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

Key Concepts

Chi-Square Goodness-of-Fit Test

This test examines whether the frequencies observed in different categories match those expected under a specified theoretical distribution. It involves computing a statistic that sums the squared differences between observed and expected counts, scaled by the expected counts, and comparing the result to a chi-square distribution.

Null and Alternative Hypotheses

In any hypothesis test, especially a chi-square goodness-of-fit test, the null hypothesis (H0) typically states that the sample data follow a specified distribution (for example, uniformly distributed), while the alternative hypothesis (Ha) states that the data do not follow that distribution. This forms the basis for determining whether any observed discrepancy is likely due to random variation or indicates a real difference.

Uniform Distribution and Expected Frequencies

When data are assumed to follow a uniform distribution, each category is expected to have an equal number of observations. The expected frequency for each category is computed by dividing the total number of observations by the number of categories, which is crucial for comparing against the observed frequencies in a chi-square test.

Degrees of Freedom

Degrees of freedom in a chi-square goodness-of-fit test are determined by subtracting one from the number of categories. This value accounts for the number of independent comparisons being made and is essential for identifying the correct critical value from the chi-square distribution.

Critical Value and Rejection Region

The critical value is the threshold obtained from the chi-square distribution corresponding to the chosen significance level (?) and the degrees of freedom. It defines the rejection region such that if the computed test statistic exceeds this critical value, the null hypothesis is rejected.

Test Statistic Calculation

The chi-square test statistic is calculated by summing the squared deviations of observed frequencies from expected frequencies for each category, divided by the respective expected frequencies. This statistic quantifies how well the observed data fit the expected model.

Decision Making and Interpretation

After computing the test statistic and comparing it to the critical value (or alternatively using a p-value), the researcher decides whether to reject or fail to reject the null hypothesis. This decision is then interpreted in the context of the research claim, determining if the data provide sufficient evidence to conclude that the observed distribution significantly deviates from the expected uniform distribution.

Transcript

00:01 Now in this question, a researcher claims that the number of homicide crimes in california by season is uniformly distributed.

00:08 Now, we have our doubts.

00:11 So to test this claim, what we do is we randomly select 1 ,200 homicides from a recent year and record the seasons in which each of them happened.

00:21 And we have a table with us.

00:23 Okay, so the first thing we do, always during solving these questions, is record the table.

00:28 So season there are four seasons spring summer fall winter so this is spring summer fall winter then we have the observed values the observed values what are the frequencies that we observed it is 309 312 290 and 289 okay now the question is saying that the researcher claims that the number of homicide crimes in california by season is uniformly distributed.

01:31 This is very important.

01:32 Uniformly distributed means what? uniformly distributed.

01:44 This means that the probability, the probability for all the seasons, for all the seasons, for all the seasons, is a.

02:09 The same.

02:13 Meaning what? overall probability is one.

02:15 There are four seasons, so all of them will have one by four probability or this turns out to be 0 .25.

02:21 So the probabilities, the probabilities are 0 .25 for all of them.

02:42 All right.

02:45 Now, what are going to be the null and the alternative hypothesis? the null hypothesis is that the distribution of homicides in california, the distribution of homicides in california is uniformly distributed, is uniformly distributed.

03:25 That is, the claim is correct.

03:27 That is, the claim seems to be correct.

03:36 Seems to be correct.

03:41 The claim seems to be correct.

03:42 The alternative hypothesis will be that the distribution of homicides is not uniformly distributed.

04:07 Is not uniformly distributed.

04:13 All right.

04:17 Now the first thing that we do while solving the chi -square statistic goodness of fit problems is calculate the expected values.

04:27 What will be the expected values? let us just make this column.

04:32 This will be the expected values.

04:36 How do you find the expected values? expected value for every category e is calculated as the formula for this is number, this is going to be the sample size.

04:48 The sample size, that is the number of observations that you have or n, multiplied by the probability for that category.

05:00 The probability for that category i now let us look at this formula in action so my sample size is 1200 this is 1200 and probability i can see is one fourth for all of them so what will be my expected values the first category will have the expected value as 1200 divided by 4 or 0 .25 into 1 ,200, which is 300.

05:33 And for the other ones also, we'll see that since the distribution is considered, is thought of to be uniform in the null hypothesis, it is going to be 300.

05:44 Right.

05:44 Now, i have my expected values.

05:46 What is the next step? the next step is calculating the kai square statistic.

05:52 So, in order to do that, i'm going to apply the formula, observed minus expected value, whole square, upon the expected value.

06:00 I'm going to do this for all the categories.

06:03 Then i'm going to sum them up and this will give me the overall kai square statistic.

06:08 All right then.

06:12 Just a moment.

06:16 I did not want this what just happened.

06:21 Okay.

06:25 All right.

06:30 Let us look at this formula in action.

06:31 This is going to be the kai square values.

06:37 So this is going to be the difference in the observed and the expected.

06:40 It's square and divided by the expected.

06:42 So the difference in the first case is.

06:44 9 square is 81 and we what we do is we divide this by the expected value of 300 this is 0 .27 0 .27 then the difference is 12 so the square is 144 divided by 300 0 .48 0 .48 then the difference is 10 square is 100 divide this by 300 this is 0 .33 0 .33 then the difference is 10 square is 100.

07:19 11 so the square is 121 divide this by 300 we get 0 .40 0 .40 now if i add all of them up 0 .27 plus 0 .48 plus 0 .33 plus 0 .30 plus 0 .40 this turns out to be 1 .48 so my kai square statistic for this problem is 1 .48 this is 1 .4 8.

07:54 Okay.

07:57 Now what do i do? now i need the degrees of freedom.

08:00 The next step is to calculate the degrees of freedom...

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