2. A random sample of 40 boys showed a mean weight of 120 lbs. with a standard deviation
of 12lbs. The standard weight for their age category is about 130lbs. Test the hypotheis that the selected boys are significantly lighter than the standard weight at0.01 level of siginificance.
The following null and alternative hypotheses need to be tested:
"H_0:\\mu\\ge130"
"H_a:\\mu<130"
This corresponds to a left-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 "\\alpha = 0.01," "df=n-1=39" degrees of freedom, and the critical value for a left-tailed test is "t_c =-2.425841."The rejection region for this left-tailed test is "R = \\{t:t<-2.425841\\}."
The t-statistic is computed as follows:
Since it is observed that "t= -5.270463<-2.425841=t_c," it is then concluded that the null hypothesis is rejected.
Using the P-value approach:
The p-value for left-tailed "df=39" degrees of freedom, "t=-5.270463" is "p=0.000003," and since "p=0.000003<0.01=\\alpha," it is concluded that the null hypothesis is rejected.
Therefore, there is enough evidence to claim that the population mean "\\mu" is less than 130, at the "\\alpha = 0.01" significance level.
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