How can I solve this?
A random sample of medical files is used to estimate the proportion p of all people who have blood type B.
If you have no preliminary estimate for p, how many medical files should you include in a random sample in order to be 85% sure that the point estimate p̂ will be within a distance of 0.04 from p? (Round your answer up to the nearest whole number.)
Answer part (a) if you use the preliminary estimate that about 10 out of 90 people have blood type B.
X medical files
To solve this problem, we need to use the margin of error formula for estimating proportions. The formula is given by:
Margin of Error = Z * sqrt(p̂ * (1 - p̂) / n)
Where:
- Z is the Z-score corresponding to the desired level of confidence (in this case, 85% confidence). The Z-score can be found using a standard normal distribution table or calculator.
- p̂ is the sample proportion (unknown in this case).
- n is the sample size (what we need to calculate).
To find the required sample size, we need to find n such that the margin of error is equal to the given distance of 0.04.
First, let's solve part (a) where the preliminary estimate is that about 10 out of 90 people have blood type B.
1. Calculate the point estimate using the preliminary estimate:
p̂ = 10/90 = 0.1111 (rounded to four decimal places)
2. Find the Z-score for 85% confidence level. Looking at the standard normal distribution table or using a calculator, we find that the Z-score is approximately 1.440.
3. Now, let's substitute the known values into the margin of error formula:
0.04 = 1.440 * sqrt(0.1111 * (1 - 0.1111) / n)
4. Rearrange the equation to solve for n:
sqrt(0.1111 * (1 - 0.1111) / n) = 0.04 / 1.440
5. Square both sides of the equation to eliminate the square root:
0.1111 * (1 - 0.1111) / n = (0.04 / 1.440)^2
6. Simplify and solve for n:
0.1111 * 0.8889 / n = 0.00107569
n = 0.1111 * 0.8889 / 0.00107569
n ≈ 812.1
7. Round up to the nearest whole number since we cannot have a fraction of a sample size:
n = 813
Therefore, if you use the preliminary estimate that about 10 out of 90 people have blood type B, you would need a sample size of 813 medical files in order to be 85% sure that the point estimate p̂ will be within a distance of 0.04 from p.
To solve this, we can use the formula for the sample size needed to estimate a population proportion. The formula is given by:
n = (Z^2 * p̂ * q̂) / (E^2)
Where:
n = sample size
Z = Z-value corresponding to the desired confidence level (85% confidence level corresponds to a Z-value of approximately 1.44)
p̂ = preliminary estimate of the population proportion
q̂ = 1 - p̂ (complement of the preliminary estimate)
E = desired margin of error
Let's calculate the sample size for part (a), where the preliminary estimate is that about 10 out of 90 people have blood type B.
Using the given values:
Z = 1.44 (corresponding to 85% confidence level)
p̂ = 10/90
q̂ = 1 - p̂
E = 0.04
Substituting these values into the formula, we get:
n = (1.44^2 * (10/90) * (80/90)) / (0.04^2)
Simplifying the expression further:
n = (2.0736 * 0.1111 * 0.8889) / 0.0016
n = 0.0245 / 0.0016
n ≈ 15.31
Thus, you would need to include at least 16 medical files in the random sample to be approximately 85% sure that the point estimate p̂ will be within a distance of 0.04 from p. (Round up to the nearest whole number)