Final answer:
For an 85% confidence level, the critical z-score (Za/2) is approximately 1.44. For an 80% confidence level with a sample size of 31 (which has 30 degrees of freedom), the critical t-score (ta/2) is approximately 1.310. These critical values are used in constructing confidence intervals.
Step-by-step explanation:
To answer the question about critical values for different confidence levels, we need to know about critical z-scores and t-scores. These scores are essential when constructing confidence intervals in statistics, which allow researchers to estimate the range in which a population parameter lies based on sample data.
For an 85% confidence level, we have an alpha (\( \alpha \)) of 1 - 0.85 = 0.15, and the alpha divided by 2 (\( \alpha/2 \)) equals 0.075 for each tail of the standard normal distribution. To find the critical value Za/2, which we denote as Z0.075, we would consult a z-table, use a calculator or statistical software. The critical z-score that leaves 0.075 in the upper tail of the standard normal distribution is approximately 1.44. Thus, Za/2 = 1.44 (rounded to two decimal places).
For an 80% confidence level with a sample size of 31, we have 30 degrees of freedom (n-1 where n is the sample size). Consulting a t-distribution table or using a calculator with the function for the inverse t-distribution, we find the critical t-score that corresponds to an 80% confidence level with 30 degrees of freedom. The critical value ta/2 for our scenario is approximately 1.310 (rounded to three decimal places).
Understanding these concepts is central to conducting statistical analyses and interpreting their results correctly. The formula for the confidence interval is (x-bar - Za/2* (standard deviation/sqrt(n)), x-bar + Za/2* (standard deviation/sqrt(n))) where x-bar is the sample mean, Za/2 is the critical z-score, and n is the sample size.