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

Key Concepts

Chi-Square Test Statistic

The chi-square test statistic is computed by summing the squared differences between each observed and expected frequency (O - E)², divided by the expected frequency for all categories. This statistic quantifies the overall deviation from the expected distribution and is compared to the critical value to make a testing decision.

Critical Value and Rejection Region

The critical value is the threshold value on the chi-square distribution corresponding to the chosen significance level and the degrees of freedom. The rejection region is defined as the set of values beyond this critical value, such that if the test statistic falls in this region, the null hypothesis is rejected.

Significance Level

The significance level, denoted by alpha (?), is the threshold probability used for deciding whether to reject the null hypothesis. It defines the probability of committing a Type I error, which is mistakenly rejecting a true null hypothesis, and is critical for setting the rigor and reliability of the test.

Decision and Conclusion

The decision in a chi-square test involves comparing the calculated chi-square statistic with the critical value. If the test statistic exceeds the critical value, the data provide sufficient evidence to reject the null hypothesis in favor of the alternative, indicating that the observed distribution significantly differs from the expected one. Otherwise, there is not enough evidence to reject the null hypothesis, leading to the conclusion that there is no significant difference.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) typically states that there is no difference between the observed data and the expected distribution. The alternative hypothesis (Ha) asserts that there is a significant difference. This framework is central to making inferences about model fit based on observed data.

Chi-Square Goodness-of-Fit Test

The chi-square goodness-of-fit test is a statistical procedure used to determine whether a set of observed frequency counts differs significantly from an expected distribution. The test is based on the chi-square statistic, which measures the cumulative squared differences between observed and expected frequencies, each divided by the corresponding expected frequency.

Expected Distribution

The expected distribution represents the frequency counts that would be observed if the null hypothesis were true. It serves as a baseline or standard against which the observed frequencies are compared in order to evaluate deviations and assess the degree of conformity between theory and observation.

Transcript

00:01 What do we have this time? now this time there is a research from that claims that the distribution of the days of the week that people order, that people are most likely to order food from for delivery is different from the distribution that is given to us in the figure.

00:16 So in order to test the claim, we randomly select 500 people and record which day of the week, each one of them is most likely to order food for delivery.

00:25 So what we have is a table.

00:27 The first step in solving these kinds of questions is always form the table.

00:32 The first column is day.

00:37 What are the different days? it starts from sunday, sunday, monday, tuesday.

00:47 Okay, let me just reload this.

00:51 So what we have in the table is the days and their respective frequencies.

00:58 So we were around here.

01:03 Where did it go? so we have sunday, monday, tuesday, wednesday, thursday, friday.

01:21 Thursday, friday.

01:26 And saturday.

01:29 All right.

01:30 What is the observed values, the frequencies? let me just write them as the observed values.

01:41 What are the observed values? 43, 16, 25, 49, 468, 158, 56, 16, 168, 153.

02:01 Now what are the probabilities that are given to us? let us look at the distribution that given to us for sunday it is 0 .07 let me say this is the probabilities this column will be called the probabilities okay so we're here on sunday it is 7 % so 0 .07 monday is 4 % 0 .04 tuesday is 6 % 0 .06 wednesday is 13 % 1 3 .13 thursday is 10 % 1 .1 friday is 36 so this is 0 .36 and then we have 0 .24 on saturday.

02:47 All right.

02:51 On friday we have 168.

02:56 Now what will be the null and the alternative hypothesis? the null hypothesis will be that the observed and the given distribution and the given distribution of food delivery are similar.

03:26 The alternative hypothesis will be the claim that the observed and the given distribution and the given distribution of food delivery are not the same.

03:57 All right.

03:58 Now, what is the first step in the kai square distribution? we are going to use the kai square statistic here.

04:01 What is the first step? the first step is always finding the expected values for all the categories.

04:09 The expected value is given as the sample size, the expected value is given as a sample size multiplied by the probability, multiplied by the probability of the category.

04:23 The probability of the category.

04:26 Okay, let us look at this formula in action.

04:30 Make a new column for the expected values.

04:34 The expected values, what we have is the probability and the observed values.

04:45 Now how do we calculate the expected values? the expected values are going to be sample size multiplied by the probability what is the sample size the sample size that we have is 500 so this is going to be 500 multiplied by 0 .07 which is 35 then we have 500 multiplied by 0 .04 which is 20 then we have 500 multiplied by 0 .06 which is 30 then we have 500 point one three which is 65 then we have 0 .1 which is 50 then we have 0 .36 .3 6 should be around 180 then we will have 0 .24 which should be around 220.

05:44 Okay now after the expected values the next thing that we do is calculate the individual kai square values the kai square values the kai square values what is happening to this okay these are the kai square values okay how do we calculate the kai square statistic for all the categories we apply the formula observed minus the expected we square it we divide this value by the expected value and in the end we sum this value that we get for all the categories and this gives us the overall high square statistic.

06:27 Let us go over here.

06:28 Look at this formula in action.

06:32 This says the difference in observed and expected values.

06:34 For the first category, let's do this.

06:36 43 minus 35 is 8.

06:38 And we squared this.

06:39 This becomes 64 and divide this by the expected value.

06:41 That is 35.

06:42 And this gives us 1 .825.

06:45 What i can write this as 1 .83.

06:48 1 .83.

06:50 Then the difference is 4 .4 squared is 16.