Putting It Together: Lupus Based on historical birthing record...

Question

Putting It Together: Lupus Based on historical birthing records, the proportion of males born worldwide is 0.51. In other words, the commonly held belief that boys are just as likely as girls is false. Systematic lupus erythematosus (SLE), or lupus for short, is a disease in which one’s immune system attacks healthy cells and tissue by mistake. It is well known that lupus tends to exist more in females than in males. Researchers wondered, however, if families with a child who had lupus had a lower ratio of males to females than the general population. If this were true, it would suggest that something happens during conception that causes males to be conceived at a lower rate when the SLE gene is present. To determine if this hypothesis is true, the researchers obtained records of families with a child who had SLE. A total of 23 males and 79 females were found to have SLE. The 23 males with SLE had a total of 23 male siblings and 22 female siblings. The 79 females with SLE had a total of 69 male siblings and 80 female siblings.
f. Does the sample evidence suggest that the proportion of male siblings in families where one of the children has SLE is less than 0.51, the accepted proportion of males born in the general population? Use the α = 0.05 level of significance.

f. Does the sample evidence suggest that the proportion of male siblings in families where one of the children has SLE is less than 0.51, the accepted proportion of males born in the general population? Use the α = 0.05 level of significance.

Accepted Answer

Hi everyone, let's take a look at the next problem. Researchers identified 27 men and 91 women with a certain inherited trait. The 27 men had 29 brothers and 25 sisters. The 91 women had 85 brothers and 97 sisters. At the alpha equals 0.05 significance level. Does the data suggest that the proportion of male siblings in these families is less than 0.50, the expected proportion in the general population? No, there's not enough evidence to conclude the proportion is less than 0.50. Yes, the proportion is significantly less than 0.50. See, yes, the proportion is significantly greater than 0.50. Or no, the proportion is exactly 0.50. Well, two of these we can rule out right away. But see, yes, the proportion is significantly greater, but that doesn't make sense as an answer, because the question is asking, is the proportion significantly less. So our answer wouldn't be yes, if it were significantly greater. This is contradictory. So we're testing a hypothesis in this case, and we would say that our null hypothesis. It would be that our proportion is equal to 0.50. The expected uh percentage, the male siblings. So our hypothesis is about our proportion of male siblings. And our alternative hypothesis is that our proportion of male siblings is less than 0.50. So, we want to test this hypothesis. It's a hypothesis about proportion. So, our test statistic that we're going to use is going to be Z. And we'll use the formula Z equals Phat. Minus P and then that all divided by. The standard error So we already know, the uh population proportion is 0.50. We can calculate our sample proportion Phat. So that's going to equal the total number. Of male siblings in this group, divided by the total number of siblings in general. So that's quite equal. Let's look at our sample. Remember, we're not counting the men and women that were have the trait. We're just looking at their siblings. So it says the 27 men have 29 brothers and 25 sisters. So, on the numerator here, for male siblings, we had 29 for the 29 brothers. I'm going to make a line for my fraction, and then a denominator, they had 29 brothers, so put 29 and 25 sisters. So we also have to add 25 down in the denominator, since that's total siblings. Now, let's look at the women in this group, says the 91 women had 85 brothers, so in the numerator, we add 85 for the women's brothers. And the denominator we add 85 of their brothers. It says they had 97 sisters, so we had 97. So, in our numerator, we have 29 plus 85, the denominator 29 plus 25, plus 85 plus 97. And when we add up those numbers, our proportion here. Will equal 114. That's the total number of male siblings, divided by 236, the total number of siblings. Again, we're not looking at the sample of people with a trait. Our group is siblings because we're looking at proportion of siblings. That's going to equal 0.483. And now we need to calculate the standard error, which we'll only need for Z value. So, we have our formula for standard error, S E equals. And we have the square root, and then under the square root sign, and the numerator, we have P, multiplied by in parentheses 1 minus P. And then that is divided by N and the denominator. That's all under a square root sign. So, that's going to equal square root. Well, in the numerator, sorry, my pen is being a little temperamental. We know that P is 0.50. And 1 minus p is also going to be 0.50. And then we're going to divide that by N, the number in our entire sample, so the total number of siblings, which was 236. Just make a little note here that that's N, so we remember. 236, and then square root of all that. So, give you a moment, you can plug this into your calculator. But that will give you erase my equal signs here, but approximately. 0. 0325. So we've got everything we need to calculate our Z value. So, as we said, Z was equal to. Draw a little bubble here. Pea hat. So our pea hat was, we calculated that. 0.483, and then we subtract P to 0.50. All that is divided by our Standard error, which we've calculated as 0.0325. So we put that all in, and we're going to end up with approximately -0.52 for our test statistic are Z value. Go ahead and highlight that in blue, so that's important. Now, to test our hypothesis, we compare the Z value to the critical value for our particular significance level, which is alpha equals 0.05. So we look at. The table, you can look at the tables in the back of your textbook for hypothesis testing. And look up the left-tailed critical value, because we're looking at a less than, so I'm up here, we're gonna say we have a need to use a Left Tailed test here. Recall we have draw a little quick distribution. I mean we're gonna have a little area to the left and to see is it significant? Is there enough evidence to reject our Z value needs to fall to the left side of the critical Z value. So the Z sub C. So left tailed. And looking at alpha equaling 0.05. And that critical Z value Z subc is going to be equal to. -1.645. Again, we look at that, look at that in our table. It's a little brackets around there and right table. So, now we need to compare these values. So we say our our own Z value, which is approximately 0.52. How does that compare with our critical Z value? I'm gonna be careful with negative signs. This is greater than -1.645, which is our critical Z value. And for a left-tailed test, that means that we fail to reject. The null hypothesis. So, go back, I should have said earlier, we could have ruled out choice D, the proportion is exactly 0.50, because we calculated the sample proportion. And we calculated that to be 0.483. So we already ruled that one out. But Again, we fail to reject our null hypothesis. So, our answer here is going to be, choice A, no, there's not enough evidence to conclude that the proportion is less than 0.50. Because we were unable to reject the null hypothesis. So, we can also then rule out choice B. Yes, the proportion is significantly less, because again, that's what we showed when we compared our Z to the critical Z value. So, overall, looking at this problem, we had a group with, you know, where we listed the, the, the siblings of the people in our group, how many brothers, how many sisters, and we wanted to know at the alpha equals 0.05 significance level. Does the data suggest that the proportion of male siblings in these families is less than 0.50, expected proportion of the general population. And we concluded choice A, no, there is not enough evidence to conclude the proportion is less than 0.50. See you in the next video.

Key Concepts

Hypothesis Testing for Proportions

Significance Level (α) and p-value

Sampling and Independence in Family Data

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