What is the value of Z for a 90 confidence interval?

As an expert in statistical analysis, I would like to clarify the concept of confidence intervals and the role of Z-scores within them. A confidence interval is a range that is likely to contain a population parameter with a certain level of confidence. The confidence level, which is expressed as a percentage, represents the frequency (in repeated sampling) with which the confidence interval would contain the true value of the parameter. The Z-score is a key component in constructing a confidence interval for a mean when the population standard deviation is known. It is a measure of how many standard deviations an element is from the mean. In the context of a confidence interval, the Z-score corresponds to the number of standard deviations one must go in either direction from the sample mean to capture the desired confidence level. When you're looking to calculate a confidence interval with a specific level of confidence, you need to identify the appropriate Z-score that corresponds to that level. This Z-score is also referred to as the critical value. For a standard normal distribution (Z-distribution), the Z-score indicates the number of standard deviations away from the mean where the desired percentage of the data lies. According to the table you've provided, for a 90% confidence interval, the corresponding Z-score is 1.645. This means that if you were to take all possible samples of a certain size from a population, calculate the mean of each sample, and construct a confidence interval using the Z-score of 1.645, then 90% of those intervals would contain the true population mean. It's important to note that the Z-score for a confidence interval is determined by the shape of the distribution of the data and the level of confidence desired. For a normal distribution, the Z-scores are well-established and can be found in Z-tables or using statistical software. However, for small sample sizes or non-normal distributions, other methods like the t-distribution might be more appropriate. In summary, to calculate a 90% confidence interval for a mean when the population standard deviation is known, you would use a Z-score of 1.645. You would then apply the formula for the confidence interval, which typically looks like this: \[ \text{CI} = \bar{x} \pm Z \times \frac{\sigma}{\sqrt{n}} \] Where: - \( \text{CI} \) is the confidence interval, - \( \bar{x} \) is the sample mean, - \( Z \) is the Z-score (1.645 for a 90% confidence level), - \( \sigma \) is the population standard deviation, - \( n \) is the sample size. Now, let's proceed to the next step as per your instructions.

Confidence IntervalsDesired Confidence IntervalZ Score90% 95% 99%1.645 1.96 2.576