Question #337689

Average Senior High School annual cost of tuition fee for all private schools last year was P46,300. A random sample of costs this year for 45 private schools indicated that the sample mean was P47,800 and a sample standard deviation was P5,600. At the 0.10 level of significance, is there sufficient evidence to conclude that the cost has increased?


1
Expert's answer
2022-05-06T13:06:03-0400

The following null and alternative hypotheses need to be tested:

H0:μ46300H_0:\mu\le 46300

Ha:μ>46300H_a:\mu>46300

This corresponds to a right-tailed test, for which a t-test for one mean, with unknown population standard deviation, using the sample standard deviation, will be used.

Based on the information provided, the significance level is α=0.10,\alpha = 0.10, df=n1=44df=n-1=44 degrees of fredom, and the critical value for a right-tailed test is tc=1.30109.t_c =1.30109.

The rejection region for this right-tailed test is R={t:t>1.30109}.R = \{t: t >1.30109\}.

The t-statistic is computed as follows:


t=xˉμs/n=47800463005600/451.79684t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{47800-46300}{5600/\sqrt{45}}\approx1.79684

Since it is observed that t=1.79684>1.30109=tc,t = 1.79684> 1.30109=t_c, it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for right-tailed, df=44df=44 degrees of freedom, t=1.79684t=1.79684 is p=0.039614,p = 0.039614, and since p=0.039614<0.10=α,p= 0.039614<0.10=\alpha, it is concluded that the null hypothesis is rejected.

Therefore, there is enough evidence to claim that the population mean μ\mu

is greater than 46300, at the α=0.10\alpha = 0.10 significance level.



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