Question

The amount of time runners take to run a marathon is believed to be uniformly distributed after running for four hours. The following is the observed number of runners and the length of time it takes to cross the finish line. Carry out the Chi Square Goodness of fit test.

Hypothesis:

Calculation: Find each expected count and find each chi square value, total value and pvalue.

	4-4.5 hours	4.5-5 hours	5-5.5 hours	5.5-6 hours	6-6.5 hours	6.5 + hours	Total
Observed	50	70	80	75	70	60	405
Expected
Chi Square

Conclusion:

Answer 1

Solution :

Null and alternative hypotheses :

The null and alternative hypotheses would be as follows :

H_{​​​​​​0} : The amount of time runners take to run a marathon follows uniform distribution.

H_{​​​​​​1} : The amount of time runners take to run a marathon do not follows uniform distribution.

Test statistic :

To test the hypothesis we shall use chi-square test of goodness of fit. The test statistic is given as follows :

Where, O_{​​​​​​i}​​​​​'s are observed frequencies and e_{​​​​​​i}​​​​​'s are expected frequencies.

We are given the observed frequencies. Now we need to obtain expected frequencies.

Under H_{​​​​​​0} we have hypothesized that run time is uniformly distributed.

Number of categories = 6

Total observed frequency = 405

Hence, frequency for each of the categories would be equal to 405 × (1/6) = 67.5

From the above table we get,

On rounding to four decimal places we get,

P-value :

The p-value is given as follows :

P-value = P(χ^{​​​​​​2} > value of the test statistic)

P-value = P(χ² > 8.7037)

P-value = 0.1215

The p-value is 0.1215.

Decision :

In our question the significance level is not given. Generally significance level 0.05 or 0.01 is used. We shall use significance level 0.05

Significance level = 0.05

P-value = 0.1215

(0.1215 > 0.05)

Since, p-value is greater than the significance level of 0.05, therefore we shall be fail to reject the null hypothesis (H_{​​​​​​0}) at 0.05 significance level.

Conclusion :

At 0.05 significance level, there is not sufficient evidence to reject the claim that the amount of time runners take to run a marathon is believed to be uniformly distributed after running for four hours.

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