hypothesis testing highway lanes problem

Question

I have the following problem which I'm trying to solve:

We have a highway with four lanes. A research with 1000 drivers is made to check whether the drivers prefer the inner lanes. There are the following results:

Lane 1: 294,
Lane 2: 276,
Lane 3: 238,
Lane 4: 192.

Do the data give enough information to prove that the proportions all the lanes are different?

Test $p_1=p_2=p_3=p_4=0.25$ when $\alpha=0.05$.

My attempt:

My interpretation is that $H_0$ claims that the distribution is Normal distribution with $\mu=2.5$ and that I have to construct a two-tailed test with $Z_{0.025}=\pm1.96$.

My test-statistic is:

$t=\frac{(\bar X-\mu)\sqrt{n}}{s}=\frac{10(2.32800-2.5)}{s}=\frac{-1.72}{s}$, where s is the sample standard deviation.

Is my attempt alright, and if so, how do I find the sample standard deviation so that I test my hypothesis?

Answer 1

Use Chi-Squared Test with the statistic

$\chi^2 = \sum_{1}^{4} \frac{(O_i-E_i)^2}{E_i} = A$

with $O_i$ is observed proportion and $E_i = .25$

Test with compute $\chi^2_{critical}$ P(X^2>=A) in a chi-square distribution with df being 3 and compare it with p = 0.05 and reject the nukl hypothesis that $H_0: p_1=p_2=p_3=p_4$ if $\chi^2_{critical}$ is greater than 0.05.

Thanks

Answer 2

You need to use test for multiple proportions e.g the Marascuillo-test.

See: http://www.itl.nist.gov/div898/handbook/prc/section4/prc474.htm