Question

1) Give an example of a research problem that you could use an independent –sample t test on.

2) Give an example of a research problem that you could use a paired-sample t test on.

Answer 1

REAL LIFE EXAMPLE(RESEARCH PROBLEM) FOR T TEST AND PAIRED T TEST

1)

RESEARCH PROBLEM

Let’s say you’re curious about whether New Yorkers and Kansans spend a different amount of money per month on movies. It’s impractical to ask every New Yorker and Kansan about their movie spending, so instead you ask a sample of each—maybe 300 New Yorkers and 300 Kansans—and the averages are $14 and $18. The t-test asks whether that difference is probably representative of a real difference between Kansans and New Yorkers generally or whether that is most likely a meaningless statistical fluke.

Technically, it asks the following: If there were in fact no difference between Kansans and New Yorkers generally, what are the chances that randomly selected groups from those populations would be as different as these randomly selected groups are? For example, if Kansans and New Yorkers as a whole actually spent the same amount of money on average, it’s very unlikely that 300 randomly selected Kansans each spend exactly $14 and 300 randomly selected New Yorkers each spend exactly $18. So if you’re sampling yielded those results, you would conclude that the difference in the sample groups is most likely representative of a meaningful difference between the populations as a whole.

2) paired sample t test

EXAMPLE: Drinking and Driving

Drunk driving is one of the main causes of car accidents. Interviews with drunk drivers who were involved in accidents and survived revealed that one of the main problems is that drivers do not realize that they are impaired, thinking “I only had 1-2 drinks … I am OK to drive.”

A sample of 20 drivers was chosen, and their reaction times in an obstacle course were measured before and after drinking two beers. The purpose of this study was to check whether drivers are impaired after drinking two beers. Here is a figure summarizing this study:

• Note that the categorical explanatory variable here is “drinking 2 beers (Yes/No)”, and the quantitative response variable is the reaction time.

• By using the matched pairs design in this study (i.e., by measuring each driver twice), the researchers isolated the effect of the two beers on the drivers and eliminated any other confounding factors that might influence the reaction times (such as the driver’s experience, age, etc.).

• For each driver, the two measurements are the total reaction time before drinking two beers, and after. You can see the data by following the links in Step 2 below.

Since the measurements are paired, we can easily reduce the raw data to a set of differences and conduct a one-sample t-test.

Here are some of the results for this data:

Step 1: State the hypotheses

We define μ_d = the population mean difference in reaction times (Before – After).

As we mentioned, the null hypothesis is:

• Ho: μ_d = 0 (indicating that the population of the differences are centered at a number that IS ZERO)

The null hypothesis claims that the differences in reaction times are centered at (or around) 0, indicating that drinking two beers has no real impact on reaction times. In other words, drivers are not impaired after drinking two beers.

Although we really want to know whether their reaction times are longer after the two beers, we will still focus on conducting two-sided hypothesis tests. We will be able to address whether the reaction times are longer after two beers when we look at the confidence interval.

Therefore, we will use the two-sided alternative:

• Ha: μ_d  ≠ 0 (indicating that the population of the differences are centered at a number that is NOT ZERO)

Step 2: Obtain data, check conditions, and summarize data

• Data: Beers SPSS format, SAS format, Excel format, CSV format

Let’s first check whether we can safely proceed with the paired t-test, by checking the two conditions.

• The sample of drivers was chosen at random.

• The sample size is not large (n = 20), so in order to proceed, we need to look at the histogram or QQ-plot of the differences and make sure there is no evidence that the normality assumption is not met.

We can see from the histogram above that there is no evidence of violation of the normality assumption (on the contrary, the histogram looks quite normal).

Also note that the vast majority of the differences are negative (i.e., the total reaction times for most of the drivers are larger after the two beers), suggesting that the data provide evidence against the null hypothesis.

The question (which the p-value will answer) is whether these data provide strong enough evidence or not against the null hypothesis. We can safely proceed to calculate the test statistic (which in practice we leave to the software to calculate for us).

Test Statistic: We will use software to calculate the test statistic which is t = -2.58.

• Recall: This indicates that the data (represented by the sample mean of the differences) are 2.58 standard errors below the null hypothesis (represented by the null value, 0).

Step 3: Find the p-value of the test by using the test statistic as follows

As a special case of the one-sample t-test, the null distribution of the paired t-test statistic is a t distribution (with n – 1 degrees of freedom), which is the distribution under which the p-values are calculated.

We will let the software find the p-value for us, and in this case, gives us a p-value of 0.0183 (SAS) or 0.018 (SPSS).

The small p-value tells us that there is very little chance of getting data like those observed (or even more extreme) if the null hypothesis were true. More specifically, there is less than a 2% chance (0.018=1.8%) of obtaining a test statistic of -2.58 (or lower) or 2.58 (or higher), assuming that 2 beers have no impact on reaction times.

Step 4: Conclusion

In our example, the p-value is 0.018, indicating that the data provide enough evidence to reject Ho.

• Conclusion: There is enough evidence that drinking two beers is associated with differences in reaction times of drivers

Thanks,

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