hailey, a researcher at the davis institute for criminal justice, is concerned that court - appointed attorneys negotiate harsher plea deals for their clients than privately retained attorneys do. hailey examines data from a randomly selected sample of cases that carried similar maximum sentences where the attorney was court - appointed, and another sample where the attorney was privately - retained, and recorded the plea deal sentences that were agreed to:
test haileys claim at $ alpha = 0.05 $. where necessary, round results to 4 decimal places.
a. determine the null and alternative hypotheses.
$ h _ { 0 } $: there is no relationship between plea deals and whether the attorney on the case is privately - retained or court - appointed.
$ h _ { a } $: there is a relationship between plea deals and whether the attorney on the case is privately - retained or court - appointed.
$ h _ { 0 } $: the distribution of plea deals arranged by court - appointed attorneys is the same as the distribution of plea deals arranged by privately - retained attorneys.
$ h _ { a } $: the distribution of plea deals arranged by court - appointed attorneys is not the same as the distribution of plea deals arranged by privately - retained attorneys.
$ h _ { 0 } $: plea deals negotiated by privately - retained attorneys are better than plea deals negotiated by court - appointed attorneys.
$ h _ { a } $: plea deals negotiated by privately - retained attorneys are not better than plea deals negotiated by court - appointed attorneys.
b. determine the test statistic.
$ x ^ { 2 } = $
c. determine the $ p $ - value.
$ p $ - value $ = $
d. make a decision.
select an answer
e. make a conclusion.
the evidence supports the claim that plea deals negotiated by privately - retained attorneys are better than plea deals negotiated by court - appointed attorneys.
the evidence is not sufficient to support the claim that plea deals negotiated by privately - retained attorneys are better than plea deals negotiated by court - appointed attorneys.
there is sufficient evidence to reject the claim that the plea deal negotiated and type of attorney are independent.
there is not sufficient evidence to reject the claim that the plea deal negotiated and type of attorney are independent.
there is sufficient evidence to conclude that the distributions of plea deals negoiated by court - appointed vs. private attorneys are not the same.
there is not sufficient evidence to conclude that the distributions of plea deals negoiated by court - appointed vs. private attorneys are not the same.

Question

Accepted Answer

# Explanation:
## Step1: Determine the null and alternative hypotheses
This is a test of homogeneity of proportions. The null hypothesis $H_0$ is that the distribution of plea deals arranged by court - appointed attorneys is the same as the distribution of plea deals arranged by privately - retained attorneys. The alternative hypothesis $H_A$ is that the distribution of plea deals arranged by court - appointed attorneys is not the same as the distribution of plea deals arranged by privately - retained attorneys.

## Step2: Calculate the row and column totals
- Row totals:
    - Appointed: $25 + 25+24 + 34=108$
    - Private: $54+43 + 33+31 = 161$
- Column totals:
    - $0 - 6$ months: $25 + 54=79$
    - $6 - 12$ months: $25+43 = 68$
    - $12 - 24$ months: $24+33 = 57$
    - More than $24$ months: $34+31=65$
- Grand total $N=108 + 161=269$

## Step3: Calculate the expected frequencies
The formula for the expected frequency $E_{ij}=\frac{R_i\times C_j}{N}$
- For $E_{11}=\frac{108\times79}{269}\approx31.6134$
- For $E_{12}=\frac{108\times68}{269}\approx27.3234$
- For $E_{13}=\frac{108\times57}{269}\approx22.9145$
- For $E_{14}=\frac{108\times65}{269}\approx26.1468$
- For $E_{21}=\frac{161\times79}{269}\approx47.3866$
- For $E_{22}=\frac{161\times68}{269}\approx40.6766$
- For $E_{23}=\frac{161\times57}{269}\approx34.0855$
- For $E_{24}=\frac{161\times65}{269}\approx38.8532$

## Step4: Calculate the $\chi^{2}$ statistic
The formula for $\chi^{2}=\sum\frac{(O_{ij}-E_{ij})^{2}}{E_{ij}}$
$$
\begin{align*}
\frac{(25 - 31.6134)^{2}}{31.6134}&+\frac{(25 - 27.3234)^{2}}{27.3234}+\frac{(24 - 22.9145)^{2}}{22.9145}+\frac{(34 - 26.1468)^{2}}{26.1468}\
&+\frac{(54 - 47.3866)^{2}}{47.3866}+\frac{(43 - 40.6766)^{2}}{40.6766}+\frac{(33 - 34.0855)^{2}}{34.0855}+\frac{(31 - 38.8532)^{2}}{38.8532}\
\end{align*}
$$
$$
\begin{align*}
\frac{(- 6.6134)^{2}}{31.6134}&+\frac{(-2.3234)^{2}}{27.3234}+\frac{(1.0855)^{2}}{22.9145}+\frac{(7.8532)^{2}}{26.1468}\
&+\frac{(6.6134)^{2}}{47.3866}+\frac{(2.3234)^{2}}{40.6766}+\frac{(-1.0855)^{2}}{34.0855}+\frac{(-7.8532)^{2}}{38.8532}\
\end{align*}
$$
$$
\begin{align*}
\frac{43.737}{31.6134}&+\frac{5.398}{27.3234}+\frac{1.178}{22.9145}+\frac{61.663}{26.1468}\
&+\frac{43.737}{47.3866}+\frac{5.398}{40.6766}+\frac{1.178}{34.0855}+\frac{61.663}{38.8532}\
\end{align*}
$$
$$
\begin{align*}
1.3835&+0.1976+0.0514 + 2.3583+0.9230+0.1327+0.0346+1.5871\
\end{align*}
$$
$\chi^{2}\approx6.6682$

## Step5: Calculate the degrees of freedom and $p -$value
- Degrees of freedom $df=(r - 1)(c - 1)=(2 - 1)(4 - 1)=3$
- Using a $\chi^{2}$ - distribution table or calculator, for $\chi^{2}=6.6682$ and $df = 3$, the $p -$value $P(\chi^{2}>6.6682)\approx0.0839$

## Step6: Make a decision
Since $\alpha = 0.05$ and $p - value=0.0839>0.05$, we fail to reject the null hypothesis.

# Answer:
a. $H_{0}$: The distribution of plea deals arranged by court - appointed attorneys is the same as the distribution of plea deals arranged by privately - retained attorneys. $H_{A}$: The distribution of plea deals arranged by court - appointed attorneys is not the same as the distribution of plea deals arranged by privately - retained attorneys.
b. $\chi^{2}\approx6.6682$
c. $p - value\approx0.0839$
d. Fail to reject the null hypothesis.
e. There IS NOT sufficient evidence to conclude that the distributions of plea deals negotiated by court - appointed vs. private attorneys are not the same.

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QUESTION IMAGE

Question

Explanation:

Step1: Determine the null and alternative hypotheses

Step2: Calculate the row and column totals

Step3: Calculate the expected frequencies

Step4: Calculate the \(\chi^{2}\) statistic

Step5: Calculate the degrees of freedom and \(p -\)value

Step6: Make a decision

Answer: