1 Each working day, Beth takes a bus to her place of work. She believes that the mean time that her journey takes is 30 minutes. In order to check this, Beth selects a random sample of 8 journeys. The times in minutes for these 8 journeys are as follows.
\(\begin{array} { l l l l l l l l } 31.9 & 28.5 & 35.9 & 31.0 & 30.2 & 34.9 & 28.9 & 31.3 \end{array}\)

1. What assumption does Beth need to make in order to construct a confidence interval for the mean journey time based on the \(t\) distribution?
2. In this question you must show detailed reasoning. Given that the assumption in part (a) is valid, determine a 95\% confidence interval for the mean journey time.
3. Explain whether the confidence interval suggests that Beth may be correct in the belief that her mean journey time is 30 minutes.