A confidence interval is a range of values calculated from a sample of data and used to estimate an unknown population parameter. The purpose of a confidence interval is to provide a range of values within which the population parameter is likely to fall, based on the sample data.
For example, if a researcher wants to estimate the mean height of all adult men in the United States, they might collect data from a sample of men and calculate a 95% confidence interval for the mean height. Based on the sample data, the confidence interval would provide a range of values within which the true mean height of all adult men in the United States is likely to fall.
The level of confidence is expressed as a percentage and is chosen by the researcher. A 95% confidence interval means that there is a 95% probability that the true population parameter lies within the confidence interval.
Introduction to the Z Score
The z score measures how many standard deviations a value is from the mean of a normal distribution. A normal distribution is a type of probability distribution that is symmetrical around the mean and is characterized by a bell-shaped curve.
It is calculated by subtracting the mean of the distribution from the value of interest and dividing the result by the standard deviation of the distribution.
For example, if the mean of a normal distribution is 100 and the standard deviation is 10, and a value of 120 is being evaluated, the z score would be calculated as follows:
z score = (120 – 100) / 10 = 2
This means that the value of 120 is two standard deviations above the mean of the distribution.
The Z Score and Confidence Intervals
In the context of confidence intervals, the z score is used to determine the range of values within which the population parameter is likely to fall.
To calculate a confidence interval, the researcher must first determine the appropriate z score for the level of confidence desired. For a 95% confidence interval, the z score is 1.96.
Next, the researcher needs to calculate the standard error of the sample mean. The standard error is a measure of the dispersion of the sample mean around the population mean. It is calculated by dividing the sample’s standard deviation by the sample size’s square root.
Once the standard error has been calculated, the confidence interval can be determined by adding and subtracting the z score multiplied by the standard error from the sample mean.
For example, if the sample mean is 100, the standard error is 10, and the desired confidence level is 95%, the confidence interval would be calculated as follows:
Lower bound of confidence interval = 100 – (1.96 * 10) = 80
Upper bound of confidence interval = 100 + (1.96 * 10) = 120
This would result in a 95% confidence interval of 80 to 120.
Conclusion
In summary, the z score is a measure of how many standard deviations a value is from the mean of a normal distribution. In the context of confidence intervals, the z score is used to determine the range of values within which the population parameter is likely to fall. The z score for a 95% confidence interval is 1.96, and the confidence interval is calculated by adding and subtracting the z score multiplied by the standard error from the sample mean.