A simple random sample of size 300 was taken from the population of all manufacturing establishments in a certain state: 11 establishments in the sample had 100 employees or more. Estimate the percentage of manufacturing establishments with 100 employees or more.5 Attach a standard error to the estimate.
Question
A simple random sample of size 300 was taken from the population of all manufacturing establishments in a certain state: 11 establishments in the sample had 100 employees or more. Estimate the percentage of manufacturing establishments with 100 employees or more.5 Attach a standard error to the estimate.
Solution
To estimate the percentage of manufacturing establishments with 100 employees or more, we can use the proportion from our sample as an estimate for the population.
Step 1: Calculate the proportion of establishments in the sample with 100 employees or more.
This is done by dividing the number of establishments with 100 employees or more by the total number of establishments in the sample.
So, 11 (establishments with 100 or more employees) / 300 (total establishments in the sample) = 0.0367.
This means that approximately 3.67% of the establishments in the sample have 100 employees or more.
Step 2: Calculate the standard error.
The formula for the standard error of a proportion is sqrt[(p(1 - p)) / n], where p is the proportion and n is the sample size.
So, sqrt[(0.0367 * (1 - 0.0367)) / 300] = 0.0108.
This means that the standard error of the estimate is approximately 0.0108, or 1.08%.
So, we estimate that about 3.67% of all manufacturing establishments in the state have 100 employees or more, with a standard error of 1.08%.
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