One Sample Z-Test Calculator
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Historical Background
The One Sample Z-Test is a statistical method developed to determine if a sample mean is significantly different from a known or hypothesized population mean. It is used when the population variance is known, and the sample size is sufficiently large (typically n > 30). The Z-Test gained prominence through the works of Karl Pearson and other statisticians in the early 20th century, who contributed to the foundational aspects of hypothesis testing.
Calculation Formula
The formula to calculate the Z-Score in a One Sample Z-Test is:
\[ Z = \frac{X̄ - μ}{\frac{σ}{\sqrt{n}}} \]
Where:
- \( X̄ \) = Sample Mean
- \( μ \) = Population Mean
- \( σ \) = Population Standard Deviation
- \( n \) = Sample Size
Example Calculation
Suppose a researcher is testing whether the average height of students in a university is different from the population average of 170 cm. A sample of 50 students shows an average height of 175 cm with a population standard deviation of 10 cm. The Z-Score can be calculated as:
\[ Z = \frac{175 - 170}{\frac{10}{\sqrt{50}}} = \frac{5}{\frac{10}{7.07}} = \frac{5}{1.41} = 3.55 \]
This Z-Score of 3.55 indicates that the sample mean is 3.55 standard deviations away from the population mean.
Importance and Usage Scenarios
The One Sample Z-Test is used in various fields to make data-driven decisions. It is useful when comparing a sample mean to a known population mean, such as in quality control (e.g., testing if the average lifespan of a product meets industry standards) or in research studies (e.g., testing if a drug’s effect differs from the expected result).
Common FAQs
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When should I use a One Sample Z-Test instead of a T-Test?
- Use the One Sample Z-Test when the population standard deviation is known and the sample size is large (n > 30). If the population standard deviation is unknown, or the sample size is small, a One Sample T-Test is more appropriate.
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What is a Z-Score?
- A Z-Score measures how many standard deviations a data point (or sample mean) is from the population mean. It helps to standardize data for comparison.
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What is the critical value for a Z-Test?
- For a 95% confidence level, the critical Z-Score value is ±1.96. If your Z-Score falls beyond this range, it is considered statistically significant.
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Can a Z-Test be used for small samples?
- No, a Z-Test is typically used for large samples. For smaller samples (n < 30), the T-Test is more suitable as it accounts for variability in smaller datasets.
This calculator helps in conducting a One Sample Z-Test quickly, providing an accessible way for researchers and analysts to test hypotheses based on statistical data.