Question

Alert nurses at the Veteran’s Affairs Medical Center in Northampton, Massachusetts, noticed an unusually high number of deaths at times when another nurse, Kristen Gilbert, was working. Those same nurses later noticed missing supplies of the drug epinephrine, which is a synthetic adrenaline that stimulates the heart. Kristen Gilbert was arrested and charged with four counts of murder and two counts of attempted murder. When seeking a grand jury indictment, prosecutors provided a key piece of evidence consisting of the table below. Use a 0.01 significance level to test the defense claim that deaths on shifts are independent of whether Gilbert was working. What does the result suggest about the guilt or innocence of Gilbert?

	Shifts With a Death	Shifts Without a Death
Gilbert Was Working	40	217
Gilbert Was Not Working	34	1350

Short answer

The deaths on shifts are dependent on whether Gilbert was working, which suggests the guilt of Gilbert.

Step-by-step solution

Given information

The data for theshifts with and without death and whether Gilbert was working is provided.

The level of significance is 0.01.

Compute the expected frequencies

Compute theexpected frequencies using the formula stated below,

\(E = \frac{{\left( {row\;total} \right)\left( {column\;total} \right)}}{{\left( {grand\;total} \right)}}\)

The counts for total rows and columns are,

	Shifts With a Death	Shifts Without a Death	Row Total
Gilbert Was Working	40	217	257
Gilbert Was Not Working	34	1350	1384
Column Total	74	1567	1641

Theexpected frequency tableis represented as,

	Shifts With a Death	Shifts Without a Death
Gilbert Was Working	11.5893	245.4107
Gilbert Was Not Working	62.4107	1321.5893

State the null and alternate hypothesis

The hypotheses are formulated as,

\({H_0}:\)The deaths on shifts are independent of whether Gilbert was working.

\({H_1}:\)The deaths on shifts are dependent on whether Gilbert was working.

Compute the test statistic

The value of the test statisticis computed as,

\(\begin{aligned}{c}{\chi ^2} = \sum {\frac{{{{\left( {O - E} \right)}^2}}}{E}} \\ = \frac{{{{\left( {40 - 11.5893} \right)}^2}}}{{11.5893}} + \frac{{{{\left( {217 - 245.4107} \right)}^2}}}{{245.4107}} + ... + \frac{{{{\left( {1350 - 1321.5893} \right)}^2}}}{{1321.5893}}\\ = 86.4809\\ \approx 86.481\end{aligned}\)

Therefore, the value of the test statistic is 86.481.

Compute the degrees of freedom

The degrees of freedomare computed as,

\(\begin{aligned}{c}\left( {r - 1} \right)\left( {c - 1} \right) = \left( {2 - 1} \right)\left( {2 - 1} \right)\\ = 1\end{aligned}\)

Therefore, the degrees of freedom are 1.

Compute the critical value

The critical value for 1 degrees of freedom and at 0.01 level of significance is 6.635.

Therefore, the critical value is 6.635.

The P-value is computed as 0.000.

State the decision

The critical (6.635) is less than the value of the test statistic (86.481). In this case, the null hypothesis is rejected.

Therefore, the decision is to reject the null hypothesis.

State the conclusion

There isinsufficient evidence to support the claimthat deaths on shifts are independent of whether Gilbert was working or not.

The results suggest that Gilbert cannot be considered as innocence for the deaths on the basis of results.