Test the claim about the difference between two population mea...

Question

Test the claim about the difference between two population means and at the level of significance α. Assume the samples are random and independent, and the populations are normally distributed.

Claim: μ1<μ2, α=0.10, Assume (σ1)^2=(σ2)^2

Sample statistics:

x̅1=0.345, s1=0.305 , n1=11 and x̅2=0.515, s2=0.215, n2=9

Accepted Answer

Hello. In this video we are told that a researcher is comparing the average recovery times for two medications. We want to test the claim that the mean recovery time for medication A is less than that for medication B at a significance level of 0.05. We want to assume independent random samples and equal population variances. Our sample statistics are that the mean of the first population is 4.2 days. Our standard deviation is 1.1, and the sample size of the first recovery time is 15. The second mean for the second recovery time for the second medication is 5.0 days with a standard deviation of 0.9 and a sample size of 12. So, just by taking a look, what we want to go ahead and do is we want to do a hypothesis test. Now, let's go ahead and figure out what the original claim is. Now the original claim is that the average recovery time for two medicines. The test claims that the mean recovery time for medication A is less than that for medication B. So what we want to go ahead and do is we want to try to set this up to where we have to reject the null hypothesis. So we're going to first state our alternate hypothesis, and our alternate hypothesis is that the mean recovery time for medication A is going to be less than the mean recovery time for medication B. And what that means is that the opposite for our non hypothesis must be true, is that the mean recovery time for medication A is going to be greater than or equal to the mean recovery time of medication B. So this is going to be our hypothesis for the for the test. Now, in this case here, because we are given 2 samples with equal population variants, what we are going to do is we are going to do a 2 sample T test. We are doing a 2 sample tea test for the differences in the means. So, the first thing we want to do is we want to go ahead and first define our degree of freedom. Now, because we are using a two sample T test, our degree of freedom is going to be defined as the sample size for medication A plus the sample size for medication B minus 2. This is going to give us a degree of freedom. Of 15. Plus 12 minus 2 and this is going to give us a degree of freedom of 25. Now the next thing we want to do is you want to go ahead and find the pooled standard deviation. Now, in this case, the pulled standard deviation is defined as the standard deviation squared of both deviations pulled. That is going to be the sample size of medication A minus 1 multiplied by the standard deviation squad of medication A plus the sample size of medication B minus 1 multiplied by the standard deviation squared of medication B. This is then going to be defined by our degree of freedom. So by plugging in the given information, we're going to define our poles squared deviation as 15 minus 1. Multiplied by 1.1 squared. Plus 12 minus 1 multiplied by 0.9 squared and this will all be divided by a degree of freedom which is 25 and so this is going to leave us with a pooled standard deviation squared of approximately 1.034. Next, what we want to do is we want to take the square root of both sides of this equation in order to solve for just the standard deviation, and by taking the square root of both sides of this equation, we get the standard deviation or the pooled standard deviation is approximately 1.01686. We are going to set this to the side for now, because now that we have our polled standard deviation, we want to go ahead and find our test statistic. Now we want to go ahead and find a T test statistic. Now the T test statistic is going to be defined as the average of medication A. Minus the average of medication B. Minus. The mean of medication A. Minus the mean of medication B. All divided By the pooled sample standard deviation multiplied by the square root of 1 divided by the sample size of medication A plus 1 divided by the sample size of medication B. Now here because we are assuming that we have equal means we can go ahead and assume that the differences between the means of A and B is zero and so if we plug in our known information we can redefine our test statistic as 4.2 minus 5.0 divided. By the quantity 1.01686 multiplied by the square root of 1 divided. By 15 plus 1 divided by 12 and by plugging this into a calculator we get an approximate T test statistic of -2.031. Finally, what we need to do is we need to find our critical value. Now we are going to look for a T critical value that has a degree of freedom of 25 and has a significance level of 0.05. And of course, we are using a left tail test. By using a t table, we will get a t critical value of approximately -1.0708. Now, if we compare our critical value to the test statistic, notice that -2.031 is less than -1.708. So what this means is that our T test statistic is going to be less than our T critical value. So what does this tell us about our hypothesis? Well, this tells us to reject the null hypothesis. And since we are rejecting the null hypothesis, what this means is that there is sufficient evidence to suggest that the mean recovery time for medication A is less than the mean recovery time for medication B. And so the final answer to this problem is going to be option A. 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.

Key Concepts

Hypothesis Testing for Two Population Means

Pooled Variance and Equal Population Variances Assumption

Significance Level and One-Tailed Test

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