Zillow.com contains several hundred postings for twoβbedroom apartments in your city. A random sample of π=16 postings from Zillow.com resulted in a standard deviation of $276. If a new study was being planned with π π=280 and the desired 95% margin of error is $100, how many postings need to be sampled? Determine the minimum appropriate value.
(Give your answer as a whole number.)
Well, let's clown around with some numbers, shall we?
We have a standard deviation of $276 from the initial sample of π=16 postings. Now we are planning a new study with a standard deviation of π π=280 and a desired 95% margin of error of $100.
To find the minimum appropriate sample size, we can use the formula:
π = (π§β
π π/πΈ)Β²
Where π is the sample size, π§ is the Z-score corresponding to the desired confidence level (95% confidence interval has a Z-score of approximately 1.96), π π is the standard deviation, and πΈ is the desired margin of error.
Substituting the given values, we get:
π = (1.96β
280/100)Β²
π = (545.6)Β²
π β 297,486.16
Now, I'm sure you don't want to go out and sample nearly 300,000 postings, do you? That's a lot of apartments to go through!
However, we need a whole number for the sample size, so we round it up to the nearest whole number. Therefore, the minimum appropriate sample size would be approximately 297,487 postings.
But hey, keep in mind that this is just a clown's attempt to answer a question. The real estate market can be a tricky beast, so it's always a good idea to consult with a statistical expert before making any decisions.
To determine the minimum number of postings that need to be sampled, we can use the formula for sample size calculation:
π = (π§ * π π / π)^2
Where π is the required sample size, π§ is the z-score corresponding to the desired confidence level (95% in this case, which has a z-score of 1.96), π π is the standard deviation of the sample from the pilot study (280), and π is the desired margin of error ($100).
Plugging in the values:
π = (1.96 * 280 / 100)^2
Simplifying the equation:
π = (1.96 * 2.8)^2
π = 4.368^2
π β 19.08
Since the number of postings needs to be a whole number, we should round up to the nearest whole number.
Therefore, the minimum appropriate value for the number of postings that should be sampled is 20.
To determine the minimum number of postings that need to be sampled, we can use the formula for margin of error:
Margin of Error = Z * (Standard Deviation / sqrt(n))
Where:
- Z is the z-score corresponding to the desired confidence level (95% confidence level corresponds to a z-score of 1.96)
- Standard Deviation is the standard deviation of the population (given as $276)
- n is the sample size
We can rearrange the formula to solve for n:
n = ((Z * Standard Deviation) / Margin of Error)^2
Given that the desired margin of error is $100 and the standard deviation of the new study is $280, we can substitute these values into the formula:
n = ((1.96 * 280) / 100)^2
Calculating this expression yields:
n = 153.664
Since the number of postings must be a whole number, we need to round up to the nearest whole number to guarantee a minimum sample size. Therefore, the minimum appropriate value for the sample size is 154 postings.