28.2 - One Continuous Response

Now, let's turn our attention towards those methods that are appropriate for the case in which we have a continuous response. You know... that means a response that falls in an interval of values... such as, weight (in pounds), temperature (in degrees Fahreneit), or grade on a statistics final exam. First, let's consider only those methods that are appropriate for the case in which we are studying just one group... such as, high school freshmen, six-year-old girls, or moray eels.

One Group with a Continuous Response

Suppose we are interested in learning about the length of the population of moray eels. In that case, we are studying just one group, namely, the population of moray eels. Then, we take a random sample of n moray eels from that population and determine the length of each specimen selected. In that situation, we have a continuous response, namely, the length of the eel. As soon as we determine that we are studying one group with a continuous response, we should be thinking means, means, means, or .... errr.... medians, medians, medians. Which is the more appropriate summary statistic, of course, depends on the distribution of the data, that is, whether it is symmetric or skewed. At any rate, the mean or the median is a natural way of summarizing the observed data, so therefore the statistical methods we should potentially use must necessarily concern either means of medians. Specifically, our options are:

performing a t-test for one mean
performing a sign test or signed rank test for one median
calculating a t-interval for one mean
calculating a distribution-free confidence interval for a median or a general percentile

Again, what we choose depends on our specific research question. If the research question is an "is it this?" question, then we'd want to conduct a hypothesis test, whereas if it's a "what is it?" question, we'd want to calculate a confidence interval. Once we determine the appropriate statistical method, Minitab can do the dirty work for us using these commands:

Stat >> Basic Stat >> 1-Sample t... to conduct a t-test for one mean or to calculate a t-interval for one mean
Stat >> Nonparametrics >> 1-Sample Sign... to conduct a sign test
Stat >> Nonparametrics >> 1-Sample Wilcoxon... to conduct a signed rank test for one median or to calculate a distribution-free confidence interval for a median

The details about how to perform the analyses in Minitab, as well as about the assumptions that must be made in each case, can be found in the relevant lessons.

Example 28-5

What is the mean length of the pointer finger of the population of college students?

Answer

The research question involves the study of one group, namely college students. The research question involves a continuous response... the length of the pointer finger of a randomly selected college student. The research question is a "what is it?" question, and therefore involves calculating a confidence interval. If $\mu$ is the (unknown) mean length of the pointer finger of college students, then we are specifically interested in estimating $\mu$. We can enter the resulting data into Minitab and then ask Minitab to calculate a t-interval for one mean for us.

Example 28-6

Is the mean IQ, as measured by the Stanford-Binet IQ test, of the population of graduating college seniors greater than 115?

Answer

The research question involves the study of one group, namely graduating college seniors. The research question involves a continuous response... the score on the Stanford-Binet IQ test. The research question is a "is it this?" question, and therefore involves conducting a hypothesis test. If $\mu$ is the (unknown) mean IQ score of graduating college seniors, then we are specifically interested in testing the null hypothesis $H_0: \mu = 115$ against the alternative hypothesis $H_A: \mu > 115$. We can enter the resulting data into Minitab and then ask Minitab to conduct a t-test for one mean for us.

Example 28-7

Is the median annual income of American households greater than \$40,000?

Answer

The research question involves the study of one group, namely American households. The research question involves a continuous response... annual income (in dollars). The research question is a "is it this?" question, and therefore involves conducting a hypothesis test. At this point, because the response is continuous we could conduct a hypothesis test about the mean or the median. However, because it is well known that the distribution of American incomes is highly skewed, the median is a better measure of the "center" of the income distribution. Therefore, our analysis should probably concern the median. That said, if m is the (unknown) median annual income of American households, then we are specifically interested in testing the null hypothesis $H_0: m = 40,000$ against the alternative hypothesis $H_A: m > 40,000$. We can enter the resulting data into Minitab and then ask Minitab to conduct either a sign test or a signed rank test for one median for us.

Two Paired Groups with a Continuous Response

Suppose we are interested in comparing the heights of first-born and second-born twins. Then, we have two groups, namely that of the first-born twins and that of the second-born twins. The groups have a special characteristic, however, in that they are not independent. As you know, we say they are paired. Therefore, any analysis we perform would have to take this dependence into account. The response, height, is of course, continuous. Therefore, our analysis involves two paired groups with a continuous response, and hence our options are:

performing a paired t-test for a mean difference
performing a sign test or signed rank test for a median difference
calculating a paired t-interval for a mean difference
calculating a distribution-free confidence interval for a median (or general percentile) difference

Again, what we choose depends on our specific research question. If the research question is an "is it this?" question, then we'd want to conduct a hypothesis test, whereas if it's a "what is it?" question, we'd want to calculate a confidence interval. Once we determine the appropriate statistical method, Minitab can do the dirty work for us using these commands:

Stat >> Basic Stat >> Paired t... to conduct a paired t-test for a mean difference or to calculate a paired t-interval for a mean difference
Stat >> Nonparametrics >> 1-Sample Sign... to conduct a sign test for a median difference
Stat >> Nonparametrics >> 1-Sample Wilcoxon... to conduct a signed rank test for a median difference or to calculate a distribution-free confidence interval for a median difference

The details about how to perform the analyses in Minitab, as well as about the assumptions that must be made in each case, can be found in the relevant lessons.

