As a manager for an advertising company, you must plan a campaign designed to increase Twitter usage. A recent survey suggests that 85% of adults know what Twitter is. How many adults should you survey in order to be 90% confident that your estimate is within 5% of the true population proportion?
Solution
Our p in this case is 0.85
Our q will be 1-p, which is 1-0.85
The Margin of Error will be 0.05,
Z*-value will be 1.645, as read from the table
Therefore, n will be given by
n= pq(z/ME)2
n= 0.85*(1-0.85) *{1.645/0.05)2
n=138.0007, approx. 138 adults
Well, if I were a manager for an advertising company, I would probably take a slightly unorthodox approach to increasing Twitter usage. How about we hire an army of birds to deliver handwritten tweets to people's doors? It's a bird-brained idea, I know, but it just might work!
But, if you want to stick with surveys, let's crunch the numbers. To calculate the sample size needed to estimate a population proportion, we can use the formula:
n = (Z^2 * p * q) / (E^2)
Where:
n = sample size
Z = Z-value corresponding to the desired confidence level (90% confidence level has a Z-value of approximately 1.645)
p = estimated population proportion (85% or 0.85)
q = 1 - p
E = margin of error (5% or 0.05)
Using this formula, we can plug in the values:
n = (1.645^2 * 0.85 * 0.15) / (0.05^2)
n ≈ 921.83
We can't have a fraction of a person, so we'll round it up to 922.
Therefore, in order to be 90% confident that our estimate is within 5% of the true population proportion, we should survey 922 adults. And who knows, maybe we can even ask them if they prefer tweets from birds or from their usual social media platforms!
To determine the sample size needed for the survey, we can use the formula for sample size calculation with a specific level of confidence. The formula is:
n = (Z^2 * P * (1-P)) / E^2
Where:
n = sample size
Z = Z-score corresponding to the desired level of confidence (90% confidence level corresponds to a Z-score of 1.645)
P = estimated proportion (0.85)
E = desired margin of error (0.05)
Using the given information, we can substitute the values into the formula:
n = (1.645^2 * 0.85 * (1-0.85)) / 0.05^2
n = (2.705*0.85*0.15) / 0.0025
n = 0.326 * 0.15 / 0.0025
n = 0.0489 / 0.0025
n ≈ 19.56
Since you cannot survey a fraction of a person, you would need to round up the sample size to the nearest whole number. Therefore, you should survey at least 20 adults in order to be 90% confident that your estimate is within 5% of the true population proportion.
To determine the number of adults you need to survey in order to be 90% confident that your estimate is within 5% of the true population proportion, you can use the formula for sample size calculation for estimating proportions.
The formula is:
n = (Z^2 * p * (1-p)) / E^2
Where:
- n is the required sample size
- Z is the z-score corresponding to the desired confidence level (90% confidence corresponds to a z-score of approximately 1.645)
- p is the estimated proportion (in this case, 85% or 0.85)
- E is the desired margin of error (in this case, 5% or 0.05)
Using this formula, let's calculate the required sample size:
n = (1.645^2 * 0.85 * (1-0.85)) / 0.05^2
n ≈ 946.34
Since you cannot have a fraction of a person, you would need to round up the sample size to the nearest whole number. Therefore, you should survey at least 947 adults to be 90% confident that your estimate is within 5% of the true population proportion.