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Samples taken in three cities, Anchorage, Atlanta, and Minneapolis, were used to learnabout the percentage of married couples with both the husband and the wife in the work-force $(U S A$ Today. January $15,2006) .$ Analyze the following data to see whether both thehusband and wife being in the workforce is independent of location. Use a .05 level of significance. What is your conclusion? What is the overall estimate of the percentage of married couples with both the husband and the wife in the workforce?

There is sufficient evidence to support the claim that the variables are notindependent.

Intro Stats / AP Statistics

Chapter 11

Comparisons Involving Proportions and a Test of Independence

Descriptive Statistics

Confidence Intervals

The Chi-Square Distribution

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this problem. We want to figure out whether the distribution of married couples and non married couples are the same across, um, or whether the distribution of married couples and location are independent of each other. So are no hypothesis is going to be that the Roman column variables air, not our independence. Our row and column variables are independent, and then the alternative hypothesis is going to be that the row and Colin variables are not independent. So what does this mean? It means that, um, the number of married couples in the workforce is independent of location. And this means that the number of married couples in the workforce is not independent of location. So the first thing we're going to do is come up with our totals. I kind of like, ignore this part up here for a little bit. We're going to figure out how much is in each, um, column and then finally, a grand total. So in the first column, we have a total of 90 people working in Anchorage, um, in Alaska, 120 in Minneapolis, 1 53 and then for the both are total is 1 90 and for only one total is 1 73 And how do I get these values I for this row or the column total? I take the sum in the column than for this row total. I take some of each row and then finally, to get a total total a grand total. I'm going to take the sum of all my column variables or call him, um, totals or my row totals. I will take some of row total. So 1 90 it's 1 73 is equal to 363. So this is our grand total now, because we're looking to do a hypothesis tests, we will be conducting a chi Square test. And this is our test statistic. And what this says is that the frequency, the observed frequency of each cell minus the expected frequency of each scale squared over the expected frequency in each cell. And the sum of all of those is going to be equal to our chi square test statistic. So how do we figure out what our expected frequency is? Our expected frequency is equal to the row total, so our row total times our column total divided by our grand total in the past. I called it the total total, but this is the formula for them. So I will do the ah, the expected, um, calculation for Anchorage and both Anchorage and both. So the expected value would be 90 times 1 90 divided by her grand total of 3 63 So we would get a value of 47 point one. Uh, that's approximately 40 7.1 47.1. So now we'll do this for all our cells, and we get, uh, this this table. So pause this video and copy this down if you want to know what's going on and if you want to Sorry s O. Now that we have this expected table, we have to come up with some the value for our test. Otis tick. And what we're going to do is take yeah, observed frequency minus the expected frequency squared, divided by the expected frequency for each cell. So for Anchorage and both this 11 cell we're going to do, um, we're going to take our are observed frequency of 57 57 minus our expected frequency of 47.1. We're going to square it divided by our expected frequency of 47.1. And here we get a value of approximately 2.8 So we do this calculation for all our cells and we get this table. We get this table over here now, with this table, we have to come up with our Chi Square test statistic and our chi square testes. Tick is the sum of all ourselves is the sum of all these values. So after re addle of these together we get a chi square approximately 13.73 But we're not done yet. We use our chi squared to come up with a P value. But to come up with R P value, we also need a degrees of freedom. So 13 0.73 and our degrees of freedom is equal to the number of rows. We have minus one times the number of columns we have minus one. We have to rose and we have three columns. So our degrees of freedom is equal to one times two, which is equal to so our p value. Given that we have degrees or high squared of 13.73 degrees of freedom of two is less than 20.5 So this is our P value. And now that we have our p value, we have to compare this to our Alfa. So because our P value is less than 0.5 which is less than our outfit, which equals 0.5 we can reject the no. We have sufficient evidence to reject the knoll. Um, so what does that mean? That means that the proportion of people working in or the proportion of married couples working in each place are independent of each other. So now we are asked to find, um the percentage of married couples with husband and wife in workforce. Um, so that is simply going to be the portion of both is simply going to be the frequency of people, all people in all locations in the workforce that are both so in this table. It would be this total right here, because that is the value for the sum of where both people in the Cup married couple are working, divided by our total of 3 63 Because that is our total sample. So this would be equal to 1 90 divided by 3 63 Since we're looking for a percent, we would multiply that by 100 and we get a value of approximately 52% of married couples with have both husband and wife in the workforce.

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