Perform the following steps. a. State the hypotheses and identify the claim. b. Find the critica

step 1 Let's start this problem by looking at the observed data. So we had officers and enlisted member

Perform the following steps. a. State the hypotheses and...

Key Concepts

Hypothesis Formulation

This concept involves stating the null hypothesis (H?), which typically asserts that there is no effect or no association, and the alternative hypothesis (H?), which claims that there is an effect or an association. In hypothesis testing, these statements are used to set up the framework for evaluating the evidence provided by data.

Significance Level

The significance level, denoted by ?, is a threshold that defines the probability of making a Type I error – rejecting the null hypothesis when it is actually true. It serves as the standard for deciding whether the test statistic indicates a statistically significant result.

Test Statistic Calculation

Calculating the test statistic involves summarizing the data into one number through a formula that measures the degree of departure from what is expected under the null hypothesis. This value is compared to a reference distribution to decide whether the observed data are extreme enough to warrant rejecting H?.

Critical Value and Rejection Region

This concept pertains to determining a cutoff point (critical value) from the appropriate statistical distribution corresponding to the set significance level. The rejection region includes all values of the test statistic that lead to rejection of the null hypothesis, guiding the decision-making process in hypothesis testing.

Decision Making in Hypothesis Testing

Decision making involves comparing the computed test statistic to the critical value or evaluating the p?value against the significance level. Based on this comparison, one either rejects or fails to reject the null hypothesis, thereby drawing conclusions about the validity of the claim under investigation.

Summarizing Results

After performing the test and reaching a decision, it is crucial to translate the statistical conclusion into context. This involves articulating the practical implications of the test outcome, whether it supports or refutes the initial claim, and discussing the evidence in simple terminology.

Transcript

00:01 Let's start this problem by looking at the observed data.

00:05 So we had officers and enlisted members, and actually they were women, of the armed forces, and some were in the army, others in the navy, marines, and air force.

00:39 And the numbers were as follows, 10 ,791, 62 ,490.

00:47 7 ,816, 42 ,750, 932, 9 ,525, 11 ,825, 11 ,819, and 54, and we are trying to test whether there is a relationship between rank and the branch of armed services.

01:23 So what we need to do is where you're going to need to run a kai square test.

01:28 And in order to run a kai squared test, we are going to need the totals of your observed data.

01:40 So therefore, i am going to add on a total row and a total column.

01:53 And our total officers was 31 ,358, and our total enlisted was 169 ,110, for a total of 200 ,000 2468 women.

02:15 Now, going across in the air force, there were 663, in the marines, there was 10 ,000, thousand four hundred fifty seven in the navy was fifty thousand five hundred sixty six and in the army there was 73 ,282.

02:44 So in order to run a kai square test, we are going to have to first write a set of hypotheses.

02:53 So that's part a of this problem.

02:56 So we are going to have our null hypothesis and our alternative hypothesis.

03:04 And we are trying to establish whether there is a relationship between rank and armed services.

03:11 So we could say rank and branch of the armed services are dependent or we can say rank and branch of the armed services are independent.

03:56 And you're null hypothesis is always the statement about independence.

04:03 And since we are trying to conclude that a relationship does exist, this is our claim.

04:15 So in part b, we need to discuss the critical value.

04:23 So when we do a kai square test, it's always nice to look at the actual kai square graph.

04:32 And the kai square graph is a curve that is skewed to the right, and the shape of the curve is dependent on the degrees of freedom.

04:43 And we find degrees of freedom by taking the number of rows minus one, multiplied by the number of columns minus one.

04:53 Well, in the data, we had four rows, and we had two columns.

05:05 So therefore, our degrees of freedom was three.

05:08 And our degrees of freedom, talks or tells us about the mean of the graph and it's always found slightly to the right of the peak.

05:16 So we have a kai square value of about three right there.

05:23 And our kai square is skewed to the right so we can say that kai square is zero where the x and the y axis intercept.

