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A poll found that 40% of 1500 people surveyed like roses. Find the margin of error and confidence interval.

User Dominix
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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.

User Christiana
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