Lifetime of $1 Bills The average lifetime of circulated $1 bills is 18 months. A researcher believes that the average lifetime is not 18 months. He researched the lifetime of 50 $1 bills and found the average lifetime was 18.8 months. The population standard deviation is 2.8 months. At α = 0.02, can it be concluded that the average lifetime of a circulated $1 bill differs from 18 months?

Answer with explanation:Let [tex]\mu[/tex] be the population mean lifetime of circulated $1 bills.By considering the given information , we have :-[tex]H_0:\mu=18\\\\H_a:\mu\neq18[/tex]Since the alternative hypotheses is two tailed so the test is a two tailed test.We assume that the lifetime of circulated $1 bills is normally distributed.Given : Sample size :  n=50 , which is greater than 30 .It means the sample is large so we use z-test.Sample mean : [tex]\overline{x}=18.8[/tex]Standard deviation : [tex]\sigma=2.8[/tex]Test statistic for population mean :-[tex]z=\dfrac{\overline{x}-\mu}{\dfrac{\sigma}{\sqrt{n}}}[/tex][tex]\Rightarrow\ z=\dfrac{18.8-18}{\dfrac{2.8}{\sqrt{50}}}\approx2.02[/tex]The p-value= [tex]2P(z>2.02)=0.0433834[/tex]Since the p-value (0.0433834) is greater than the significance level (0.02) , so we do not reject the null hypothesis.Hence, we conclude that we do not have enough evidence to support the alternative hypothesis that the average lifetime of a circulated $1 bill differs from 18 months.