Question

In Exercises 5–20, conduct the hypothesis test and provide the test statistic and the P-value and, or critical value, and state the conclusion.

Police Calls Repeat Exercise 11 using these observed frequencies for police calls received during the month of March: Monday (208); Tuesday (224); Wednesday (246); Thursday (173); Friday (210); Saturday (236); Sunday (154). What is a fundamental error with this analysis?

Short answer

There is enough evidence to conclude that the police calls do not occur equally frequently on the different days of the week.

Because March has 31 days, there is a fundamental inaccuracy in the presented observations: not all of the days occur an equal number of times. Some days will appear four times in a month, while others will appear five times.

As a result, the number of calls on days that occur 5 times will be higher than on other days. As a result, the provided analysis does not appear to be very applicable.

Step-by-step solution

Given information

The observed frequencies of the police calls on the 7 days of the week of in March are recorded.

It is expected that the calls occur equally frequently on the 7 days of the week.

Check the requirements

Assume the recordings are taken from randomly selected experimental units.

Let the serial numbers from 1 to 7 denote the seven days of the week starting from Monday.

Let O denote the observed frequencies of the police calls.

The following values are obtained:

\(\begin{aligned}{l}{O_1} = 208\\{O_2} = 224\\{O_3} = 246\\{O_4} = 173\\{O_5} = 210\\{O_6} = 236\\{O_7} = 154\end{aligned}\)

The sum of all observed frequencies is computed below:

\(\begin{aligned}{c}n = 208 + 224 + .... + 154\\ = 1451\end{aligned}\)

Let E denote the expected frequencies. It is given that the days are expected to occur with the same frequency on each day.

The expected frequencies for each of the 7 days are equal to:

\(\begin{aligned}{c}E = \frac{{1451}}{7}\\ = 207.2857\end{aligned}\)

As the expected frequencies are all greater than 5, the requirements of the test are satisfied.

State the hypotheses

The null hypothesis for conducting the given test is as follows:

The police calls occur equally frequently on the different days of the week.

The alternative hypothesis is as follows:

The police calls do not occur equally frequently on the different days of the week.

Compute the test statistic

The table below shows the necessary calculations:

Day	O	E	\(\left( {O - E} \right)\)	\({\left( {O - E} \right)^2}\)	\(\frac{{{{\left( {O - E} \right)}^2}}}{E}\)
Monday	208	207.2857	0.714286	0.510204	0.002461
Tuesday	224	207.2857	16.71429	279.3673	1.34774
Wednesday	246	207.2857	38.71429	1498.796	7.23058
Thursday	173	207.2857	-34.2857	1175.51	5.670966
Friday	210	207.2857	2.714286	7.367347	0.035542
Saturday	236	207.2857	28.71429	824.5102	3.977651
Sunday	154	207.2857	-53.2857	2839.367	13.69784

The value of the test statistic is equal to:

\[\begin{aligned}{c}{\chi ^2} = \sum {\frac{{{{\left( {O - E} \right)}^2}}}{E}} \;\\ = 0.002461 + 1.34774 + ... + 13.69784\\ = 31.96278\end{aligned}\]

Thus,\({\chi ^2} = 31.963\).

Let k be the number of days which are 7.

The degrees of freedom for\({\chi ^2}\)is computed below:

\(\begin{aligned}{c}df = k - 1\\ = 7 - 1\\ = 6\end{aligned}\)

The critical value of\({\chi ^2}\)at\(\alpha = 0.01\)with 6 degrees of freedom is equal to 16.812.

The p-value is equal to,

\(\begin{aligned}{c}p - value = P\left( {{\chi ^2} > 31.963} \right)\\ = 0.000\end{aligned}\)

Since the test statistic value is greater than the critical value and the p-value is less than 0.01, the null hypothesis is rejected.

State the conclusion

There is enough evidence to conclude that the police calls do not occur evenly on the different days of the week.

Error in the analysis

A fundamental error in the given observations is that not all the days occur an equal number of times because March has 31 days. Some days will occur 4 times while others will occur 5 times in the month.

Thus, the number of calls on the days that occur 5 times will be more as compared to the other days.

So, the given analysis does not seem appropriate.