Table of Contents
Confidence Interval: A confidence interval is a statistical concept used to estimate the range within which a true population parameter lies. It is based on the sample data. It provides a measure of uncertainty around the sample estimate and is expressed with a certain level of confidence, typically 95%, although other levels such as 90% and 99% are also common.
To put it simply, when we say we have a 95% confidence interval, it means that if we were to take 100 different samples and calculate the confidence interval for each, we would expect 95 of those intervals to contain the true population parameter. This does not mean that there is a 95% chance that the true parameter lies within our specific interval, rather, it reflects the reliability of the method used to calculate the interval.
Confidence Interval Formula
A confidence interval provides a range within which the true value of a parameter is likely to fall. This range is determined by the confidence level, which is chosen by the investigator and expressed as a percentage. A higher confidence level results in a wider interval, meaning the estimate is less precise. To fully grasp confidence intervals, it’s important to first understand basic statistical formulas, including the z-score formula. The formula for calculating a CI is as follows:
- If n ≥ 30:
Confidence Interval = x̄ ± zc(σ/√n) - If n < 30:
Confidence Interval = x̄ ± tc(S/√n)
Where:
n = Number of terms (sample size)
x̄ = Sample Mean
σ = Population Standard Deviation
zc = Value corresponding to the desired confidence level in the z-table
tc = Value corresponding to the desired confidence level in the t-table
These formulas help determine the range within which the true parameter value is likely to fall, depending on whether the sample size is large or small.
How Confidence Intervals Are Calculated
CI are calculated using a specific formula that takes into account the sample mean, the standard deviation, and the size of the sample. The general formula for a confidence interval is:
- Confidence Interval = x̄ ± zc(σ/√n) (if n ≥ 30)
- Confidence Interval = x̄ ± tc(S/√n) (If n < 30)
Where: n = Number of terms (sample size)
x̄ = Sample Mean
σ = Population Standard Deviation
zc = Value corresponding to the desired confidence level in the z-table
tc = Value corresponding to the desired confidence level in the t-table
Choosing the Confidence Level
The confidence level is selected before the data is examined and reflects how certain we want to be about our estimates. A higher confidence level, such as 99%, will produce a wider interval, indicating more uncertainty but greater confidence that the interval contains the true parameter. Conversely, a lower confidence level, like 90%, results in a narrower interval but with less confidence that it includes the true parameter.
Applications of Confidence Intervals
Confidence intervals are widely used in various fields such as medicine, economics, and social sciences to estimate population parameters like means, proportions, and differences between groups. They provide valuable insight into the precision of sample estimates and help in making predictions, comparing groups, and assessing the impact of interventions.
For an example, in clinical trials, confidence intervals are used to determine the effectiveness of a new drug compared to a placebo. If the confidence interval for the difference in recovery rates between the drug and the placebo does not include zero, it suggests a statistically significant difference. It means that the drug likely has an effect.
| Confidence Interval | Z Value |
| 80% | 1.282 |
| 85% | 1.440 |
| 90% | 1.645 |
| 95% | 1.960 |
| 99% | 2.576 |
| 99.5% | 2.807 |
| 99.9% | 3.291 |
Common Misconceptions
- One common misconception is that the confidence interval provides a range where the true parameter lies with a certain probability.
- But in reality, the true parameter is either inside or outside the interval; the confidence level reflects the long-term success rate of the interval construction method.
- Another misconception is that increasing the sample size always leads to a narrower confidence interval.
- While a larger sample size does reduce the standard error, the width of the confidence interval also depends on the variability of the data.
- Highly variable data might still result in a wide interval even with a large sample.
Examples of Confidence Interval
Example 1: A random sample of 40 oranges was taken from a large population. The sample’s mean weight was found to be 150 grams, with a standard deviation of 12 grams. Calculate the 90% confidence limits for the mean weight of the entire population of oranges.
Solution: For 90% confidence, Z = 1.645.
