To construct a confidence interval for the proportion of all caterpillars that lived to become a butterfly, we can use the following formula:
Confidence interval = sample proportion ± margin of error
where the sample proportion is the number of caterpillars that lived to become a butterfly divided by the total number of observed caterpillars, and the margin of error is calculated using the following formula:
Margin of error = critical value × standard error
The critical value can be found using a z-table or calculator, and is based on the level of confidence and the degrees of freedom (which is n - 1 for a proportion). For a 90% confidence interval with 422 degrees of freedom, the critical value is approximately 1.645.
The standard error can be calculated using the following formula:
Standard error = sqrt((sample proportion × (1 - sample proportion)) / sample size)
Plugging in the values we have, we get:
Sample proportion = 52 / 423 = 0.123
Standard error = sqrt((0.123 × (1 - 0.123)) / 423) = 0.024
So the margin of error is:
Margin of error = 1.645 × 0.024 = 0.039
Therefore, the 90% confidence interval for the proportion of all caterpillars that lived to become a butterfly is:
0.123 - 0.039 < p < 0.123 + 0.039
0.084 < p < 0.162
So with 90% confidence, the proportion of all caterpillars that lived to become a butterfly is between 0.084 and 0.162.
Therefore, the answer is between 0.084 and 0.162.