05:33 Now to find our critical value, we have to look at the significance level or the alpha that we are trying to work with.

05:42 And we are trying to run a hypothesis test at a 0 .05 significance level.

05:48 So what we're going to do is we're going to put 0 .05 in that right tail.

05:55 So we need to find the boundary between these two parts of our kai square image.

06:05 So this here on the right side of that boundary line is the fail to reject the null hypothesis region.

06:21 And when you're in the part with the 0 .05, that is going to be your reject the null hypothesis region.

06:33 And we're just trying to figure out what's that boundary line that's going to separate them.

06:38 And in order to find that boundary line, you're going to have to utilize your chart in the back of the book.

06:46 So you're going to go to the kai score distribution in the back of the book.

06:52 And down the side, it talks about the degrees of freedom.

06:56 And across the top, it talks about your level of significance.

07:00 So we're going to go across the top to 0 .05.

07:04 And we're going to go down the side to we get to about 3.

07:09 And where those two line up is going to be our critical value.

07:14 And where those two line up, it's going to be 7 .815.

07:20 So that's saying a kai square of 7 .815 is the boundary line from when we pass from the reject region into the fail to reject region or vice versa.

07:33 So that would be your answer for part b.

07:39 And part c, sorry.

07:47 Next thing is going to be to find your kai square test value or your kai square test statistic.

07:58 So part c, we're trying to find our kai square test value.

08:09 And some books say that it's a test statistic.

08:15 And in order to find that, you're going to utilize the formula, the sum of all the observed values minus their corresponding expected values, that difference is squared, and then we divide it by the expected value.

08:32 So that means we've got to go back up to our problem, our chart, and we've got to put together our expected values.

08:42 So again, i'm going to draw a chart, and i'm only going to do a two -column chart here.

08:55 We're going to have the officers, and i'm just going to back up and move it over a little bit.

09:06 We're going to have our officers and our enlisted.

09:19 And we've got army, navy, marines, and air force.

09:27 And to find your expected values, to find expected values, what you do is you take the sum of the row, multiply it by the sum of the column and divide it by the total surveyed.

09:57 So if we're trying to calculate the value that's going to go right here, well, that's in this row, and actually i'm going to take it further, that's in this row, and it's in this column.

10:17 So i'm going to take the sum of the row, and i'm going to take the sum of the row, and i'm going to bring in my calculator as i do it.

10:24 So i'm going to take the sum of the row, which is going to be 73 ,282.

10:31 I'm going to multiply it by the sum of the column, which is 3158, and i'm going to divide it by the total surveyed.

10:41 And there were 200 ,000 -4168 total surveyed.

10:48 And i'm going to just round to two decimal places, so i get a value of 11 ,463 .06 .11 ,463 .06.

11:02 Now i'm going to continue to do that.

11:04 So now i want to fill in this space.

11:10 And that space is in this row and this column.

11:19 So i'm going to bring in my calculator and i'm going to take the row total, 50566.

11:25 Multiply it by the column total 31358 and divided by the total number surveyed 200 ,468.

11:34 And i'm going to get an answer of 7 ,909 .73.

11:45 And now i'm going to do the next one.

11:49 So i want this position right here, which is the marines row and the officer's column.

11:58 So i'm going to take 10 ,457, multiplied by 31 ,358, divided by 200 ,468, and i will get 1 ,6355 .73.

12:22 1635 .73.

12:24 I'm going to keep that pattern going.

12:27 So now i'm going to complete this one.

12:31 So that is the air force row, the officer column.

12:42 I'll bring in my calculator.

12:44 So i'll do 6613 times 31358 divided by 20468.

12:53 And i'm going to get 10 ,349 .48.

13:04 I've got to keep it going.

13:06 Now i've got to do the enlisted army numbers.

13:10 So army and enlisted so army sum is 7382 times the enlisted sum is 169 -110 divided by total surveyed and we get 61818 .94 618181894 next i want to do right here so that's the navy row and the enlisted column...

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