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

Key Concepts

Interpretation in Context

Interpreting the results of the chi-square test in context involves explaining what the decision (rejecting or failing to reject the null hypothesis) means for the real-world scenario being studied. This step translates the statistical conclusion back into the context of the original claim, indicating whether the observed distribution provides support for the claim of a different distribution.

Decision Making

Decision making in hypothesis testing involves comparing the computed test statistic to the critical value. If the test statistic exceeds the critical value, the observed discrepancies are considered statistically significant, leading to the rejection of the null hypothesis. Otherwise, there is insufficient evidence to reject the null hypothesis.

Test Statistic

The chi-square test statistic is calculated using the formula which sums the squared difference between observed and expected counts divided by the expected counts for each category. This statistic provides a summary measure of the overall discrepancy between the observed data and the expected model under the null hypothesis.

Critical Value and Rejection Region

The critical value is a threshold obtained from the chi-square distribution table based on the chosen significance level (alpha) and the degrees of freedom. The rejection region consists of all test statistic values greater than the critical value, and if the computed chi-square test statistic falls in this region, the null hypothesis is rejected.

Null and Alternative Hypotheses

In hypothesis testing, the null hypothesis (H0) represents the baseline or default assumption about a population parameter—in this case, that the observed distribution fits the expected distribution. The alternative hypothesis (Ha) posits that there is a significant difference between the observed and expected distributions, indicating that the distribution does not match the claimed model.

Expected Frequencies

Expected frequencies are the theoretical counts calculated under the assumption that the null hypothesis is true. They are derived from the proportions specified by the expected distribution and are essential for computing the chi-square test statistic by providing a basis of comparison for the observed frequencies.

Chi-Square Goodness-of-Fit Test

The chi-square goodness-of-fit test is a statistical test used to determine how well an observed distribution of categorical data matches an expected distribution. This test calculates the chi-square test statistic, which measures the cumulative discrepancy between observed and expected frequencies, and compares it to a critical value based on the chi-square distribution.

Hypothesis Testing

Hypothesis testing is a systematic method used to decide whether there is enough evidence to reject a null hypothesis about a population parameter. It involves comparing observed data to what would be expected if a certain claim were true, and deciding if discrepancies are due to random chance or indicate a significant effect.

Degrees of Freedom

Degrees of freedom in the chi-square goodness-of-fit test are computed as the number of categories minus one. They represent the number of independent pieces of information upon which the estimation of parameters is based, and are used to determine the appropriate critical value from the chi-square distribution.

Transcript

00:01 What does this question at hand have to say? a personal director claims that the distribution of the reasons workers leave for their jobs is different from a given distribution.

00:13 A distribution is given to us in a figure.

00:15 So in order to see if his claim is true or not, what we do is we randomly select 200 workers who recently left their jobs and we record their reasons for doing so.

00:26 So what we have with us is survey results.

00:29 All right so let us just draw the diagram which is of always the first step that we should do when solving these questions the first one is response okay then we will have the observed values the observed values okay the first response is limited advancement potential so let me just write this as l a and this frequency was 78 then lack of recognition then we had lack of recognition lack of recognition for this there were 52 observations then we have low salary or benefits then we have low salary or benefits okay for this we have 30 then we have unhappy with the management then we have unhappy people and this is 25 then we have board don't know okay the last category is that of board people these are 15 in number and if i add this 78 plus 52 plus 30 plus 25 plus 15 is 200 this adds up to 200 so our sample size n is 200 all right now what is the claim the personal director claims that the distribution of the reasons workers leave their job is different from the distribution shown in the figure.

02:33 Okay.

02:35 So we are asked about the null and the alternative hypothesis.

02:38 The null hypothesis in goodness of fit test is that the distributions are same.

02:44 So what we can write over here is the distribution, the distribution of the reasons, of the reasons workers leave their jobs, leave their jobs and the given distribution and the given distribution are same or are similar are similar what will be the alternative hypothesis the alternative hypothesis is the claim of the personal directive so this is going to be that the distribution of the reasons the distribution of the reasons and the given distribution and the given distribution differ they are different all right so we are going to use the kai square statistic kai square goodness of fit test what is the first step the first step is to find the expected values the expected values for the categories how do you find the expected values expected value for a category is given by the formula sample size that is n which happens to be 200 in this case multiplied by the probability of the category multiplied by the probability of the category so this is going to be pi okay let's apply this over here first we need to find the probability what were the probabilities okay so you will have to erase this first will come the probability column okay so let these be the probabilities probabilities okay so what other probabilities limited advancement and potential that is lap is 41 so this is 0 .41 then lack of recognition is 25 % so this is 0 .25 then low salary benefits is 15 % 0 .15 then unhappy is 10 % so this is 0 .1 1 and unhappy is 10 % so this is 0 .1 and board is 9 % so this is 0 .09 all right now we'll have the expected values the expected values okay so 200 and 2 .41 this is going to be 82 then we have 1 4th of 200 which will be 50 then we have 15 % this will be 30 then we have 10 % which is 20 and then we have 10 % which is 20 and then we have of 9 % which is 18.

06:26 All right.

06:27 Now we have the expected values.

06:29 What is the next step? the next step is finding the kai square statistic.

06:34 How do you do that? for all the categories, we are going to find the difference between observed and expected values.

06:41 We are going to square that difference.

06:44 Divide the squared value by the expected value.

06:46 And in the end, we will sum this for all the categories.

06:49 And this will give us the overall kai square statistic.

06:53 Okay.

06:54 Let us look at this formula and action these will be the kai square values all right if i use my calculator for this one i have 78 and 82 the difference is 4 the square becomes 16 and i divide this by 82 this becomes 01951 so let me just write this as 0 .195 then the difference over here is 2 the square will become 4 so 4 divided by 50 4 divided by 50 this becomes 0 .08 0 .08 then over here the difference is 0 then over here the difference is 5 5 square is 25 and 25 divided by 20 turns out to be 1 .25 this should be 1 .25 and then the difference is 3 3 square is 9 9 divided by 18 is 0 .5 so this is 0 .5 now if i add all of these up this is 0 .25.

08:09 15195 plus 0 .08 plus 0 plus 0 .25 plus 0 .5.

08:20 The answer is 2 .025.

08:23 My kai square statistic is 2 .025.

08:29 All right.

08:31 Now i have my kai's statistic but i also need the degrees of freedom...

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