The waiting time for a bus during peak hours in a city is \(X\) minutes and it is known that the standard deviation of \(X\) is \(1.2\) minutes. The Metropolitan Transport Board (MTB) which runs a large fleet of buses claims that the mean waiting time for buses during peak hours in the city is \(6.5\) minutes.
With various changes to the infrastructure in the city, MTB wants to review whether the mean waiting time has changed since the claim was made. MTB records a random sample of \(20\) waiting times and found that the mean waiting time was \(\overline{x}\) minutes.
Given that \(\overline{x}=6.9\), test, at the \(5\%\) level of significance, whether the mean waiting time differs from \(6.5\) minutes.
State an assumption for the test to be valid.[4]
Explain what you understand by the phrase ‘at the \(5\%\) level of significance’ in the context of this question.[1]
Find the range of values of \(\overline{x}\) for which there is sufficient evidence at \(1\%\) significance level that the mean waiting time differs from \(6.5\) minutes.[2]
MTB operates the train system in the city as well and has received public feedback that the mean waiting time for trains has been getting worse. It is known that the waiting times for trains are normally distributed. MTB wishes to test, at the \(5\%\) level of significance, if the mean waiting time for trains is more than \(3.7\) minutes. A random sample of \(n\) observations, where \(n\geq30\), was taken and a mean of \(3.95\) minutes and standard deviation of \(0.9\) minutes were obtained.
Given that the null hypothesis is not rejected, obtain an inequality involving \(n\), and hence find the set of values that \(n\) can take.[5]
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Answer:(a)(i) \(z=1.49,\ p=0.136\); do not reject \(H_0\). Assume normal population waiting times. (ii) A \(5\%\) chance of concluding the mean differs when it is actually \(6.5\) minutes. (iii) \(\overline{x}\leq5.808832469\ldots\) or \(\overline{x}\geq7.191167531\ldots\). (b) \(\dfrac{0.25\sqrt{n-1}}{0.9}<1.644853627\ldots\); \(n\in\{30,31,32,33,34,35,36\}\).