Hi there, I'm an expert in statistics and data analysis. I'm here to help you understand how to calculate and interpret z-scores for different confidence intervals, which are crucial in statistical hypothesis testing and estimation.

When we talk about a confidence interval, we're referring to a range that's likely to contain the true value of a population parameter. The confidence level is the probability that the interval contains the true value. For example, a 95% confidence interval means that if we were to take many samples and construct a confidence interval from each, 95% of those intervals would contain the true population parameter.

The z-score is a standard score that tells you how many standard deviations a data point is from the mean of a normally distributed dataset. In the context of confidence intervals, the z-score corresponds to the number of standard deviations that you would need to go from the mean to the critical value that defines the interval at a given confidence level.

For a 99.9% confidence interval, you're looking for a very high level of certainty that the true value lies within your interval. To find the z-score that corresponds to this level, you would typically look at a standard normal (z-) distribution table or use statistical software that can calculate this for you.

The critical z-value for a 99.9% confidence interval is typically around 3.89 or higher because you want to capture 99.9% of the data in the normal distribution, which means you're looking at the extreme tails of the distribution. This is a two-tailed test because you're looking at both ends of the distribution, not just one.

Now, regarding the statement you provided: "For small samples the t value is higher than the Z value what logically means that the confidence interval for smaller samples with the same confidence level is larger." This is indeed correct. The t-distribution is used instead of the z-distribution when dealing with small sample sizes because it is more accurate for those situations. The t-distribution is wider, especially in the tails, which means that for a given confidence level, the confidence interval will be wider when using the t-distribution than when using the z-distribution.

Table 1 would typically list the z-scores for various confidence levels. For example:
- A 90% confidence level might correspond to a z-score of 1.645.
- A 95% confidence level corresponds to a z-score of 1.96.
- A 99% confidence level corresponds to a z-score of 2.576.
- And as mentioned, a 99.9% confidence level corresponds to a z-score of approximately 3.89.

It's important to note that these values are for large samples where the z-distribution can be used. For small samples, you would use the t-distribution, and the critical values would be different.

Now, let's proceed to the next step.

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For small samples the t value is higher than the Z value what logically means that the confidence interval for smaller samples with the same confidence level is larger. Z values for matching 90%, 95%, 99% and 99.9% confidence levels are listed in the Table 1.read more >>