A electronics manufacturer has developed a new type of remote control button that is designed to operate longer before failing to work consistently. A random sample of 23 of the new buttons is selected and each is tested in continuous operation until it fails to work consistently. The resulting lifetimes are found to have a sample mean of x¯ = 1274.2 hours and a sample standard deviation of s = 114. Independent tests reveal that the mean lifetime of the best remote control button on the market is 1210 hours. Conduct a hypothesis test to determine if the new button's mean lifetime exceeds 1210 hours. Round all calculated answers to four decimal places.

Answer:Claim :  if the new button's mean lifetime exceeds 1210 hours. [tex]H_0:\mu = 1210\\H_a:\mu > 1210[/tex]Sample mean = [tex]\bar{x}=1274.2[/tex]Sample standard deviation s = 114 n = 23Since n < 30 and sample standard deviation is given . So, we will use t test Formula : [tex]t=\frac{x-\mu}{\frac{s}{\sqrt{n}}}}[/tex]Substitute the values :[tex]t=\frac{1274.2-1210}{\frac{114}{\sqrt{23}}}[/tex][tex]t=2.7008[/tex]degree of freedom = n-1 = 23-1 =22confidence level = 95%Significance level = 5%[tex]t_{df,\frac{\alpha}{2}}=t_{22,\frac{0.05}{2}}=1.7170[/tex]t calculated > t criticalSo, we failed to accept null hypothesis Thus the new button's mean lifetime exceeds 1210 hours.