**Analysis of Sentencing and Defendant Race**The data below explores the potential relationship between the type of sentence received by defendants and their race. This study was conducted on 260 defendants within a local jurisdiction.**Table: Type of Sentence Received by Race of Defendant**| Type of Sentence Received | White | Black | Other | Total ||---------------------------|-------|-------|-------|-------|| Jail Only | 10 | 25 | 18 | 53 || Fine and Jail | 20 | 37 | 28 | 85 || Probation | 24 | 30 | 20 | 74 || Community Service | 16 | 18 | 14 | 48 || **Total** | 70 | 110 | 80 | 260 |**Explanation:**- **Jail Only:** Out of 53 defendants, 10 were White, 25 were Black, and 18 were from other racial groups.- **Fine and Jail:** Of the 85 defendants given this sentence, 20 were White, 37 were Black, and 28 belonged to other racial categories.- **Probation:** This type of sentence was issued to 74 defendants, comprising 24 White, 30 Black, and 20 others.- **Community Service:** Among the 48 sentenced, 16 were White, 18 were Black, and 14 were from other groups.The total number of defendants in this study was 260, distributed as 70 White, 110 Black, and 80 from other racial groups. This data serves as a basis for hypothesis testing on racial disparity in sentencing. 1. What is the independent and what is the dependent variable, and the total sample size?2. Calculate the cell percentages.3. Calculate the expected frequencies for each cell in the table.4. Test the Null Hypothesis. If you reject the Null hypothesis run the correct measure of association to report the strength of the relationship between the two variables.

The given contingency table for the race of the defendant and the type of received sentence is as follows:Type of Sentence ReceivedRace of DefendantTotalWhiteBlackOtherJail Only10251853Fine and Jail20372885Probation24302074Community Service16181448Total7011080260Part (1)As the objective is to determine if the type of received sentence is independent of the race of the defendant, i.e., if the received conviction is essentially influenced by the race of the defendant, the variable "race of the defendant" is considered as the independent variable, whereas, the variable "type of received sentence" is considered as the dependent variable.The total sample size is the considered number of the convicted defendants. That is the sample size is 260.Part (2)The cell percentage is calculated as the percentage of defendants of a certain race with corresponding specified types of conviction. The cell percentage for each cell is calculated as follows:cell percentage=(corresponding cell frequencytotal sample size×100)%Calculation Table:Type of Sentence ReceivedRace of DefendantWhiteBlackOtherJail Only3.846% (=(10260×100)%)9.615% (=(25260×100)%)6.923%Fine and Jail7.692% (=(20260×100)%)14.231%10.769%Probation9.231%11.538%7.692%Community Service6.154%6.923%5.385%Part (3)The expected frequencies for each cell are calculated as follows:Emn=mth row total×nth column totaltotal sample sizeCalculation Table:Type of Sentence ReceivedRace of DefendantWhiteBlackOtherJail Only14.269 (=53×70260)22.423 (=53×110260)16.308 (=53×80260)Fine and Jail22.885 (=85×70260)35.96226.154Probation19.923 (=74×70260)31.30822.769Community Service12.923 (=48×70260)20.30814.769Part (4)It is set up that:H0: the type of received sentence is hypothesized to be independent of the race of the defendant.H1: the type of received sentence is hypothesized to be not independent of the race of the defendant.In order to validate if the deviation between the theoretical frequencies of convicted defendants of certain races and the experimental frequencies of convicted defendants of certain races is purely due to chance, the Chi-square test statistic is formulated.The test statistic is:χ2=(Oi−Ei)2Ei; Oi: observed frequency, Ei: expected frequencyThe test statistic calculation table:RaceSentence TypeOiEi(Oi−Ei)2EiWhiteJail Only1014.2691.2773Fine and Jail2022.8850.3636Probation2419.9230.8343Community Service1612.9230.7326Black Jail Only2522.4230.2961Fine and Jail3735.9620.0300Probation3031.3080.0546Community Service1820.3080.2622OtherJail Only1816.3080.1756Fine and Jail2826.1540.1303Probation2022.7690.3368Community Service1414.7690.0401Total4.5336The test statistic is χcalculated2=4.5336.The degrees of freedom, df=(no. of rows-1)×(no. of columns-1)=(4−1)×(3−1)=6The p-value for the Chi-square score of 4.5336 with distribution df=6 is obtained using excel as follows:The p-value is 0.6049.The decision rule is "reject H0, if p-value≤α".For any standard significance level of α=0.01, 0.05, and 0.10, p-value (=0.6049)>α. Thus H0 is failed to be rejected.The type of received sentence is concluded to be independent of the race of the defendant, i.e., no relation between the race of the defendant and the type of sentence is claimed. (1) The variable "race of the defendant" is the independent variable, and the variable "type of received sentence" is the dependent variable. The total sample size is 260.(2) The cell percentages are calculated as follows:Type of Sentence ReceivedRace of DefendantWhiteBlackOtherJail Only3.846%9.615%6.923%Fine and Jail7.692%14.231%10.769%Probation9.231%11.538%7.692%Community Service6.154%6.923%5.385%(3) The expected frequencies are as follows:Type of Sentence ReceivedRace of DefendantWhiteBlackOtherJail Only14.26922.42316.308Fine and Jail22.88535.96226.154Probation19.92331.30822.769Community Service12.92320.30814.769(4) The null hypothesis is failed to be rejected. No relation between the race of the defendant and the type of sentence is claimed. No additional measure of association is needed.