90% confidence; the sample size is 1150 of which 40% are successes.
To determine the 90% confidence interval for a sample proportion, you need to use the following formula:
CI = p̂ ± Z * √( p̂(1-p̂) / n )
Where:
- p̂ is the sample proportion (in decimal form), which is 40% or 0.4 in this case.
- n is the sample size, which is 1150 in this case.
- Z is the Z-score corresponding to the desired confidence level. For a 90% confidence level, the Z-score is approximately 1.645.
Let's plug in these values into the formula:
CI = 0.4 ± 1.645 * √( 0.4(1-0.4) / 1150 )
Now let's calculate the confidence interval:
CI = 0.4 ± 1.645 * √( 0.24 / 1150 )
CI = 0.4 ± 1.645 * 0.0157
CI = 0.4 ± 0.0259
Therefore, the 90% confidence interval for the proportion of successes is approximately 0.3741 to 0.4259.
This means we can be 90% confident that the true proportion of successes in the population lies between 37.41% and 42.59%, based on the given sample.