How to Run a One Sample t-test in Stata: A Step-by-Step Guide

Introduction to the One Sample t-Test

A One Sample t-test is a parametric hypothesis test that examines whether the mean of a single population differs significantly from a predetermined constant. This test is appropriate when the population standard deviation is unknown (which is usually the case in real-world research) and the sample size is relatively small, although it performs well even with larger samples. Researchers must ensure that the data meet the underlying assumptions of the t-test, primarily that the data are randomly sampled and approximately normally distributed. Violations of these assumptions, especially major deviations from normality in small samples, might necessitate the use of non-parametric alternatives.

The core utility of this test lies in its ability to validate assumptions or benchmarks. For instance, a quality control team might use it to check if the average weight of a product batch meets the specified industry standard, or educators might test if student performance scores align with the national average. By comparing the observed sample mean ($bar{x}$) to the hypothesized population mean ($mu_0$), we quantify the difference relative to the variability within the sample. This comparison yields the t-statistic, which serves as the measure of how many standard errors the sample mean is away from the hypothesized value.

The Core Hypothesis: Null vs. Alternative

Every hypothesis test is built upon two competing statements: the null hypothesis ($H_0$) and the alternative hypothesis ($H_a$ or $H_1$). The null hypothesis assumes no effect or no difference, stating that the population mean is exactly equal to the hypothesized value ($mu = mu_0$). Conversely, the alternative hypothesis posits that the population mean is different from the hypothesized value. This alternative can be two-sided ($mu neq mu_0$) or one-sided ($mu mu_0$), depending on the specific research question being addressed.

The choice between a one-sided or two-sided test is crucial as it affects the resulting p-value and, consequently, the conclusion. A two-sided test checks for differences in either direction, dividing the significance level ($alpha$) across both tails of the t-distribution. A one-sided test, used when researchers have a directional prediction, concentrates all of $alpha$ in one tail. The decision to reject the null hypothesis hinges on comparing the calculated p-value to the chosen significance level (commonly $alpha = 0.05$). If $p < alpha$, we reject $H_0$, concluding that there is sufficient evidence of a difference, thus achieving statistical significance.

Case Study: Investigating Automobile MPG in Stata

To illustrate the practical application of the one sample t-test, we will use a common scenario involving vehicle efficiency data. Researchers are interested in determining if the average fuel efficiency of automobiles in a specific population meets a benchmark of 20 miles per gallon (mpg). They have collected a representative sample consisting of 74 cars and intend to conduct a one sample t-test to rigorously assess whether the true average mpg ($mu$) differs statistically from the target value of 20.

In this scenario, our hypotheses are formally defined as follows: the null hypothesis ($H_0$) states that the true population mean mpg is 20 ($mu = 20$). The alternative hypothesis ($H_a$), reflecting the general interest in knowing if the average is different from 20, is two-sided: the true population mean mpg is not equal to 20 ($mu neq 20$). We will proceed using the statistical software Stata to manage the data and perform the calculations necessary to test this claim.

Step 1: Preparing and Loading the Dataset

The initial step in any data analysis workflow in Stata is to load the relevant data into the active session. For this specific example, we utilize a built-in dataset commonly used for demonstrations, accessible directly via a URL. To load this dataset, type the following command precisely into the Stata Command window and execute it by pressing Enter. This constitutes the first concrete step in our analysis:

use http://www.stata-press.com/data/r13/auto

Upon execution, Stata confirms that the dataset, which contains information on 74 automobiles, has been successfully loaded into memory. This step is crucial, as all subsequent statistical operations will reference variables within this loaded dataset. It is good practice to ensure the data is loaded correctly before proceeding to data inspection or analysis.

Step 2: Inspecting the Raw Data Structure

Before running the formal test, it is highly recommended to inspect the raw structure of the data to verify variable names, identify potential data entry errors, and understand the range of values. In Stata, this can be achieved interactively through the graphical user interface (GUI). Navigate to the top menu bar and select Data > Data Editor > Data Editor (Browse). This action opens a read-only viewer displaying all the variables and observations.

While the dataset contains various details about the 74 cars (such as make, weight, and price), our focus for the One Sample t-test is exclusively on the miles per gallon variable, designated as mpg. Reviewing this column ensures that the variable is correctly formatted (numeric) and that the data appears reasonable. This visual confirmation step is vital for ensuring data integrity and minimizing analytical errors later on.

Step 3: Executing the t-Test Using Stata’s GUI

With the data loaded and inspected, the next phase is to execute the One Sample t-test. The most intuitive way for new users to perform this analysis is through Stata’s menu system. Begin by navigating the top menu: Statistics > Summaries, tables, and tests > Classical tests of hypotheses > t test (mean-comparison test). This sequence opens the dedicated dialogue box for mean comparison tests.

