Question

A town-planning sub-committee in Tshwane wanted to know if there is any difference in the mean travelling time to work of car and Train commuters. They there carried out a survey amongst car and bus commuters and with the following sample statistics:
Car Commuters
Train Commuters
X1= 29.6 min
X2 = 25.2 min
S1= 5.2 min
S2= 2.8 min
N1=22 drivers
N2=36 passengers
4.1 Test the hypothesis at the 5% significance level that it takes car commuters to get to work earlier than Train commuters.

Answer 1

Answer on Question #40842 – Math – Statistics and Probability

A town-planning sub-committee in Tshwane wanted to know if there is any difference in the mean travelling time to work of car and Train commuters. They there carried out a survey amongst car and bus commuters and with the following sample statistics:

Car Commuters

Train Commuters

$$X _ {1} = 29.6 \text{ min}$$ $$X _ {2} = 25.2 \text{ min}$$ $$S _ {1} = 5.2 \text{ min}$$ $$S _ {2} = 2.8 \text{ min}$$ $$N _ {1} = 22 \text{ drivers}$$ $$N _ {2} = 36 \text{ passengers}$$

4.1 Test the hypothesis at the 5% significance level that it takes car commuters to get to work earlier than Train commuters.

Solution

The population variances are not assumed to be equal. The population variances are unknown, $N_{1}$ and $N_{1}$ are small, so we need to use t-test.

$$H _ {0}: \mu_ {1} = \mu_ {2}; H _ {\alpha}: \mu_ {1} < \mu_ {2}.$$

The test statistic is

$$t = \frac {X _ {1} - X _ {2}}{\sqrt {\frac {S _ {1} ^ {2}}{N _ {1}} + \frac {S _ {2} ^ {2}}{N _ {2}}}} = \frac {29.6 - 25.2}{\sqrt {\frac {5.2 ^ {2}}{22} + \frac {2.8 ^ {2}}{36}}} = 3.66.$$

Number of degrees of freedom is smaller of $N_{1} - 1$ and $N_{2} - 1$: $df = 22 - 1 = 21$.

For $df = 21$, the tabled value is $t_{\alpha} = t_{0.05} = 1.72$. The rejection region: $t < -1.72$. The observed value of the test statistic is $t = 3.66 > -1.72$. Therefore, at the $\alpha = 0.05$ level of significance the null hypothesis $H_0$ is accepted and we conclude that the claim (it takes car commuters to get to work earlier than Train commuters) is not substantiated by the data.

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