Question

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 meantime 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 $3$years with a standard deviation of $1.8$years. Suppose that it is somehow known that the population standard deviation is $1.5$. If you were conducting a hypothesis test to determine if the mean length of jail time has increased, what would the null and alternative hypotheses be? The distribution of the population is normal.

$a.H_o$_______
$b.H_a$_______

Short answer

a.$H_0:\mu \le 2.5$

b.$H_a:\mu >2.5$

Step-by-step solution

Given Information

A first-time convicted burglar spends an average of$2.5$ years in prison. The goal is to see if the average amount of time spent in prison has grown.

Explanation part a

• The null hypothesis is
• $H_0$: A first-time convicted burglar spends less than or equal to $2.5$years in prison.
• The Null Hypothesis ($H_0$): It's a declaration that there's no difference between sample means or proportions, or between a sample mean or proportions and a population mean or proportion.

Step 3: Explanation part b

• The alternative hypothesis is:
• $H_a$: A first-time convicted burglar spends more than $2.5$years in jail on average.
• Here's an alternative theory: It is the population assertion that contradicts $H_0$, and it is what we conclude when we reject $H_0$.

Step 4: Conclusion

For the above data, the null and alternative hypotheses are provided by:

$H_0:\mu \le 2.5$

$H_a:\mu >2.5$