For Exercises 9 through 24, perform the following steps. Assume that all variables are normally

For Exercises 9 through 24, perform the following steps....

Key Concepts

Hypothesis Testing Framework

Hypothesis testing is a systematic method for making decisions about population parameters based on sample data. It involves formulating a null hypothesis that represents a baseline assumption and an alternative hypothesis that represents a claim to be tested, then using test statistics and significance levels to accept or reject the null hypothesis.

Null and Alternative Hypotheses

These statements form the foundation of the test. The null hypothesis typically asserts that there is no difference or effect (for example, equal variances between groups), while the alternative hypothesis claims that a difference exists. Clearly defining these hypotheses is crucial because they determine the direction and nature of the statistical test.

Significance Level

The significance level, denoted by alpha (?), is the threshold for deciding whether the observed effect is statistically significant. It represents the probability of rejecting the null hypothesis when it is actually true (a Type I error). A common choice, as in this example, is 0.05, meaning there is a 5% risk of a Type I error.

F-Test for Equality of Variances

The F-test is used to compare the variances of two independent normally distributed populations. It involves computing the ratio of the two sample variances, with the larger variance placed in the numerator, and determining whether this ratio is significantly different from 1. Under the null hypothesis of equal variances, this ratio follows an F-distribution, making it possible to assess the claim of variance difference.

Critical Value

A critical value is the cutoff point on the test statistic's probability distribution that defines the boundary of the rejection region for the null hypothesis. In hypothesis testing, if the computed test statistic falls beyond this critical value, the null hypothesis is rejected. The critical value is determined based on the significance level and the degrees of freedom of the test.

Test Statistic

The test statistic is a standardized value calculated from sample data that is used to determine whether to reject the null hypothesis. For an F-test, the test statistic is the ratio of sample variances, and its value is compared against the critical value from the F-distribution to conclude if the difference in variances is statistically significant.

Decision Rule and Conclusion

The decision rule involves comparing the computed test statistic with the critical value: if the test statistic exceeds the critical value (or falls in the rejection region), the null hypothesis is rejected. This step concludes the hypothesis test by summarizing the results, indicating whether the evidence supports a significant difference in variances, based on the predetermined level of significance.

Transcript

00:01 What's up, statcast? my name is erin, and in this video, we're going to be given an example, and we're going to be performing the following steps for it.

00:10 So step a is we're going to state our hypothesis and identify the claim.

00:15 Then we're going to find our critical value.

00:17 Then we're going to compute our test value.

00:19 We're going to make a decision on our test, and then we're going to summarize the results.

00:23 And this is known as the traditional method of hypothesis testing.

00:27 So the example we're given, we're given raw data, and this is a number of pups per pack of montana and idaho wolf packs, and our alpha level is 0 .05.

00:42 So what we want to know is does the variance and the number of pups per pack differ in between these two different locations? so step a, we're going to state our hypothesis and identify our claim.

01:00 And because we just want to know if the variances in the number of pups per pack differ, that is going to tell us something about our alternative hypothesis.

01:13 So it's going to look something like this.

01:17 So this is our null hypothesis, and it's going to be that our variances are equal because the null hypothesis is always that there is no change.

01:32 So our variances are equal to each other.

01:34 Now, our alternative hypothesis is pretty much the exact opposite in this case.

01:40 It's going to be that our variances are not equal to each other.

01:48 And because this is a not equal symbol, this tells us that we're going to be doing a two, two -tailed test.

02:02 So because it's a two -tailed test, our alpha level is going to be divided in two.

02:13 And this is the value that we're going to use for our each table in the next step.

02:22 Sorry, i have a little frog in my throat.

02:26 So step b is find the critical, oh, i'm sorry.

02:30 So step a, so we stated our hypothesis.

02:32 Now we have to identify our plane and our claim is the alternative hypothesis.

02:42 So step b we're going to find our critical value.

02:46 So because we have a two -tail test, our alpha level is now 0 .025 and that's the alpha level we're going to use for our h table.

02:55 So to find the critical value we need to look at the h table.

03:05 And if we pull this up, i'm going to pull the 0 .025 h table up.

03:17 So here's that alpha level, you can see that we're going to need degrees of freedom of the numerator and degrees of freedom of the denominator before we can actually find that critical value.

03:28 So to find that, we have to look at the sample sizes for montana and idaho.

03:33 This is the sample size for montana and this is the sample size for idaho.

03:37 So we have 13 observations for montana and 12 observations for idaho.

03:44 And to find which of these we are going to use as the degrees of freedom for the numerator, we have to calculate the variances of the sample because the larger variance, the larger variance always goes in the numerator.

04:13 So before we find the degrees of freedom that we're going to use on the h table, we have to find the variances of our samples because as you can see we were just given the raw data.

04:27 So we have to calculate those numbers ourselves.

04:34 So what i'm going to do is i'm going to pull up an excel spreadsheet and i'm just going to walk you guys how to calculate these numbers on excel.

04:43 So this is going to be our variance for montana and this is going to be our variance for idaho.

04:48 So i'm going to pull up that excel sheet.

04:53 And i already entered the data.

04:55 So these are our observations for montana, and these are our observations for idaho.

05:02 And i'm just going to go ahead and delete these.

05:04 So i can just show you guys step by step.

05:08 So in this cell, i'm going to calculate the standard deviation because we find the variance by squaring that number.

05:18 So i'm going to put an equal sign.

05:21 I'm going to type in standard deviation.

05:24 And then i'm going to highlight these examples.

05:28 Close those parentheses.

05:31 Enter.