The 95 confidence interval equation is a statistical tool used to estimate the population parameter based on a sample of data. It provides a range of values within which the true population parameter is likely to lie. The equation for the 95 confidence interval is given by:
Confidence Interval Formula
The formula for the confidence interval is: x̄ - (Z * (σ / √n)) ≤ μ ≤ x̄ + (Z * (σ / √n)), where x̄ is the sample mean, μ is the population mean, σ is the population standard deviation, n is the sample size, and Z is the Z-score corresponding to the desired confidence level.
Calculating the Z-Score
The Z-score is a critical component of the confidence interval equation. For a 95% confidence interval, the Z-score is typically 1.96, which corresponds to a probability of 0.025 in each tail of the standard normal distribution. However, this value can be obtained from a standard normal distribution table or calculated using statistical software.
| Confidence Level | Z-Score |
|---|---|
| 90% | 1.645 |
| 95% | 1.96 |
| 99% | 2.576 |
Example Calculation
Suppose we want to estimate the average height of a population of adults based on a sample of 100 individuals. The sample mean height is 175 cm, and the population standard deviation is 5 cm. To calculate the 95% confidence interval, we would use the following values: x̄ = 175, σ = 5, n = 100, and Z = 1.96. Plugging these values into the equation, we get:
175 - (1.96 * (5 / √100)) ≤ μ ≤ 175 + (1.96 * (5 / √100))
Simplifying the equation, we get:
175 - (1.96 * 0.5) ≤ μ ≤ 175 + (1.96 * 0.5)
175 - 0.98 ≤ μ ≤ 175 + 0.98
174.02 ≤ μ ≤ 175.98
Therefore, we can conclude that the true population mean height is likely to lie between 174.02 cm and 175.98 cm with 95% confidence.
Interpretation of Results
The confidence interval provides a range of values within which the true population parameter is likely to lie. In this example, the 95% confidence interval for the population mean height is (174.02, 175.98). This means that if we were to repeat the sampling process many times, we would expect the true population mean height to lie within this interval 95% of the time.
What is the purpose of the confidence interval equation?
+The confidence interval equation is used to estimate the population parameter based on a sample of data. It provides a range of values within which the true population parameter is likely to lie.
What is the difference between a 90% and 95% confidence interval?
+A 90% confidence interval has a narrower range of values than a 95% confidence interval. This means that the 90% confidence interval is less likely to include the true population parameter, but it is also more precise.
How do I calculate the Z-score for a 95% confidence interval?
+The Z-score for a 95% confidence interval can be obtained from a standard normal distribution table or calculated using statistical software. The Z-score is typically 1.96, which corresponds to a probability of 0.025 in each tail of the standard normal distribution.