Example 28-8

Do people's pulse rates increase after exercise?

Answer

The research question involves the study of one group, namely people. Oops, actually if you think about it, the question involves the study of two groups, people, before exercise and people after exercise. Although the research question doesn't specifically suggest this, we should all know by now that it would be a good idea to collect the data in a paired way, that is, to measure the same people before and after exercise. Doing otherwise would introduce needless variability into the data. By measuring the same people twice, of course, removes the independence of the groups, and hence we should be thinking paired, paired, paired.

Because the research question involves a continuous response... the pulse rate, we should be thinking mean, mean, mean or median, median, median. So, we have a paired, paired, paired, mean, mean, mean or a paired, paired, paired, median, median, median. (I've been clearly writing too long today.) At any rate, the research question is clearly a "is it this?" question. It is? Clearly? Well, if $\mu_D = \mu_{After} − \mu_{Before}$, is the (unknown) mean difference in the pulse rates, then we are specifically interested in testing the null hypothesis $H_0: \mu_D = 0$ against the alternative hypothesis $H_A: \mu_D > 0$. We can enter the resulting data into Minitab and then ask Minitab to conduct either a paired t-test for one mean or, alternatively, a sign test or signed-rank test for the median difference. Of course, if we went a step further, we could also ask Minitab to calculate a confidence interval for us, so that we can quantify how different the pulse rates are before and after exercise.

Two Independent Groups with a Continuous Response

Suppose we are interested in comparing the gas mileage of two different vehicles, Toyota Camry and Volkswagen Passat, say. In this case, we have two independent groups, namely that of Toyota Camry vehicles and that of Volkswagen Passat vehicles. The response, gas mileage, is a continuous measurement. Therefore, our analysis would involve two independent groups with a continuous response, and hence our options are:

performing a two-sample t-test for the difference in two means
performing a two-sample Wilcoxon test for a difference in two medians
calculating a two-sample t-interval for the difference in two means

Again, what we choose depends on our specific research question. If the research question is an "is it this?" question, then we'd want to conduct a hypothesis test, whereas if it's a "what is it?" question, we'd want to calculate a confidence interval. Once we determine the appropriate statistical method, Minitab can do the dirty work for us using these commands:

Stat >> Basic Stat >> 2-Sample t... to conduct a t-test for the difference in two means or to calculate a t-interval for the difference in two means
Stat >> Nonparametrics >> Mann-Whitney... to conduct a variation of the two-sample Wilcoxon test for a difference in two medians

The details about how to perform the analyses in Minitab, as well as about the assumptions that must be made in each case, can be found in the relevant lessons.

Example 28-9

Do the resting pulse rates of adult males and females differ?

Answer

The research question involves the study of two independent groups, namely that of adult males and females. The research question involves a continuous response... resting pulse rates. The research question is a "is it this?" question, and therefore involves conducting a hypothesis test. If $\mu_M$ is the (unknown) mean pulse rate of adult males, and $\mu_F$ is the (unknown) mean pulse rate of adult females, then we are specifically interested in testing the null hypothesis:

$H_0: \mu_M − \mu_F = 0$

against the alternative hypothesis:

$H_A: \mu_M − \mu_F ≠ 0$

We can enter the resulting data into Minitab and then ask Minitab to conduct a two-sample t-test for one mean for us. Of course, we should check, as always, for the normality of the data and the equality of the population variances.

More than Two Independent Groups with a Continuous Response??

Suppose we are interested in comparing the average 5-kilometer race times of four different age groups. Because we have four independent groups and one continuous response, namely the race times, we would want to conduct a one-factor analysis of variance. If we were interested in testing whether a second factor, such as gender, had an effect on race times, then we would want to conduct a two-factor analysis of variance. We'd, of course, have to check the necessary assumptions, but once we did that, we could let Minitab do the analysis for us using these commands:

Stat >> ANOVA >> One-way... to conduct a one-factor analysis of variance with the grouping variable in one column and the response in a second column
Stat >> ANOVA >> One-way (Unstacked)... to conduct a one-factor analysis of variance with each group's responses being recorded in a different column
Stat >> ANOVA >> Two-way... to conduct a two-factor analysis of variance

The details about how to perform the analyses in Minitab, as well as about the assumptions that must be made in each case, can be found in the relevant lessons.

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