A survey of Davao Region finds the average commute time of employees on one way is 30 minutes. The Digos Chamber of Commerce feels that in their city is greater and want to publicize this. They randomly select 28 commuters and find the average is 35 minutes with a standard deviation of 6 minutes. At 𝛼 = 0.05, are they correct?

$H_0:a=a_0=30$

$H_1:a>a_0$

Test statistic:

$T={\frac {(x-a)*\sqrt{n}} S}$ , where x - sample mean, n - sample size,  S - sample standard deviation. So, in the given case we have

$T={\frac {(35-30)*\sqrt{28}} 6}\approx4.41$

Since population standard deviation is unknown and sample size is not big(<30), then it is aapropriate to use t-statisctic. The critical value is such C, that

$P(T(n-1)>C)=0.05\implies P(T(27)>C)=0.05\implies C=2.052$

T>C, so we should reject the null hypothesis and admit that average commute time is greater than 30 minutes