E. The T-Test

One-Sample T-Test

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It is perhaps easiest to demonstrate the ideas and methods of the one-sample t-test by working through an example. To reiterate, the one-sample t-test compares the mean score of a sample to a known value, usually the population mean (the average for the outcome of some population of interest). The basic idea of the test is a comparison of the average of the sample (observed average) and the population (expected average), with an adjustment for the number of cases in the sample and the standard deviation of the average. Working through an example can help to highlight the issues involved and demonstrate how to conduct a t-test using actual data.

Example - Prenatal Care and Birthweight

One of the best indicators of the health of a baby is his/her weight at birth. Birthweight is an outcome that is sensitive to the conditions in which mothers experienced pregnancy, particularly to issues of deprivation and poor diet, which are tied to lower birthweight. It is also an excellent predictor of some difficulties that infants may experience in their first weeks of life. The National Center for Health Statistics reports that although infants weighing 5 1/2 pounds (88 ounces) or less account for only 7% of births, they account for nearly 2/3 of infant deaths.

In the United States, mothers who live in poverty generally have babies with lower birthweight than those who do not live in poverty. While the average birthweight for babies born in the U.S. is approximately 3300 grams, the average birthweight for women living in poverty is 2800 grams.

Eliminating the linkage between poverty and low birthweight status has been a prominent dimension of health policy for the past decade. Recently, a local hospital introduced an innovative new prenatal care program to reduce the number of low birthweight babies born in the hospital. In the first year, 25 mothers, all of whom live in poverty, participated in this program. Data drawn from hospital records reveals that the babies born to these women had a birthweight of 3075 grams, with a standard deviation of 500 grams.

The question posed to you, the researcher, is whether this program has been effective at improving the birthweights of babies born to poor women.

1. Establish Hypotheses

The first step to examining this question is to establish the specific hypotheses we wish to examine. Recall from the unit on hypothesis testing that most social science research involves the development (based on theory) of a null hypothesis and an alternative hypothesis - some test statistic is then calculated to determine whether to reject the null hypothesis or not.

For this example, what is the null hypothesis? What is the alternative hypothesis?

In this case:

  • Null hypothesis is that the difference between the birthweights of babies born to mothers who participated in the program and those born to other poor mothers is 0. Another way of stating the null hypothesis is that the difference between the observed mean of birthweight for program babies and the expected mean of birthweight for poor women is zero.
  • Alternative hypothesis - the difference between the observed mean of birthweight for program babies and the expected mean of birthweight for poor women is not zero.

2. Calculate Test Statistic

Calculation of the test statistic requires four components:

  1. The average of the sample (observed average)
  2. The population average or other known value (expected average)
  3. The standard deviation (SD) of the sample average
  4. The number of observations.

    With this example, the components are as follows:
  1. Sample average = 3075 grams
  2. Population average (poor women - remember we're interested in whether this program improves birth outcomes relative to those of poor women) = 2800 grams
  3. SD of the sample average = 300 grams
  4. Number of observations = 25

    With these four pieces of information, we calculate the following statistic, t:

Mathematical Markup
In the case of our example,

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3. Use This Value To Determine P-Value

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Having calculated the t-statistic, compare the t-value with a standard table of t-values to determine whether the t-statistic reaches the threshold of statistical significance.

Plugging in the values of t (.898) and n (number of cases = 25) yields a p-value of .378. Generally speaking, we require p-values of .05 or less in order to reject the null hypothesis. With a value of .378, we cannot reject the null. Therefore, we conclude that the intervention did not successfully improve birthweight.

Extension Exercise

Although the prenatal care program appears to have been successful in improving infants' birthweights significantly above those of other mothers born to poverty, the question remains whether the program alleviated the disadvantage infants born to poorer women have in birthweight. The same source tells us that the birthweight of all babies born in the United States in X was 3,339 grams.

Are the birthweights of the babies born to the participants of the prenatal care program significantly different from the average for the overall national average?

Summary

To calculate a one-sample t-test, do the following steps:

1. Establish Hypotheses

Null hypothesis - difference between observed and expected is 0
Alternative hypothesis - difference between observed and expected is not 0.

2. Calculate Test Statistic

Calculation of the test statistic requires four components:

  1. The average of the sample (observed average)
  2. The population average or other known value (expected average)
  3. The standard deviation of the average
  4. The number of observations.

With these four pieces of information, we calculate the following statistic, t:

Mathematical Markup

3. Use This Value to Determine P-Value

Having calculated the t-statistic, compare the t-value with a standard table of t-values to determine whether the t-statistic reaches the threshold of statistical significance.