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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's standard deviation is $1.5.$

A hypothesis test is used to determine whether the average length of time spent in prison has increased. The allocation of jail time is usually done in a fair manner. According to the information provided, the tests will be administered to $26$ first-time convicted burglars. The following is an example. The random variable is the average amount of time these criminals spent in jail. It is assumed that the sample's mean age is three years.

As a result, the value of $\stackrel{¯}{X}$ is three years.

A hypothesis test is used to determine whether the average length of time spent in prison has increased. The allocation of jail time is usually done in a fair manner. The population standard deviation describes the variation in population statistics. The population standard deviation is $1.5,$ according to the data.

As a result, the value of$\mathrm{σ}$ is$1.5.$

A hypothesis test is used to determine whether the average length of time spent in prison has increased. The allocation of jail time is usually done in a fair manner. The sample standard deviation represents the variation in the data sample. A sample of $26$ burglars is used in this case. These $26$ robbers have a standard deviation of $1.8.$

As a result, $S_x$ has a value of$1.8.$

A hypothesis test is used to determine whether the average length of time spent in prison has increased. The allocation of jail time is usually done in a fair manner. The goal of this study is to see if first-time convicted burglars spend more than $2.5$ years in jail or less than $2.5$years. It is not possible to capture the entire population of burglars here. In this scenario, a sample of $26$ first-time burglars is taken into account.

As a result, the value ofn is $26$.