A z-score is a standardized score that measures the distance between a data point and the mean of a distribution in terms of standard deviations. Specifically, a z-score is calculated as the difference between a data point and the mean of the distribution divided by the standard deviation of the distribution.
The formula for calculating a z-score is:
z = (x – μ) / σ
Where x is the data point, μ is the mean of the distribution, and σ is the standard deviation of the distribution.
For example, suppose we have a sample of test scores for a population of students and the mean score is 75 with a standard deviation of 10. If a student scores 85 on the test, and we can calculate the z-score as follows:
z = (85 – 75) / 10 = 1
It means that the student’s score is one standard deviation above the mean.
How is a z-score used?
A z-score is useful because it allows us to compare data points from different distributions, even if the means and standard deviations of the distributions are different. By standardizing the data, we can more easily compare how far apart different data points are from their respective means.
One common use of z-scores is in hypothesis testing. In hypothesis testing, we compare a sample mean to a known or assumed population mean to resolve whether the sample mean is statistically different from the population mean. We can calculate a z-score for the sample mean and use that score to determine the probability of obtaining that sample mean if the population mean were true.
How to calculate the z-score for a 90% confidence interval
We need first to understand what a confidence interval is to calculate the z-score for a 90% confidence interval. A confidence interval is a range of values within which we can reasonably ensure that the true population parameter lies. The confidence level refers to the percentage of times the interval will contain the true parameter if the same study were conducted multiple times.
For a 90% confidence interval, the z-score is based on the standard normal distribution, which has a mean of 0 and a standard deviation of 1. A table of standard normal probabilities can be used to determine the z-score corresponding to a given confidence level.
For a 90% confidence interval, the z-score is 1.645. It means that the interval extends 1.645 standard deviations from the mean in both directions. To calculate the upper and lower bounds of the confidence interval for a given sample mean and sample standard deviation, we would use the following formula:
CI = X ± (z-score) *
(Standard deviation / sqrt(n))
Where X is the sample mean, z-score is 1.645 for a 90% confidence interval, the standard deviation is the sample standard deviation, and n is the sample size.
The z-values for different confidence levels are based on the standard normal distribution with a mean of 0 and a standard deviation of 1.
The table below shows the z-values for common confidence levels:
| Confidence Level | Z-value |
| 90% | 1.645 |
| 95% | 1.96 |
| 99% | 2.576 |
Let’s take a random sample from a population and construct a confidence interval based on that sample. We can be 90% confident that the true population parameter lies within the interval calculated using the sample mean and standard deviation. The z-value is 1.645 for a 99% confidence level.
Let’s take a sample from a population and construct a confidence interval based on that sample. We can be 95% confident that the true population parameter lies within the interval calculated using the sample mean and standard deviation. For a 95% confidence level, the z-value is 1.96.
Here is a random sample from a population and construct a confidence interval based on that sample. We can be 99% confident that the true population parameter lies within the interval calculated using the sample mean and standard deviation. The z-value is 2.576 for a 99% confidence level.
What is the formula for the 90% confidence interval?
The formula for a 90% confidence interval for a population mean is:
CI = X ± (z-score) * (s / sqrt(n))
Where:
- CI is the confidence interval
- X is the sample mean
Z-score is the critical value from the standard normal distribution, which is 1.645 for a 90% confidence interval
- s is the sample standard deviation
- n is the sample size
This formula calculates the range of values within which we can be 90% confident that the true population mean lies.
For example, let’s take a random sample of 50 students and find that their average GPA is 3.5 with a standard deviation of 0.5. We can construct a 90% confidence interval for the population mean GPA using the formula:
CI = 3.5 ± (1.645) * (0.5 / sqrt(50))
Simplifying this expression, we get:
CI = 3.5 ± 0.115
Therefore, the 90% confidence interval for the population means GPA is (3.385, 3.615). We can be 90% confident that the true population means GPA lies within this range.
Conclusion: What is the Z score for 90 confidence interval?
The z-score for a 90% confidence interval is 1.645. It means that if we take a random sample from a population and construct a confidence interval based on that sample, we can be 90% confident that the true population parameter lies within the interval calculated using the sample mean and standard deviation, with a margin of error determined by the z-score. The z-score is based on the standard normal distribution and is a critical value used in determining the range of values for a confidence interval. The formula for a 90% confidence interval includes the z-score, the sample mean, the sample standard deviation, and the sample size. The confidence interval provides valuable information about the range of values within which we can be confident that the true population parameter lies based on a sample taken from that population.