To find the margin of error and confidence interval, we first need to calculate the sample size and the standard deviation.
Given that 40% of the 1500 people surveyed like roses, we can calculate the sample size using the formula:
Sample size = (Z-score)^2 * (p)(1-p) / (E)^2
Where:
- Z-score represents the desired level of confidence. For example, for a 95% confidence level, the Z-score is 1.96.
- p represents the proportion of people who like roses (40% or 0.4).
- E represents the margin of error, which is the maximum amount by which the sample proportion might differ from the true population proportion.
Now, let's calculate the sample size:
Sample size = (1.96)^2 * (0.4)(1-0.4) / (0.05)^2
Sample size = 384.16
Since we can't have a fraction of a person, we need to round up the sample size to the nearest whole number. Therefore, the sample size is 385.
The margin of error is calculated by using the formula:
Margin of Error = Z-score * sqrt(p * (1-p) / n)
Where:
- Z-score is the same as before (1.96 for a 95% confidence level).
- p is the proportion of people who like roses (0.4).
- n is the sample size (385).
Let's calculate the margin of error:
Margin of Error = 1.96 * sqrt(0.4 * (1-0.4) / 385)
Margin of Error = 0.05
Therefore, the margin of error is 0.05 or 5%.
To calculate the confidence interval, we use the formula:
Confidence Interval = Sample proportion ± Margin of Error
Given that the sample proportion is 0.4 and the margin of error is 0.05, we can calculate the confidence interval:
Confidence Interval = 0.4 ± 0.05
Confidence Interval = (0.35, 0.45)
So, the confidence interval for this survey is (0.35, 0.45), which means we can be 95% confident that the true proportion of people who like roses falls within this range.