Conduct the hypothesis test and provide the test statistic and the P-value and/or critical value

Conduct the hypothesis test and provide the test statistic...

Key Concepts

Significance Level and P-value

The significance level (commonly set at 0.05) is the threshold for rejecting the null hypothesis, representing the probability of making a Type I error. The P-value is the probability of obtaining a test statistic as extreme as, or more extreme than, the one computed from the sample data, assuming that the null hypothesis is true. Comparing the P-value to the significance level (or comparing the test statistic to a critical value) helps in deciding whether to reject the null hypothesis.

Conclusion of the Test

Based on the chi-square test statistic and the corresponding P-value (or by comparing it to the critical value), a decision is made regarding the null hypothesis. If the test statistic exceeds the critical value or the P-value is below the significance level, the null hypothesis is rejected, suggesting that the observed data do not fit the assumed model. Otherwise, the null hypothesis is not rejected, implying that there is insufficient evidence to conclude that the data differ from the expected distribution.

Degrees of Freedom

Degrees of freedom refer to the number of values in the final calculation of a statistic that are free to vary. In the context of a chi-square goodness-of-fit test, the degrees of freedom are typically calculated as the number of categories minus the number of parameters estimated from the data (and often minus one more) and are essential for determining the critical value from the chi-square distribution.

Hypothesis Testing

Hypothesis testing is a statistical method used to decide whether there is enough evidence in a sample of data to infer that a certain condition holds for the entire population. It involves establishing a null hypothesis (typically a position of no difference or no effect) and an alternative hypothesis, then determining whether the observed data are sufficiently inconsistent with the null hypothesis to warrant rejecting it.

Goodness-of-Fit Test

A goodness-of-fit test is used to assess how well observed frequency data match expected frequencies from a specified probability distribution. It determines whether any observed differences are due to random chance or if they indicate a significant lack of fit between the data and the proposed model.

Poisson Distribution

The Poisson distribution is a discrete probability distribution that models the number of events occurring within a fixed interval of time or space when these events happen with a known constant mean rate and independently of the time since the last event. In fitting data to a Poisson model, one examines if the observed event frequencies are consistent with those predicted by the distribution.

Chi-Square Test Statistic

The chi-square test statistic is calculated by summing the squared differences between the observed and expected frequencies, divided by the expected frequencies for each category. This statistic quantifies the discrepancy between the observed data and the model's predictions, providing a basis for testing the validity of the assumed distribution.

Transcript

00:01 We want to run a hypothesis test based on data from world war ii in south london.

00:08 So south london was divided into regions, and those regions were a quarter square kilometer, and they analyzed how many bombs hit the regions.

00:23 Some of the regions had no bombs hit.

00:27 Others had one, two, three, or four plus.

00:31 And they found 229 of those quarter square kilometer regions had no bombs hit.

00:41 211 had 1, 93 had 2, 35 had 3, and 8 had 4 or more.

00:54 They then ran a pusan distribution, and they calculated the expected number of hits, and found 227 .5 of the regions were expected to have no bombs hit.

01:15 211 .4, we're expected to have 1 .97 .9 for 2, 30 .5 for 3, and 8 .7 for 4.

01:30 Now we're going to test a claim, and the claim is going to be what becomes our null hypothesis.

01:40 And the claim is that the actual frequencies of hits to each region fits the calculated poisson distribution.

02:19 Now, when you run a hypothesis test, you need an alternative hypothesis to fall back on.

02:26 So our alternative is going to be the actual frequencies of hits to each region.

02:44 Do not fit the calculated poisson distribution.

03:06 And in order to run this test, we are going to have to run the goodness of fit test, because we are asking how well did the actual numbers, or did the actual numbers, fit the expected numbers? and in order to run the goodness of fit test, you will need to generate, calculate a kai square test statistic and to calculate that kai square test statistic you are going to sum up observed minus expected values squared divided by expected so let's go back up to our chart and these are your observed values and here are your expected values so we are going to add a column on to our chart and we're going to call it o minus e quantity square divided by e.

04:14 So we'll take the observed value minus its corresponding expected value.

04:19 The answer we get will square before dividing it by the expected value.

04:25 Now the fastest approach to this would be to put the data into your graphing calculator.

04:30 So i'm going to just clear out a list for a second here and then i'm going to go stat, edit, and as you can see, i have list one with all the observed values, and list two already has the expected values.

04:51 So i'm going to sit on top of list three, and i'm going to tell the calculator, please take all the observed values from list one, subtract their corresponding expected values from list two, square that deviation, and then, then divide it by all the expected values in list 2.

05:13 And you will get these decimals.

05:16 Now for the sake of recording them, i'm going to record them out to three non -zero decimals.

05:23 So the first one would be .00989, 0 .000757.

05:34 0 .245 .6 .4.

05:40 And 0 .056.

05:43 0 .0563.

05:44 Now, in order to calculate that kai square test statistic, i must add all of these values together.

05:52 Again, the fastest approach would be to come back to that calculator, and i'm going to tell the calculator to add up everything in list 3.

06:02 So in order to do that, i'm going to quit out of there.

06:05 I'm going to hit second stat.

06:08 I'm going to scoot over to math, and i'm going to hit number five to sum up list three.

06:15 And in doing so, i get my kai square test statistic to be approximately 0 .976.

06:24 So i'm going to come back to our screen here, and our kai square test statistic was 0 .976.

06:34 16153, 4, 8 ,9, 6.

06:39 Now we're ready to find our p value.

06:46 And our p value, we're really asking what's the probability that kai square is greater than that test statistic.

06:54 So our test statistic was 0 .976153, 4, 8, 96.

07:02 And to get a better handle on that, i recommend that we draw a picture.

07:10 So this is what the kai square distribution would look like.

07:14 It is a skewed right distribution.

07:19 And the shape of it is dependent on the degrees of freedom.

07:24 And the degrees of freedom are found by doing k minus 1.

07:28 And k represents the number of categories we have split our data into.

07:34 And if we go back to our chart, you can see we've split.

07:39 Split our data into five different categories.

07:46 So our k value will be five, and our degrees of freedom will be four.

07:53 Now, not only does our degrees of freedom kind of dictate the shape of our curve, it also tells us what the mean of our distribution is.

08:04 So the mean of our kai square distribution will also be four.

08:08 And you find that mean always slightly to the, the right of the peak of that curve.

08:15 So we know the four is right here.

08:19 Now, we came up with a test statistic of .976 -ish, so that's all the way back here.

08:29 So our p value is our likelihood of being greater than that.

08:34 So we're talking about the area of the curve into the right tail.

08:39 Now, the fastest approach is to use our kai square cumulative density function...

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