We have:
x̄ =150 grams
s=12 grams
Z=1.645
n=40
Substituting into the confidence interval formula:
x̄ ± Z(sn)
Therefore, the 90% confidence limits are:
150 ± 1.645 (1240) =150 ± 1.645 (126.325) = 150 ± 3.1
Therefore, the 90% confidence limits are 150 ± 3.1
Example 2: A random sample of 50 eggs was selected from a large batch. The mean length of the eggs in the sample was 55 millimeters, with a standard deviation of 5 millimeters. Calculate the 95% confidence limits for the mean length of the entire population of eggs.
Solution: For 95% confidence, Z = 1.96.
We have:
x̄ =55 millimeters
s=5 millimeters
Z=1.96
n=50
Substituting into the confidence interval formula:
x̄ ± Z(sn)
Therefore, the 95% confidence limits are:
55 ± 1.96 (550) =55 ± 1.96 (57.071) = 55 ± 1.4
Therefore, the 95% confidence limits are 55 ± 1.4 millimeters.
Practice Questions on Confidence Interval
Q 1: A random sample of 25 strawberries was taken from a large batch. The mean weight of the sample was found to be 18 grams, with a standard deviation of 3 grams. Calculate the 90% confidence limits for the mean weight of the entire population of strawberries.
Q 2: A researcher selects a random sample of 35 students to measure their average study time per week. The sample mean is 12 hours, with a standard deviation of 2.5 hours. Calculate the 95% confidence limits for the mean study time of the entire student population.
Q 3: In a factory, a random sample of 45 bolts is selected. The average length of the bolts in the sample is 10 centimeters, with a standard deviation of 0.6 centimeters. Calculate the 99% confidence limits for the mean length of the bolts produced by the factory.
Q 4: A sample of 28 light bulbs from a production line is tested for lifespan. The sample mean lifespan is 800 hours, with a standard deviation of 50 hours. Calculate the 85% confidence limits for the mean lifespan of the entire population of light bulbs.
Q 5: A random sample of 32 tomatoes is taken from a farm. The mean diameter of the tomatoes in the sample is 70 millimeters, with a standard deviation of 9 millimeters. Calculate the 92% confidence limits for the mean diameter of the entire population of tomatoes.
Q 6: A quality control engineer selects a random sample of 20 batteries from a large batch. The mean voltage of the sample is 1.5 volts, with a standard deviation of 0.1 volts. Calculate the 95% confidence limits for the mean voltage of the entire population of batteries.
Confidence Interval: FAQs
What Is the Z-Score for a 95% Confidence Interval?
The z-score for a 95% confidence interval is 1.96
What Is a Good Confidence Interval?
A good confidence interval is one that is narrow (tight) with a confidence level of 95% or higher, as this provides a reliable estimate with a smaller margin of error.
How Do You Find the Confidence Interval?
Ans. To calculate a Confidence Interval, you can use the formula: If n ≥ 30: Confidence Interval = x̄ ± zc(σ/√n) If n < 30: Confidence Interval = x̄ ± tc(S/√n) Where: n = Number of terms (sample size) x̄ = Sample Mean σ = Population Standard Deviation zc = Value corresponding to the desired confidence level in the z-table tc = Value corresponding to the desired confidence level in the t-table
What is confidence level and significance level?
Confidence Level and Significance Level are key concepts in statistics, often used in hypothesis testing. The Confidence Level indicates how sure we are that the results of a study or experiment are reliable. It is usually expressed as a percentage, such as 90%, 95%, or 99%. For example, if a study has a 95% confidence level, it means that if the same study were repeated 100 times, the results would be expected to be similar in 95 out of those 100 times. The higher the confidence level, the more confident we are that the results are accurate. The Significance Level (denoted as α) measures how likely it is that the results observed in a study are due to chance rather than a real effect. It is usually expressed as a decimal, like 0.05 or 0.01, and is the threshold used to decide whether to reject the null hypothesis in a statistical test. For instance, if the significance level is 0.05, there is a 5% chance that the results are due to random variation. If the probability of the observed result is less than the significance level, the result is considered statistically significant, meaning it is unlikely to have occurred by chance.
