Answer: c. Yes, because –2.77 falls in the critical region
Step-by-step explanation:
Let
be the population mean.
As per given , we have to test hypothesis :
Since the alternative hypothesis is left-tailed , so test is a left-tailed test.
Also, population standard deviation is unknown , so we will perform a left tailed t-test.
Sample size : n= 48
Sample mean : ![\overline{x}=46](https://tex.z-dn.net/?f=%5Coverline%7Bx%7D%3D46)
sample standard deviation : s= 10
Test statistics : ![t=\dfrac{\overline{x}-\mu}{\dfrac{s}{\sqrt{n}}}](https://tex.z-dn.net/?f=t%3D%5Cdfrac%7B%5Coverline%7Bx%7D-%5Cmu%7D%7B%5Cdfrac%7Bs%7D%7B%5Csqrt%7Bn%7D%7D%7D)
![t=\dfrac{46-50}{\dfrac{10}{\sqrt{48}}}](https://tex.z-dn.net/?f=t%3D%5Cdfrac%7B46-50%7D%7B%5Cdfrac%7B10%7D%7B%5Csqrt%7B48%7D%7D%7D)
![t=\dfrac{-4}{\dfrac{10}{6.92820323028}}\approx-2.77](https://tex.z-dn.net/?f=t%3D%5Cdfrac%7B-4%7D%7B%5Cdfrac%7B10%7D%7B6.92820323028%7D%7D%5Capprox-2.77)
Degree of freedom : df = n-1 = 47
For significance level 0.025 and degree of freedom 47 , we have
Critical t-value for left-tailed test =
[Using student's t-distribution table]
Rejection region : we reject null hypothesis for t<-2.012
Decision : Since -2.77(calculated t- value)< -2.012(Critical value) , it means it falls under rejection region.
i.e. We reject the null hypothesis.
We have sufficient evidence to support the alternative hypothesis μ < 5.7 ounces .
i.e. The company have sufficient evidence at α = 0.025 to believe that these inspectors are slower than average.
Hence , the correct answer = c. Yes, because –2.77 falls in the critical region