how to work this math problemAn archaeologist has 250 animal bone fragments from prehistoric site -of these 195 are white-tail deer bones. Construct the 90% confidence interval for the percentage of bone fragments from this site that are from white-tail deer
To construct the 90% confidence interval for the percentage of bone fragments from this site that are from white-tail deer, you can use the formula for calculating confidence intervals for proportions.
The formula is:
Confidence Interval = Sample Proportion ± (Z * Standard Error)
1. Calculate the Sample Proportion:
Sample Proportion = (Number of white-tail deer bones) / (Total number of bone fragments)
Sample Proportion = 195 / 250 = 0.78 (or 78%)
2. Determine the Z-score corresponding to a 90% confidence level. Since the confidence level is 90%, the alpha level is 0.1, which means you need to find the Z-score that corresponds to a cumulative probability of 0.05 on each side of the normal distribution. You can use a Z-table or a calculator to find this value. The Z-score for a 90% confidence level is approximately 1.645.
3. Calculate the Standard Error:
Standard Error = sqrt((Sample Proportion * (1 - Sample Proportion)) / Sample Size)
Standard Error = sqrt((0.78 * (1 - 0.78)) / 250) ≈ 0.027
4. Plug the values into the formula:
Confidence Interval = 0.78 ± (1.645 * 0.027)
5. Calculate the lower and upper bounds of the confidence interval:
Lower Bound = 0.78 - (1.645 * 0.027) ≈ 0.74 (or 74%)
Upper Bound = 0.78 + (1.645 * 0.027) ≈ 0.82 (or 82%)
Therefore, the 90% confidence interval for the percentage of bone fragments from this site that are from white-tail deer is approximately 74% to 82%.