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A confidence interval is a statistical measure used to estimate the range of values within which a population parameter, such as a mean or proportion, is likely to lie. It is calculated using a sample from the population and provides a range of values with a certain level of confidence. Three common methods for calculating confidence intervals are the t-test, z-test, and bootstrap method.
The t-test is used when the population standard deviation is unknown and the sample size is small. It calculates the confidence interval by taking into account the sample mean, sample standard deviation, and sample size. This method is commonly used in medical research and social sciences.
The z-test is used when the population standard deviation is known and the sample size is large. It is based on the standard normal distribution and provides a more precise estimate of the confidence interval compared to the t-test. This method is commonly used in quality control and manufacturing processes.
The bootstrap method is a non-parametric approach to calculating confidence intervals. It involves creating multiple samples from the original sample and calculating the confidence interval for each sample. The average of these intervals provides an estimate of the true confidence interval. This method is useful when the underlying population distribution is unknown or not normally distributed. It is commonly used in finance, economics, and environmental studies.
In conclusion, the t-test, z-test, and bootstrap method are three examples of calculating confidence intervals. Each method has its own advantages and is suitable for different scenarios depending on the sample size and availability of population parameters. By providing a range of values with a certain level of confidence, confidence intervals help researchers and decision-makers make more informed conclusions about a population based on a sample.
Calculate Confidence Intervals: 3 Example Problems
A confidence interval for a mean is a range of values that is likely to contain a with a certain level of confidence.
We use the following formula to calculate a confidence interval for a mean:
Confidence Interval = x +/- t*(s/√n)
where:
- x: sample mean
- t: the t critical value
- s: sample standard deviation
- n: sample size
Note: We replace a t critical value with a z critical value in the formula if the population standard deviation (σ) is known and the sample size is greater than 30.
The following examples show how to construct a confidence interval for a mean in three different scenarios:
- Population standard deviation (σ) is unknown
- Population standard deviation (σ) is known but n ≤ 30
- Population standard deviation (σ) is known and n > 30
Let’s jump in!
Example 1: Confidence Interval when σ is Unknown
Suppose we would like to calculate a 95% confidence interval for the mean height (in inches) of a certain species of plant.
Suppose we collect a simple random sample with the following information:
- sample mean (x)= 12
- sample size (n) = 19
- sample standard deviation (s) = 6.3
We can use the following formula to construct this confidence interval:
- 95% C.I. = x +/- t*(s/√n)
- 95% C.I. = 12 +/- tn-1, α/2*(6.3/√19)
- 95% C.I. = 12 +/- t18, .025*(6.3/√19)
- 95% C.I. = 12 +/- 2.1009*(6.3/√19)
- 95% C.I. = (8.964 , 15.037)
The 95% confidence interval for the population mean height for this particular species of plant is (8.964 inches, 15.037 inches).
Note #1: We used the to find the t critical value associated with 18 degrees of freedom and a confidence level of 0.95.
Example 2: Confidence Interval when σ is Known but n ≤ 30
Suppose we would like to calculate a 99% confidence interval for the mean exam score on a certain college entrance exam.
Suppose we collect a simple random sample with the following information:
- sample mean (x)= 85
- sample size (n) = 25
- population standard deviation (σ) = 3.5
We can use the following formula to construct this confidence interval:
- 99% C.I. = x +/- t*(s/√n)
- 99% C.I. = 85 +/- tn-1, α/2*(3.5/√25)
- 99% C.I. = 85 +/- t24, .005*(3.5/√25)
- 99% C.I. = 85 +/- 2.7969*(3.5/√25)
- 99% C.I. = (83.042, 86.958)
The 99% confidence interval for the population mean exam score on this particular college entrance exam is (83.042, 86.958).
Note #1: We used the to find the t critical value associated with 24 degrees of freedom and a confidence level of 0.99.
Note #2: Since the population standard deviation (σ) was known but the sample size (n) was less than 30, we used the t critical value when calculating the confidence interval.
Example 3: Confidence Interval when σ is Known and n > 30
Suppose we would like to calculate a 90% confidence interval for the mean weight of a certain species of turtle.
Suppose we collect a simple random sample with the following information:
- sample mean (x)= 300
- sample size (n) = 40
- population standard deviation (σ) = 15
We can use the following formula to construct this confidence interval:
- 90% C.I. = x +/- z*(σ/√n)
- 90% C.I. = 300 +/- 1.645*(15/√40)
- 90% C.I. = (296.099, 303.901)
The 90% confidence interval for the population mean weight of this particular species of turtle is (83.042, 86.958).
Note #1: We used the to find the z critical value associated with a significance level of 0.1.
Note #2: Since the population standard deviation (σ) was known and the sample size (n) was greater than 30, we used the z critical value when calculating the confidence interval.
Additional Resources
The following tutorials provide additional information about confidence intervals: