A delinquency offense is an act committed by a juvenile for...

A delinquency offense is an act committed by a juvenile for which an adult could be prosecuted in a criminal court. The following data represent the number of various types of delinquencies by gender in a random sample of 750 delinquencies. (a) Does the evidence suggest that gender is independent of type of delinquency at the $\alpha=0.05$ level of significance? (b) Compute the $P$ -value for this test by finding the area under the chi-square distribution to the right of the test statistic. (c) Construct a conditional distribution by type of delinquency and draw a bar graph. Does this evidence support your conclusion in part (a)?

A delinquency offense is an act committed by a juvenile for which an adult could be prosecuted in a criminal court. The following data represent the number of various types of delinquencies by gender in a random sample of 750 delinquencies.
(a) Does the evidence suggest that gender is independent of type of delinquency at the $\alpha=0.05$ level of significance?
(b) Compute the $P$ -value for this test by finding the area under the chi-square distribution to the right of the test statistic.
(c) Construct a conditional distribution by type of delinquency and draw a bar graph. Does this evidence support your conclusion in part (a)?

Key Concepts

Conditional Distribution

A conditional distribution breaks down the distribution of one variable within the categories of another variable. It helps in understanding the pattern of one variable when conditioned on the value of another, thereby providing insights into the nature of the association between the variables.

Contingency Table

A contingency table is a type of table in a matrix format that displays the frequency distribution of the variables. It is used in the chi-square test to summarize the relationship between two categorical variables, such as gender and various types of delinquency, facilitating the calculation of expected frequencies and the overall test statistic.

Degrees of Freedom

Degrees of freedom in the context of a chi-square test for independence relate to the number of categories in the contingency table and are calculated as (number of rows - 1) multiplied by (number of columns - 1). They determine the shape of the chi-square distribution used in hypothesis testing.

Bar Graph for Categorical Data

A bar graph is a graphical representation used to display data from categorical variables. Each category is represented by a bar, and the length or height of the bar corresponds to the frequency or proportion of observations in that category. When constructed based on conditional distributions, a bar graph can visually compare the distributions of one variable across the levels of another, reinforcing insights gained from statistical tests.

Significance Level (?)

The significance level, often denoted by ?, is the threshold used to decide whether to reject the null hypothesis in a hypothesis test. It represents the probability of committing a Type I error—rejecting the null hypothesis when it is actually true. In many tests, including the chi-square test mentioned, a common level used is 0.05.

Hypothesis Testing

Hypothesis testing is a systematic method used in statistics to decide whether there is enough evidence to reject a null hypothesis, which typically states that there is no effect or no association. In the context of the chi-square test, the null hypothesis assumes that the variables (such as gender and type of delinquency) are independent, and the alternative hypothesis posits that they are associated.

Chi-square Test for Independence

This is a statistical test used to determine whether there is a significant association between two categorical variables. The test compares the observed frequencies in each category with the frequencies expected if the two variables were truly independent. The result helps determine whether any deviations from expected frequencies are due to random chance or indicate an actual relationship between the variables.

P-value

The P-value is the probability, assuming the null hypothesis is true, of obtaining a test statistic at least as extreme as the one observed. It provides a measure of the evidence against the null hypothesis. In a chi-square test, if the P-value is less than the chosen significance level, it indicates strong evidence against the null hypothesis, suggesting that the variables may not be independent.

Transcript

00:01 So we're going to be assuming that there is independence between the gender and the type of delinquency.

00:17 And alternately, there's a dependent relationship.

00:21 And that's all i'm going to put as dependent relationship.

00:24 Now, when we calculate our k -squared statistic, and by the way, there were the number of degrees of freedom, whereas there are two rows because we have male and female, so there's one less than that.

00:34 And then we have one, two, three, four.

00:37 So we'd have three different columns.

00:40 And, well, four minus one.

00:42 So we have three degrees of freedom.

00:44 And the test statistic for this comes out to be.

00:48 And let me quick get.

00:49 I had it here.

00:50 And i got rid of it.

00:54 So the test statistic comes out to be only 0 .946.

01:02 And if we're using the critical number, we were using a 5 % significance level.

01:12 And so for three degrees of freedom, the 5 % significance cutoff point would end up being at 7 .81.

01:20 And we have a value that's way back here at 0 .946.

01:25 So we would have insufficient evidence to reject the null.

01:30 So that would mean that it appears as though there is this independent relationship between gender and the type of delinquency.

01:40 So they don't have a different pattern.

01:43 And then part b asked us for the p value and the likelihood of having that type of a test statistic come up for a distribution that is independent.

01:54 That has a 0 .845 as its p value.

01:59 So this area right here is 0 .845.

02:03 Then we are asked to look at the conditional distribution...

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