occur in each 3-hour time period.

Step 1: Set up hypotheses
The null hypothesis (H0) is that fatal bicycle accidents are equally likely to occur in each 3-hour time period.
The alternative hypothesis (HA) is that fatal bicycle accidents are not equally likely to occur in each 3-hour time period.

H0: The proportions of accidents in each time period are equal.
HA: The proportions of accidents in each time period are not equal.

Step 2: Define the significance level
The significance level is given as 0.05, which means we are willing to accept a 5% chance of making a Type I error (rejecting the null hypothesis when it is true).

Step 3: Calculate the test statistic
To calculate the test statistic, we use the formula:
χ2 = Σ((Observed - Expected)^2 / Expected)

First, we need to calculate the expected number of accidents for each time period. The expected number is given by (total number of accidents / total number of time periods) multiplied by the number of time periods in each category.

Expected number of accidents for each time period:
Midnight to 3 a.m.: (711/8) * 38 = 267.375
3 a.m. to 6 a.m.: (711/8) * 29 = 222.375
6 a.m. to 9 a.m.: (711/8) * 64 = 571.5
9 a.m. to Noon: (711/8) * 78 = 679.875
Noon to 3 p.m.: (711/8) * 98 = 839.25
3 p.m. to 6 p.m.: (711/8) * 127 = 1156.875
6 p.m. to 9 p.m.: (711/8) * 164 = 1325.25
9 p.m. to Midnight: (711/8) * 113 = 969.375

Next, we calculate the test statistic using the formula:
χ2 = Σ((Observed - Expected)^2 / Expected)
χ2 = ((38 - 267.375)^2 / 267.375) + ((29 - 222.375)^2 / 222.375) + ((64 - 571.5)^2 / 571.5) + ((78 - 679.875)^2 / 679.875) + ((98 - 839.25)^2 / 839.25) + ((127 - 1156.875)^2 / 1156.875) + ((164 - 1325.25)^2 / 1325.25) + ((113 - 969.375)^2 / 969.375)

Calculate this sum to obtain the χ2 value.

Step 4: Find the degrees of freedom
The degrees of freedom (df) for a chi-square test in this case is (number of categories - 1).

In this case, we have 8 categories, so df = 8 - 1 = 7.

Step 5: Find the p-value
Use the χ2 value and degrees of freedom to find the p-value in a chi-square distribution table or by using statistical software.

The p-value is the probability of observing a test statistic as extreme as, or more extreme than, the one calculated, assuming the null hypothesis is true.

Step 6: Make a decision
If the p-value is less than the significance level (0.05), we reject the null hypothesis. If the p-value is greater than or equal to the significance level, we fail to reject the null hypothesis.

Finally, report the χ2 value and the p-value.

For part (b) of the question, we only need to consider two time categories: midnight to noon and noon to midnight.

Repeat steps 1-6 using the new expected values and observed values for these two categories, with the null hypothesis being 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.

The chi-square test statistic and p-value obtained will determine if the given data support or are consistent with the proposed hypothesis.

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