Question #334218

The CEO of a battery manufacturing company claimed that their batteries would last an average of 280 hours under normal use. A researcher randomly selected 20 batteries from the production line and tested them. The tested batteries had a mean life span of 250 hours with a standard deviation of 40 hours. Do we have enough evidence to suggest that the claim of an average of 280 hours is false?


1
Expert's answer
2022-04-27T17:05:51-0400

The following null and alternative hypotheses need to be tested:

H0:μ=280H_0:\mu=280

H1:μ280H_1:\mu\not=280

This corresponds to a two-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=201=19df=n-1=20-1=19 degrees of freedom, and the critical value for a two-tailed test is tc=2.093024t_c = 2.093024

The rejection region for this two-tailed test isR={t:t>2.023024}.R = \{t: |t| > 2.023024\}.

The t-statistic is computed as follows:


t=xˉμs/n=25028040/203.354102t=\dfrac{\bar{x}-\mu}{s/\sqrt{n}}=\dfrac{250-280}{40/\sqrt{20}}\approx-3.354102

Since it is observed that t=3.354102>2.093024=tc,|t| = 3.354102 >2.093024= t_c, it is then concluded that the null hypothesis is rejected.

Using the P-value approach:

The p-value for two-tailed, df=19df=19 degrees of freedom, t=3.354102t=-3.354102 is p=0.003333,p = 0.003333, and since p=0.003333<0.05=α,p=0.003333<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 280, at the α=0.05\alpha = 0.05 significance level.


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