Question #345093

2. A simple random sample of 15 people from a certain population has a


mean age of 35 with a standard deviation of 20. Can we conclude that


the mean age of the population is younger than 35? Let alpha = .05.

1
Expert's answer
2022-05-26T17:10:36-0400

The following null and alternative hypotheses need to be tested:

H0:μ35H_0:\mu\ge35

H1:μ<35H_1:\mu<35

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 α=0.05,\alpha = 0.05, df=n1=14df=n-1=14 and the critical value for a left-tailed test is tc=1.76131.t_c =-1.76131.

The rejection region for this left-tailed test is R={t:t<1.76131}.R = \{t:t<-1.76131\}.

The t-statistic is computed as follows:


t=xˉμs/n=353520/15=0t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{35-35}{20/\sqrt{15}}=0

Since it is observed that t=0>1.76131=tc,t=0>-1.76131=t_c, it is then concluded that the null hypothesis is not rejected.

Using the P-value approach:

The p-value for left-tailed, df=14df=14 degrees of freedom, t=0t=0 is p=0.5,p=0.5, and since p=0.5>0.05=α,p=0.5>0.05=\alpha, it is concluded that the null hypothesis is notrejected.

Therefore, there is not enough evidence to claim that the population mean μ\mu

is less than 35, at the α=0.05\alpha = 0.05 significance level.



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