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As per the given information, a random survey is conducted on 75 death row inmates revealed that the mean length time of death row inmates is 17.4 years with a deviation of 6.3 years. Conduct a hypothesis test to determine if the population's meantime on death row could likely be 15 years.

The hypothesis test is usually performed on sample data taken from a larger population. As per the given problem the researcher wants to determine the population mean time on death row of jail inmates therefore, the above test is of one mean.

The null hypothesis states that there is no statistical connection between the two variables.

The null hypothesis is the meantime on death row could likely be 15 years.

$H_0:渭=15$

The alternate hypothesis states that there is a significant relationship between the two variables.

The alternative hypothesis is the meantime on death row could not likely be 15 years.

$H_a:渭鈮�15$

The alternate hypothesis states that the meantime on death row could not likely be 15 years i.e,

$H_a:渭鈮�15$

Thus when the sign of the alternative hypothesis is$鈮�$then it is a two-tailed test.

In the above case, the random variable is represented as X.

The random variable in words is the mean length time of 75 death row jail inmates o death row.

Generally, population standard deviation is calculated on the total population sample taken by the researcher, and in the above case the population standard deviation is given in the information as 6.3 years

According to the information, the random variable which is X is the mean length time of 75 death row jail inmates on death row which is likely to be 15 years.

The s is the sample standard deviation which indicates any variation in the sample data. It is calculated as:

$$\begin{gathered} s=\frac{蟽}{\sqrt{n}} \\ =\frac{6.3}{\sqrt{75}} \\ =\frac{6.3}{8.6602} \\ =0.73 \end{gathered}$$

The n is the total number of the researcher has decided to take as a sample for his survey. Therefore the n is 75 respondents.

Thus the value of $\stackrel{炉}{X}$is $15$years, s is $0.73$and n is $75$.

As per the given problem the researcher wants to determine the population mean time on death row of jail inmates therefore single population mean test should be used.

In the above case, we know that the population and the sample data are normally distributed but we do not know the population standard deviation therefore when the population variance is unknown we follow the student's t-distribution.

P-value $<$alpha value

Thus the p-value is$0.000$.

The decision is to reject the null hypothesis.

The p-value is less than the alpha value therefore the null hypothesis is rejected.

Since the null hypothesis is rejected we have enough evidence to support the alternate hypothesis which states that the meantime on death row could not likely be 15 years.

Thus the null hypothesis is rejected since the p-value is less than the alpha value therefore we accept the alternate hypothesis.