(a) identify the expected distribution and state H0 and Ha,...

Key Concepts

Interpretation of Hypothesis Test

This concept involves drawing a conclusion from the test results in the context of the research claim. It includes determining whether the discrepancy between observed and expected frequencies is statistically significant, and hence, whether there is support to either reject the null hypothesis or fail to reject it, impacting the original claim.

Test Statistic Calculation

The chi-square test statistic is computed by summing over all categories the squared difference between observed and expected frequencies divided by the expected frequency. This value quantifies how far the observed data deviate from what would be expected under the null hypothesis.

Critical Value and Rejection Region

The critical value is the threshold value from the chi-square distribution corresponding to the chosen significance level and degrees of freedom. The rejection region is the set of all values for the test statistic that leads to rejecting the null hypothesis. If the computed chi-square statistic exceeds the critical value, the null hypothesis is rejected.

Significance Level

The significance level, denoted by alpha (?), is the probability threshold used for deciding whether to reject the null hypothesis. It defines the risk of committing a Type I error, which is the incorrect rejection of a true null hypothesis.

Degrees of Freedom

Degrees of freedom in the chi-square goodness-of-fit test are calculated as the number of categories minus one (k - 1). They represent the number of values in the final calculation of a statistic that are free to vary and are used to determine the appropriate critical value from the chi-square distribution.

Uniform Distribution

A uniform distribution is one in which all outcomes are equally likely. In the context of the chi-square test, under the null hypothesis, it implies that each category (or month, here) should have the same expected frequency of events.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H?) is a statement of no effect or no difference, while the alternative hypothesis (H?) suggests that there is an effect or a difference. For the chi-square goodness-of-fit test, H? typically states that the observed frequencies follow the expected distribution, and H? contends that they do not.

Chi-Square Goodness-of-Fit Test

This is a statistical test used to determine whether there is a significant difference between an observed frequency distribution and a theoretical (expected) distribution. It is particularly useful for categorical data and helps in assessing if data follow a specified distribution, such as the uniform distribution in this context.

Transcript

00:01 Okay, so in this question, a researcher claimed that the number of homicide crimes in california by month is uniformly distributed.

00:09 So in order to test this claim, we randomly select 1 ,200 homicides and we have the data for that.

00:16 So, as always, the first thing that we do is make the table.

00:22 The first column is month.

00:29 Okay, this is going to be a long table.

00:33 All right.

00:33 So we have january, february, march, april, may, june, july, august, september, october, november, and december.

01:16 Okay.

01:18 Then we have the observed values.

01:19 Let me just write this as observed values.

01:31 Okay all right what are the observed values for jan we have 115 75 90 okay this is 115 75 95 90 98 121 29 99 115 we have 98 again 92 108 then this is 90 and 90 and this is 90 and this is 99.

02:14 All right.

02:15 So we have the observed values as well.

02:19 Now what we do is in order to test the claim, we randomly select 1 ,100, 1 ,200, so this addition is 120, 200.

02:27 1200 homicides from a recent year and record the month when each happened.

02:33 So we have all the observed values.

02:36 Now, what is the null and alternative hypothesis over here? this is very important.

02:41 This is our first question.

02:42 We have to write the null and the alternative hypothesis.

02:44 The claim is that the number of homicides by month is uniformly distributed.

02:50 So our null hypothesis is the distribution of the number of homicides of the number of homicides by month is uniform, is uniformly distributed.

03:24 Uniformly distributed.

03:32 Okay, this is going to be our null hypothesis.

03:36 After that, we have the alternative hypothesis.

03:38 What will be the alternative hypothesis? it will be that the distribution of the number of homicides of the number of homicides by month is not uniform.

04:11 Is not uniform okay now if we say that it is uniformly distributed it means that all the 12 values should be same so it should be 1 by 12 right the probabilities should be 1 by 12 so 1 divided by 12 if i use my calculator i get 0 .0833 so what should be the probabilities the probability should be 0 .0833 3 3 so what should be 0 .0833 and this is going to be for all of them.

04:49 0 .0833.

05:00 Just a moment.

05:04 This table is actually very long so it might take some time to fill this but this is very important.

05:13 Table building is always an important part of solving the question.

05:17 0 .08333.

05:38 Now, now that we have the probabilities, the next step in finding the kai square is finding the expected values and how do you find the expected values this is done for all the categories the expected value for a category is given by the formula sample size okay this is given by sample size which happens to be 1200 in this case multiplied by the probability the probability of that category so let us let us look at at this formula and action what is the sample size the sample size is 1200 and the probability for each one of them is 0 .0833 so this turns out to be 99 .96 or i can write this as 100 so my expected values my expected values column will have 100 now since the probability is same for all the categories this will be 100 for all of them.

07:04 So let me just fill this one as well.

07:17 Okay, so these are all hundreds.

07:25 All right.

07:31 This last one should also be 100.

07:36 Okay.

07:38 Now the next step is finding the kai square statistic and how do you do that? we are going to do observed value minus the expected value.

07:46 We are going to do this for all the categories.

07:48 We are going to square this, divide this by the expected value.

07:52 And in the end we are going to sum all of these values the entire column and this is going to give us the overall kai square statistic for our problem so let us go here these will be the kai square values column okay so the steps are find the difference between the observed and the expected values the difference is 15 right 115 minus 100 which is 15 i square this this is 225 and i divide this by the expected value which is 100 so this is 2 .25 so this is 2 .25.

08:28 2 .25.

08:31 All right.

08:33 Then i have the difference as 25.

08:35 The square is 625 and divide this by 100.

08:38 This is 6 .25.

08:40 6 .25.

08:44 Then the difference is 10.

08:46 10 square is 100.

08:47 100 by 100 is 1 .00 is 1 .00.

08:50 Then the difference is 21.

08:50 In the next case, the difference is 2 .2.

08:54 4 .4.