Answer to Question #184418 in Statistics and Probability for tsepang

Question

Question 4 The SA Department of Education in its annual report stated that the tardiness rates (in days) of children differ between the inner city and the suburbs neighborhoods’ schools. Unconvinced with this assertion, you proceed to take random samples of 25 and 20 children in these neighborhoods, respectively. Your findings yielded the following. In the inner city neighborhood the mean and standard error of tardiness were 4 years and 1.8 years. For the suburbs, the tardiness mean rate is 5.2 years while its standard error is 1.2 years. • Which test would you use? (1 mark) • Is it a 1-tail or 2-tail test? (1 mark) • At α = 0.05 is there any significance difference between the variances of the two population? (7 marks). • At α = 0.05 is there any significance difference between the means of the two population? Suppose α = 0.01, would your answer change?

Answer 1

We will used t-test for two samples.

It is single tail test.

Let $H_o$ : $s_1^2=s_2^2$ There are not any significant difference between the variances of two popuation.

and alternative hypothesis-

$H_a:s_1^2\neq s_2^2$

As the two smples are are given, So we use t-test for two independent solution.

Using t-test for two samples-

The overall standard error-

$S=\dfrac{n_1s_1^2+n_2s_2^2}{n_1+n_2-2}=\dfrac{25\times 1.8+20\times 1.2}{25+20-2}=\dfrac{69}{43}=1.60$

$t=\dfrac{\bar{x_1}-\bar{x_2}}{S\sqrt{\dfrac{1}{n_1}+\dfrac{1}{n_2}}}=\dfrac{5.2-4}{1.6\sqrt{\dfrac{1}{20}+\dfrac{1}{25}}}=\dfrac{1.2\times \sqrt{500}}{1.6\times \sqrt{45}}=2.5$

The standrd value of t at $\alpha=0.05$ and degrre of freedom $(n_1+n_2-2)i.e.(20+25-2)=43 \text{ is } 1.659$

Conclusion: The calculated value of t is greater than the standard value so $H_o$ is rejected i.e.$s_1^2\neq s_2^2$ There is significant difference between the variances of the population.

At 0.01 level of significance, The standard value for t is 2.326 which is greater than the calculated value so Hypothesis is accepted i.e. $s_1^2=s_2^2.$ means There is not any significant difference between the variances of two popuation.