Within the dialogue box, ensure that the radio button for One-sample test is selected, confirming our objective of comparing a single sample mean against a constant. For the Variable name dropdown, select mpg. Critically, in the field labeled Hypothesized mean, input the value of 20, as this is the benchmark against which we are testing the population mean. Lastly, the Confidence level setting allows adjustment of the desired level of certainty for the confidence interval; the default setting of 95% corresponds to a statistical significance level ($alpha$) of 0.05. After configuring these parameters, clicking OK executes the test and generates the detailed statistical output.

Understanding the t-Test Output

The results window in Stata provides a concise summary table followed by the three possible hypothesis test outcomes (two-tailed and two one-tailed tests). Interpreting this output requires careful attention to each statistic provided. The summary section details the characteristics of the sample used in the analysis:

• Obs: This denotes the number of observations ($n$), which is 74, representing the total number of cars in the sample.
• Mean: This is the calculated sample mean ($bar{x}$) for mpg. In this specific output, the mean mpg is reported as 21.2973 miles per gallon.
• Std. Err: The Standard Error of the Mean, which estimates the variability of the sample mean if the sampling process were repeated. It is calculated as the standard deviation divided by the square root of the sample size: $text{Std Err} = sigma / sqrt{n} = 5.785503 / sqrt{74} approx textbf{0.6725511}$.
• Std. Dev: The standard deviation of the mpg variable, which measures the dispersion of the data around the sample mean. Here, it is 5.785503.
• 95% Conf. Interval: The 95% confidence interval for the true population mean ($mu$). We are 95% confident that the true average mpg for the population falls within the range (19.9569, 22.63769).

Below the summary statistics, Stata reports the actual test results: the calculated t-statistic and the degrees of freedom. The t-statistic, denoted by t, quantifies the difference between the sample mean and the hypothesized mean in standard error units. It is calculated as: $t = (bar{x} – mu_0) / text{Std Err} = (21.2973 – 20) / 0.6725511 approx textbf{1.9289}$. The degrees of freedom (df) for a one sample t-test are calculated simply as $n-1$, which in this case is $74 – 1 = textbf{73}$.

Interpreting the Statistical Significance

The most critical part of the output is the bottom section, which provides the p-values corresponding to the three possible alternative hypotheses. Since our initial research question was whether the true average mpg is 20 or not (a two-sided test), we must look at the results for $H_a: text{mean} neq 20$. This two-tailed test is located in the center of the output table. The calculated p-value for this test is 0.0576.

We compare this p-value (0.0576) against our predetermined significance level ($alpha = 0.05$). Since $0.0576$ is greater than $0.05$, we do not meet the threshold required to reject the null hypothesis ($H_0$). Therefore, based on the collected sample of 74 cars, we conclude that there is insufficient evidence, at the 5% statistical significance level, to assert that the true population mean mpg is different from 20 mpg. Although the sample mean (21.2973) is numerically higher than 20, the difference is not large enough, relative to the data variability, to be deemed statistically significant.

Alternative Method: Using the Stata Command Line

While the graphical user interface is excellent for beginners, experienced Stata users often prefer the command line for efficiency and reproducibility. The equivalent command to perform the one sample t-test used in the previous steps is highly succinct and easy to recall. The general syntax requires the command ttest, followed by the variable name, and then the option == followed by the hypothesized mean value.

For our specific case study testing the mpg against a hypothesized mean of 20, the command is:

ttest mpg == 20

Executing this single line of code in the Command window will produce the exact same output generated by navigating the menus, confirming the t-statistic of 1.9289 and the two-tailed p-value of 0.0576. Using command syntax is the preferred method for constructing reproducible do-files, ensuring that analyses can be easily shared and validated by others.

Step 4: Reporting the Findings

The final and crucial step in any statistical analysis is clearly and accurately reporting the findings in a format that conveys the methodology and conclusions to the relevant audience. When reporting a One Sample t-test, standard conventions require including the test statistic (t), the degrees of freedom (df), and the exact p-value. The context, including the sample size and the hypothesized value, must also be clearly stated.

The following is an example of how the results from our automobile mpg analysis should be summarized in a formal report, adhering to statistical reporting standards:

A One Sample t-test was conducted on a sample of 74 automobiles to evaluate whether the true population mean miles per gallon (mpg) differed significantly from the hypothesized value of 20 mpg.

Results showed that the true population mean was not different than 20 mpg (t(73) = 1.9289, p = .0576) at a significance level of 0.05. Consequently, we failed to reject the null hypothesis.

A 95% confidence interval for the true population mean resulted in the interval of (19.9569, 22.63769). Since this confidence interval includes the hypothesized value of 20, this finding reinforces the conclusion that the evidence for a significant difference is insufficient.