Final answer:
The appropriate t critical value for a two-tailed 95% confidence interval with 19 degrees of freedom is 2.093. For estimating a population proportion to be within four percentage points at a 95% confidence level, a sample size of 1,068 is needed.
Step-by-step explanation:
The appropriate t critical value depends on the confidence level and the sample size, which defines the degrees of freedom (df). For a two-tailed 95% confidence interval with 19 degrees of freedom (df), the t value is 2.093. This value is found using a t-distribution table or calculator functions like invT(0.975, 19) because you are looking for the t-value at the upper tail that leaves 2.5% in each tail when the confidence level is 95%. The levels of confidence and corresponding critical values change based on the specific requirements of the confidence interval being calculated.
Sample size calculations for estimating a population proportion use a different approach. For example, to estimate a population proportion within four percentage points with 95% confidence, and an estimated population proportion of 0.5, a sample size of approximately 1,068 would be required. This ensures adequate precision of the confidence interval.