A one-sample t-test in data science is a statistical hypothesis test used to determine whether the mean of a single sample is significantly different from a known or hypothesized population mean. It's a parametric test, meaning it relies on assumptions about the underlying distribution of the data (namely, that it is approximately normally distributed).
Here's a breakdown:
Purpose
The primary goal of a one-sample t-test is to answer the question: "Is the mean of my sample significantly different from a specific value?" This value is often a pre-existing standard, a theoretical value, or a target value.
Key Components
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Sample Mean: The average value calculated from your data sample.
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Hypothesized Population Mean (μ₀): The value you are comparing your sample mean to.
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Sample Standard Deviation (s): A measure of the spread or variability within your sample data.
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Sample Size (n): The number of data points in your sample.
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T-Statistic: A calculated value that summarizes the difference between the sample mean and the hypothesized population mean, relative to the sample's variability. The formula is:
t = (x̄ - μ₀) / (s / √n)
where:
- x̄ is the sample mean
- μ₀ is the hypothesized population mean
- s is the sample standard deviation
- n is the sample size
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Degrees of Freedom (df): Represents the number of independent pieces of information available to estimate a parameter. For a one-sample t-test, df = n - 1.
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P-value: The probability of observing a test statistic as extreme as, or more extreme than, the one calculated from your sample data, assuming that the null hypothesis is true.
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Null Hypothesis (H₀): The statement that there is no significant difference between the sample mean and the hypothesized population mean. (x̄ = μ₀)
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Alternative Hypothesis (H₁): The statement that there is a significant difference between the sample mean and the hypothesized population mean. This can be two-sided (x̄ ≠ μ₀), right-tailed (x̄ > μ₀), or left-tailed (x̄ < μ₀).
Assumptions
The one-sample t-test relies on the following assumptions:
- Independence: The data points in the sample are independent of each other.
- Normality: The sample data is approximately normally distributed. This is especially important for small sample sizes. The Central Limit Theorem can help mitigate this concern for larger sample sizes (n > 30).
- Random Sampling: The data was obtained through a random sampling method.
Steps in Performing a One-Sample t-Test
- State the Hypotheses: Define the null (H₀) and alternative (H₁) hypotheses.
- Choose a Significance Level (α): This is the probability of rejecting the null hypothesis when it is actually true (Type I error). Common values are 0.05 or 0.01.
- Calculate the t-Statistic: Use the formula mentioned above.
- Determine the Degrees of Freedom: df = n - 1.
- Find the P-value: Use a t-distribution table or statistical software to find the p-value associated with the calculated t-statistic and degrees of freedom.
- Make a Decision:
- If the p-value is less than or equal to the significance level (p ≤ α), reject the null hypothesis. This suggests that there is a statistically significant difference between the sample mean and the hypothesized population mean.
- If the p-value is greater than the significance level (p > α), fail to reject the null hypothesis. This suggests that there is not enough evidence to conclude that a statistically significant difference exists.
Example
Let's say you want to determine if the average height of students at a particular university is different from the national average height of 67 inches. You collect a random sample of 30 students and find that their average height is 69 inches, with a standard deviation of 2.5 inches.
- H₀: μ = 67 (The average height of students at the university is equal to the national average)
- H₁: μ ≠ 67 (The average height of students at the university is different from the national average)
- α = 0.05
- t = (69 - 67) / (2.5 / √30) ≈ 4.38
- df = 30 - 1 = 29
Using a t-distribution table or statistical software, you find that the p-value for a two-tailed test with t = 4.38 and df = 29 is very small (much less than 0.05). Therefore, you would reject the null hypothesis and conclude that the average height of students at the university is significantly different from the national average.
Use Cases in Data Science
- A/B Testing: While A/B tests often use two-sample t-tests, a one-sample t-test can be used if you want to compare the performance of a new feature to a pre-existing baseline metric.
- Quality Control: Determining if the average measurement of a manufactured product meets a specific standard.
- Sensor Calibration: Verifying that a sensor's readings are accurate by comparing its average output to a known standard.
- Analyzing Survey Data: Comparing the average response to a survey question against a neutral or expected value.
