Vitamin A Supplements in Low-Birth-Weight Babies Low-birth-wei...

Question

Vitamin A Supplements in Low-Birth-Weight Babies Low-birth-weight babies are at increased risk of respiratory infections in the first few months of life and have low liver stores of vitamin A. In a randomized, double-blind experiment, 130 low-birth-weight babies were randomly divided into two groups. Subjects in group 1 (the treatment group, n1=65) were given 25,000 IU of vitamin A on study days 1, 4, and 8 where study day 1 was between 36 and 60 hours after delivery. Subjects in group 2 (the control group, n2=65) were given a placebo. The treatment group had a mean serum retinol concentration of 45.77 micrograms per deciliter (μg/dL), with a standard deviation of 17.07 μg/dL. The control group had a mean serum retinol concentration of 12.88 μg/dL, with a standard deviation of 6.48 μg/dL. Does the treatment group have a higher standard deviation for serum retinol concentration than the control group at the α=0.01 level of significance? It is known that serum retinol concentration is normally distributed.

Accepted Answer

Hello. In this video we are told that a researcher is studying the variability in blood glucose levels between two groups of diabetic patients. Group A of sample size 51 receives the new medication, while Group B of sample size 51 receives a placebo. The standard deviation of blood glucose levels in Group A is 22.5 mg, and Group B is 15.3. Assumed blood glucose levels are normally distributed at a 0.01 significance level. Is the standard deviation of Group A significantly higher than in Group B. OK, so let's go ahead and first list out the given information in the problem. For Group A, we know that the sample size of Group A is 51. And we have a standard deviation of group A. And that standard deviation is 22.5. For Group B, we have a sample size of 51 as well, and the standard deviation of Group B is 15.3. Now we also want to work with a significance level of 0.01. Now, the claim here is that the standard deviation in group A is significantly higher than the standard deviation of group B. So we have the standard deviation of A is greater than the standard deviation of B. Now in this case here, because we know the standard deviations of both of the samples and they are normally distributed, this tells us that this is going to be our alternate hypothesis. And with that being said, the null hypothesis is going to be the complement of this claim, and the complement of this claim is that the standard deviation of group A is less than or equal to the standard deviation of Group B. So this is going to be our null and alternate hypothesis. Now here, because we are trying to compare the population variances of two independent groups, and we are told that the blood glucose levels are normally distributed, we are going to approach this problem by using the F test. So let's go ahead and start this problem by finding our F test statistic. Now the F test statistic is defined as the following. It is going to be the standard deviation of group A 2 divided by the standard deviation of group B2. Now we do know these standard deviations. So by plugging in the known values, this is going to be 22.5 quantity squared divided by 15.3 quantity squared, and this is going to give us an approximate F test of 1.95 or 2.16. So let's go ahead and set that to the side for now. Now what we want to do is you want to find the critical values of the F test, but in order to find the critical value, we need to find the degree of freedom of both Group A and Group B. So let's go ahead and define the degrees of freedom. Now, the degree of freedom for group A is defined as the sample size of group A minus 1. That is going to be 51 minus 1, which gives us a degree of freedom of 50. The same can be said for the degree of freedom of Group B. The degree of freedom of Group B is going to be the sample size of Group B minus 1, which is also 51 minus 1, so 50. So we have equal degree of freedoms. The degree of freedom of Group A is equal to the degree of freedom of Group B, and they are both equal to 50. Now that we've calculated the degree of freedoms of the independent groups, we can go ahead and find the critical value by using the F critical charts. So for the critical value, we know that we are working with a significance level of 0.01, and we are working with a degree of freedoms, both of them being 50. So by the F chart, when both of the degree of freedoms are 50, and we have a significance level of 0.01, the F critical value is going to be 1.95. Now that we've found the critical value, we're going to compare the test value with the critical value. Now here we can see that 2.16 is greater than 1.95. So the F test test statistic is greater than the F critical value. And this tells us that we're going to reject the null hypothesis. Since we are rejecting the null hypothesis, what does this tell us about the original claim? Well, this tells us that there is sufficient evidence to conclude that the standard deviation of the blood glucose levels of Group A is significantly higher than the blood glucose levels of Group B. And so the solution to this problem is going to be Option B. So I hope this video helps you in understanding how to approach this problem, and we will go ahead and see you all in the next video.

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