Question #350898

A researcher wishes to determine the performance level of students with pre-school education and with no preschool education. Fifteen senior high school students with pre-school education and fifteen senior high school students with no pre-school education were randomly selected and the results are tabulated below.

Education

 (mean)

 (standard deviation)

 (sample size)

With pre-school education

89.4

3.25

15

With no pre-school education

85.2

2.85

15

Is it safe to conclude that those with pre-school education have a better performance than those with no pre-school education? Use 99% level of confidence



1
Expert's answer
2022-06-20T18:02:26-0400

Testing for Equality of Variances

A F-test is used to test for the equality of variances. The following F-ratio is obtained:


F=s12s22=3.2522.852=1.3004F=\dfrac{s_1^2}{s_2^2}=\dfrac{3.25^2}{2.85^2}=1.3004

The critical values for α=0.01,df1n1=1=14\alpha=0.01, df_1-n_1=1=14degrees o freedom, df2=n21=14df_2=n_2-1=14 degrees o freedom, are FL=0.2326F_L = 0.2326 and FU=4.2993,F_U = 4.2993, and since F=1.3004,F = 1.3004, then the null hypothesis of equal variances is not rejected.

The following null and alternative hypotheses need to be tested:

H0:μ1μ2H_0:\mu_1\le\mu_2

Ha:μ1>μ2H_a:\mu_1>\mu_2

This corresponds to a right-tailed test, for which a t-test for two population means, with two independent samples, with unknown population standard deviations will be used.

The degrees of freedom are computed as follows, assuming that the population variances are equal:


df=n1+n2=15+152=28df=n_1+n_2=15+15-2=28

Based on the information provided, the significance level is α=0.01,\alpha = 0.01,

and the degrees of freedom are df=28df = 28 degrees of freedom.

Hence, it is found that the critical value for this right-tailed test is tc=2.46714,t_c = 2.46714, for α=0.01\alpha = 0.01 and df=28.df = 28.

The rejection region for this right-tailed test is R={t:t>2.46714}.R = \{t: t > 2.46714\}.

Since it is assumed that the population variances are equal, the t-statistic is computed as follows:


t=Xˉ1Xˉ2(n11)s12+(n21)s22n1+n22(1n1+1n2)t=\dfrac{\bar{X}_1-\bar{X}_2}{\sqrt{\dfrac{(n_1-1)s_1^2+(n_2-1)s_2^2}{n_1+n_2-2}(\dfrac{1}{n_1}+\dfrac{1}{n_2})}}=89.485.2(151)(3.25)2+(151)(2.85)215+152(115+115)=\dfrac{89.4-85.2}{\sqrt{\dfrac{(15-1)(3.25)^2+(15-1)(2.85)^2}{15+15-2}(\dfrac{1}{15}+\dfrac{1}{15})}}3.7631\approx3.7631

Since it is observed that t=3.7631>tc=2.46714,t = 3.7631> t_c = 2.46714, it is then concluded that the null hypothesis is rejected.

Using the P-value approach: The p-value for right-tailed test, df=28df=28 degrees of freedom, t=3.7631,t=3.7631, is p=0.000395,p=0.000395, and since p=0.000395<0.01=α,p = 0.000395< 0.01=\alpha, it is concluded that the null hypothes is rejected.

Therefore, there is enough evidence to claim that the population mean μ1\mu_1

is greater than μ2,\mu_2, at the α=0.01\alpha = 0.01 significance level.


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