Use the following information to answer the next seven exercises: Suppose that a recent article stated that the mean time spent in jail by a first-time convicted burglar is $2.5$years. A study was then done to see if the mean time has increased in the new century. A random sample of $26$ first-time convicted burglars in a recent year was picked. The mean length of time in jail from the survey was three years with a standard deviation ofyears. Suppose that it is somehow known that the population standard deviation is $1.5.$ Conduct a hypothesis test to determine if the mean length of jail time has increased. Assume the distribution of the jail times is approximately normal.

Is this a test of means or proportions?

The average sentence for a first-time burglar is $2.5$ years in prison.

There were $26$first-time burglars identified. The average length of time spent in prison is three years, with a standard variation of $1.8$ years. The population standard deviation is $1.5.$

A hypothesis test is carried out to see if the average length of time spent in prison has increased. The distribution of jail time is customarily done. The average sentence for a first-time burglar is $2.5$ years. This test determines whether the recent century's meantime is more than or equal to $2.5$ years. This demonstrates that the exam was conducted on the basis of means.

To determine whether the mean is equal to or greater than $2.5$ years, the hypothesis for this test is as follows:

$$\begin{gathered} H_0:\mu =2.5 \\ {H}_a:\mu >2.5 \end{gathered}$$