Answer to Question #90820 in Statistics and Probability for leonardo barrios

Question #90820
A survey of 100 similar-sized hospitals revealed a mean daily census in the pediatrics service of 27with a standard deviation of 6.5. Do these data provide sufficient evidence to indicate that thepopulation mean is greater than 25? Let a: 0,05
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Expert's answer
2019-06-27T13:34:55-0400

Solution.  We define null and alternative hypothesis:


H0:μ25,H_0: \mu \leq 25,

Ha:μ>25.H_a: \mu >25.

Next we find a critical value. We will use a right-tailed test, for which a t-test for one mean, with unknown population standard deviation will be used.

The significance level is α=0.05\alpha = 0.05, and the critical value for a right-tailed test is 

tc=t(α,n1)=t(0.05,99)=1.66.t_c = t(\alpha, n-1)= t(0.05, 99) = 1.66.

The rejection region for this right-tailed test is t>1.66t > 1.66 .

The t-statistic is computed as follows:


t=xˉμsn=27256.5100=26.510=3.077.t = \frac{\bar{x} - \mu}{s}\cdot \sqrt{n} = \frac{27 - 25}{6.5}\cdot \sqrt{100} = \frac{2}{6.5}\cdot 10 = 3.077.

Since it is observed that t=3.077>tc=1.66t = 3.077 > t_c = 1.66, it is then concluded that the null hypothesis is rejected.

Conclusion:

It is concluded that the null hypothesis H0H_0  is rejected. Therefore, there is enough evidence to claim that the population mean μ\mu  is greater than 25, at the 0.05 significance level.




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