Question #334844

A sociologist believes that it costs more than Php90,000 to raise a child from birth to age one. A random sample of 49 families, each with a child is selected to see if this figure is correct. The average expenses for these families reveal a mean of Php92,000 with a standard deviation of Php4,500. 


1
Expert's answer
2022-04-29T09:44:14-0400

The following null and alternative hypotheses need to be tested:

H0:μ90000H_0:\mu\le90000

Ha:μ>90000H_a:\mu>90000

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=0.05,\alpha = 0.05, df=n1=491=48df=n-1=49-1=48 degrees of freedom, and the critical value for a right-tailed test is tc=2.010635.t_c = 2.010635.

The rejection region for this right-tailed test isR={t:t>2.010635}.R = \{t: t > 2.010635\}.

The t-statistic is computed as follows:


t=xˉμs/n=92000900004500/493.111111t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{92000-90000}{4500/\sqrt{49}}\approx3.111111

Since it is observed that t=3.111111>2.010635=tc,t =3.111111> 2.010635= t_c , it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for right-tailed, df=48df=48 degrees of freedom, t=3.111111t=3.111111 is p=0.001567,p=0.001567, and since p=0.001567<0.05=α,p=0.001567<0.05=\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 90000, at the α=0.05\alpha = 0.05 significance level.



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