Question #349921

The following output from MINITAB presents the results of a hypothesis test. Test of mu=49 vs. not=49, The assumed standard deviation 10.4, N 58, Mean 46.32, SE 1.412088, 95%CI (43.5523077, 49.0878923, Z -1.962526, P 0,049701, Do you reject H0 at the


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Expert's answer
2022-06-13T17:54:06-0400

The following null and alternative hypotheses need to be tested:

H0:μ=49H_0:\mu=49

H1:μ49H_1:\mu\not=49

This corresponds to a two-tailed test, for which a z-test for one mean, with known population standard deviation will be used.

Based on the information provided, the significance level is α=0.05,\alpha = 0.05, and the critical value for a two-tailed test is zc=1.96.z_c = 1.96.

The rejection region for this two-tailed test is R={z:z>1.96}.R = \{z:|z|>1.96\}.

The z-statistic is computed as follows:


z=xˉμσ/n=46.324910.4/58=1.962526z=\dfrac{\bar{x}-\mu}{\sigma/\sqrt{n}}=\dfrac{46.32-49}{10.4/\sqrt{58}}=-1.962526

Since it is observed that z=1.962526>1.96=zc,|z|=1.962526>1.96=z_c, it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value is p=2P(z<1.962526)=0.049701,p=2P(z<-1.962526)=0.049701, and since p=0.049701<0.05=α,p=0.049701<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 different than 49, at the α=0.05\alpha = 0.05 significance level.


SE=σn=10.4581.365587SE=\dfrac{\sigma}{\sqrt{n}}=\dfrac{10.4}{\sqrt{58}}\approx1.365587

The corresponding 95% confidence interval is


CI=(xˉzc×SE,xˉ+zc×SE)CI=(\bar{x}-z_c\times SE, \bar{x}+z_c\times SE)

=(46.321.96×1.365587,=(46.32-1.96\times 1.365587,

46.32+1.96×1.365587)46.32+1.96\times 1.365587)

=(43.6434497,48.9965503)=(43.6434497, 48.9965503)

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