Formula used to calculate confidence interval..please give example. Thanks
For 95% confidence, mean ± (1.96 standard deviations)/sq rt. of n. For other levels of confidence, the number of standard deviations will vary. It indicates that, if we drew 100 samples, the true population mean will be within this range 95% of the time. Of course, the larger you n, the smaller the range of your estimate is.
I'll let you think of an example.
I hope this helps.
Eric has computed that it takes an average (mean) of 17 minutes with a standard deviation of 3 minutes to drive from home, park the car, and walk to his job. One day it took Eric 21 minutes to get to work. You would use the formula for transforming a raw score in a sample into a z-score to determine how many standard deviations the raw score represents. Since his "score" is 21, you would subtract the mean of 17 from 21 and divide that result (4) by the standard deviation of 3. The z-score of 1.33 tells you that Eric’s time to get to work is 1.33 standard deviations from the mean.
The formula used to calculate a confidence interval depends on the type of data and the situation. Different formulas are used for estimating the confidence interval of a population mean, population proportion, or the difference between two means or proportions.
1. Confidence Interval for Population Mean (Large Sample)
The formula used in this case is:
confidence interval = sample mean ± (z * standard error)
where:
- sample mean is the mean of the sample data
- z is the z-score for the desired confidence level (e.g., 1.96 for 95% confidence)
- standard error is the standard deviation of the sample divided by the square root of the sample size
Example:
Let's say we have a sample of 100 individuals and want to estimate the average height of the population. The sample mean is 175 cm, and the standard deviation is 5 cm (population standard deviation is usually unknown). We want a 95% confidence interval, so the z-value is 1.96 (from the standard normal distribution). Now we can calculate the confidence interval using the formula:
confidence interval = 175 ± (1.96 * (5 / √100))
confidence interval = 175 ± (1.96 * 0.5)
confidence interval = 175 ± 0.98
confidence interval = (174.02, 175.98)
Therefore, we can be 95% confident that the population mean height falls within the range of 174.02 cm and 175.98 cm.
2. Confidence Interval for Population Proportion (Large Sample)
The formula used in this case is:
confidence interval = sample proportion ± (z * standard error)
where:
- sample proportion is the proportion of successes in the sample
- z is the z-score for the desired confidence level (e.g., 1.96 for 95% confidence)
- standard error is the square root of (sample proportion * (1 - sample proportion) / sample size)
Example:
Suppose we want to estimate the proportion of people in a city who own a smartphone. In a sample of 200 individuals, 150 of them own a smartphone. The sample proportion is 150/200 = 0.75. Let's calculate the 95% confidence interval:
confidence interval = 0.75 ± (1.96 * √(0.75 * (1 - 0.75) / 200))
confidence interval = 0.75 ± (1.96 * 0.032)
confidence interval = 0.75 ± 0.063
confidence interval = (0.687, 0.813)
Hence, we can be 95% confident that the true proportion of smartphone owners in the city is within the range of 0.687 and 0.813.
These are just a couple of examples to calculate confidence intervals. The formulas can vary depending on the specific situation and type of data.