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

Key Concepts

Significance Level and Critical Value

The significance level (?) is the threshold for determining whether the observed difference between the observed and expected data is statistically significant. The critical value is derived from the chi-square distribution table using the chosen ? level and the degrees of freedom, establishing the rejection region for the test.

Decision Rule and Rejection Region

The decision rule involves comparing the calculated chi-square test statistic to the critical value. If the test statistic exceeds the critical value (i.e., it falls in the rejection region), then the null hypothesis is rejected; otherwise, there is insufficient evidence to do so.

Degrees of Freedom

Degrees of freedom in a goodness-of-fit test are generally determined by subtracting one from the number of categories. They are critical for determining the appropriate critical values from the chi-square distribution and thereby for interpreting the test statistic.

Interpretation in Context

Once a decision is made regarding the null hypothesis, it must be interpreted within the context of the problem. This involves explaining what the statistical result means in practical or real-world terms, such as whether the observed data conforms to the hypothesized distribution of a particular characteristic.

Expected Frequencies

Expected frequencies are the counts predicted under the assumption that the null hypothesis is true. They are calculated based on the theoretical distribution provided and are essential for comparing with the actual observed counts using the chi-square test.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H?) typically states that the observed distribution fits the expected theoretical distribution, while the alternative hypothesis (H?) posits that the observed distribution deviates from this expectation. Formulating these hypotheses is the first step in conducting any statistical test.

Chi-Square Goodness-of-Fit Test

This test is used to evaluate whether the observed frequencies in different categories match the expected frequencies derived from a specific distribution. It compares the differences between observed counts and expected counts to determine if any significant discrepancies exist.

Chi-Square Test Statistic

The chi-square test statistic is calculated using the formula ?[(Observed - Expected)² / Expected] across all categories. This statistic quantifies how much the observed frequencies differ from the expected frequencies, providing a measure to assess the fit between the data and the assumed distribution.

Transcript

00:01 Okay, so this is question number seven.

00:04 Now, in this question, a researcher claims that the ages of people who go to movies at least once a month are distributed in a particular way.

00:14 What we have is the frequency and we can find the distribution from that.

00:19 We have been given a table.

00:20 So what we do is we select 1 ,000 people randomly who go to the movies at least once a month and record the age of each one of them.

00:30 The table that we have, first of all, let us just draw the table.

00:34 What are the different age groups that are considered? just a moment, i'll just bring up the table.

00:45 Okay, so over here we are going to have the first column as age.

00:53 What are the different age groups? the first one is 2 to 17.

00:57 The next one is 18 to 24.

01:02 After that, we have 25 to 39, 25 to 39.

01:06 After that we have 40 to 49 and then we have 50 plus then we have 40 to 49 and then we have 50 plus.

01:17 All right.

01:18 What are the frequencies? the observed frequencies, the observed values, let me just write it like this, the observed values.

01:30 The observed values are 240, 214, and then we have 183, then we have 156.

01:45 And then we have 207.

01:49 Now, what is the distribution that is given to us? the distribution that is given to us.

01:56 So let me just write this as the probabilities.

01:58 The probabilities, okay? they are, for 2 to 17, this is 22%.

02:03 So let me just write this as 0 .22.

02:05 For 18 to 24, it is 21 % or 0 .21.

02:10 For 25 to 39 it is 24 % 0 .24.

02:15 For 40 to 49 it is 14 % 0 .14.

02:19 For 50 plus it is 19 % 0 .19.

02:25 Okay.

02:27 Now what is going to be the null hypothesis? for a goodness of fit test, the null hypothesis is always considered to be that the observed distribution matches the given distribution.

02:40 So over here, if we look at this question, the researcher claims that the ages of people who go to move at least once a month are distributed as shown in the figure.

02:52 We can say that the observed distribution, the observed distribution, this is very important to write, the observed distribution fits the the observed distribution and the distribution and the distribution in the distribution in the figure are similar are similar okay this is going to be our null hypothesis and what will be the alternative hypothesis the observed and the given distribution and the given distribution are not the same are not the same all right now the first step in calculating the kai square statistic is we need to find the expected value.

04:13 And how do we find the expected value? the expected value e for all the categories is given by the formula, sample size, sample size multiplied by the probability for each category, the probability for the probability of the category, the probability of the category.

04:43 Multiply the probability of the category.

04:46 So what is the sample size in this case? 1000.

04:49 And the probabilities will differ.

04:52 From category to category.

04:55 So now if i use my calculator, for the first one, it is going to be 1000 multiplied by 0 .22, which is 220, which is 220.

05:10 This should be the expected value.

05:11 This is the column of expected values.

05:17 These are the expected values.

05:19 Then over here we are going to have 210, then we'll have 240, then we'll have 140, and then we will have 190.

05:29 These are the expected values.

05:31 Now, what is the next step? the next step is calculating the kai square statistic.

05:36 How do we do that? well, we apply this formula over you.