In each exercise,
e. interpret the decision in the conte...

Question

In each exercise,

e. interpret the decision in the context of the original claim.

[APPLET] In Exercises 1–3, use the data, which list the hourly wages (in dollars) for randomly selected surgical technologists from three states. Assume the wages are normally distributed and that the samples are independent. (Adapted from U.S. Bureau of Labor Statistics)

Maine: 22.76, 27.60, 25.08, 17.01, 30.15, 27.09, 20.95, 25.52, 20.11, 23.67, 24.32

Oklahoma: 24.64, 21.66, 19.38, 18.19, 23.14, 20.58, 19.53, 30.77, 27.46, 23.80

Massachusetts: 27.07, 24.71, 32.80, 28.34, 33.45, 33.36, 36.81, 30.04, 29.01, 24.30, 29.22, 29.50

Are the mean hourly wages of surgical technologists the same for all three states? Use α=0.01. Assume that the population variances are equal.

Accepted Answer

Hello. In this video, we are told to suppose that we are comparing the average sleep hours per night for individuals in 3 different occupations. The independent random samples of size 7 were taken from each occupation, and the computed test statistic is F is equal to 2.45 for testing whether the mean sleep hours differ among the three occupations. We want to assume normality of the populations and equal population variances, and we want to use a significance level of 0.05. What conclusion can be drawn about the mean sleep hours across the populations or the occupations. OK, so in order to approach this problem, the first thing we want to do is we want to create our null and alternate hypothesis. Now the problem tells us is that we are comparing the average sleep hours per night between 3 different occupations. So what this means is that our null hypothesis is going to state that the average sleep hours per night between the three occupations are going to be exactly the same. There's no differences. But that means that our alternate hypothesis is going to claim the complement which states that at least one occupation. Has a different average of sleep hours per night from each from the other groups, so at least one occupation differs from the others. So this is going to be our null and alternate hypothesis. Now let's take a look at the given information. Now what's important here is that we are taking random samples of size 7 from each occupation. And that means that our total sample size is going to be 7 x 3, because we are taking 7 individuals from 3 different groups. That is going to give us a total sample size of 21. We are also given the F test statistic for this problem. The F test statistic is given to us as 2.45, and we want to work with the significance level of 0.05. So, what we really need to do here is we need to find our critical value. But in order to find our critical value, we first need to find two different degrees of freedom. We need the degree of freedom between groups. And we need the degree of freedom within the groups. So in order to find the critical value, we need to find these first these degree of freedoms first. Let's go ahead and start with the degree of freedom between the groups. Now, the degree of freedom between groups is defined as the following. It is the number of groups. -1. Now, because we are comparing the mean average sleep hours between the three different occupations, that means that the number of groups we are working with is 3. So the degree of freedom is going to be 3 minus 1, which gives us a degree of freedom of 2. This is going to be the first degree of freedom that we need for the critical value. Now we repeat this process, but with the degree of freedom within the groups. Now the degree of freedom within the groups is defined as the following. It is the total sample size. Minus the number of groups. Now we know that the number of groups is going to be 3. But earlier we stated that we're taking 7 independent samples from each of the groups, so our total sample size is going to be 21. So the degree of freedom within the groups is going to be 21 minus 3. Which is going to give us a degree of freedom of 18. That will be the 2nd degree of freedom we need for the critical value. Now that we have the 2 degrees of freedom, we can calculate our critical value by using an F distribution table. Now again, we are working with a significance level of 0.05. The first degree of freedom we calculated is 2, and the second degree of degree of freedom we calculated is 18. So by using an F distribution table, the approximate F critical value is going to be 3.55. Now that we have the F critical value, we're going to compare the critical value to the test statistic. Now notice that 2.45 is less than 3.55. So what this tells us is that our test statistic is less than our critical value. So what does this tell us about the conclusion? Well, since the test statistic is less than the critical value, we fail. To reject. Are null hypothesis. Since we fail to reject our null hypothesis, what this means is that there is not enough sufficient evidence to suggest that the mean sleep hours differ among the three groups of occupations. And so with that being said, the correct solution 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.

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