Final answer:
The 95% confidence interval for the proportion of all adults whose IQ score is greater than 100 is approximately 0.4678 to 0.6064 and the lower limit is 0.4678, and the upper limit is 0.6064.
Step-by-step explanation:
To compute a 95% confidence interval for the proportion of all adults whose IQ score is greater than 100, we can use the formula:
Confidence Interval = sample proportion ± (critical value) * (standard error)
1. Calculate the sample proportion:
The sample proportion is the number of adults who scored higher than 100 divided by the total sample size:
Sample proportion = 94/175 = 0.5371 (rounded to four decimal places)
2. Determine the critical value:
The critical value corresponds to the desired confidence level. For a 95% confidence level, the critical value can be found using a Z-table or a statistical calculator. In this case, the critical value is approximately 1.96 (rounded to two decimal places).
3. Calculate the standard error:
The standard error measures the variability of the sample proportion. It is computed using the formula:
- Standard error = √((sample proportion * (1 - sample proportion)) / sample size)
- Standard error = √((0.5371 * (1 - 0.5371)) / 175) = 0.0356 (rounded to four decimal places)
4. Compute the confidence interval:
- Confidence Interval = sample proportion ± (critical value) * (standard error)
- Confidence Interval = 0.5371 ± (1.96 * 0.0356)
- Lower Limit = 0.5371 - (1.96 * 0.0356)
- Lower Limit ≈ 0.4678 (rounded to four decimal places)
- Upper Limit = 0.5371 + (1.96 * 0.0356)
- Upper Limit ≈ 0.6064 (rounded to four decimal places)
Therefore, the 95% confidence interval for the proportion of all adults whose IQ score is greater than 100 is approximately 0.4678 to 0.6064. The lower limit is 0.4678, and the upper limit is 0.6064.