Final answer:
To estimate the population mean within 2 units with a 95% confidence level when the population standard deviation is 8, the margin of error formula is used to calculate the minimum sample size. The calculation shows that we need a sample size of at least 16, which exceeds the provided option numbers, indicating a possible error in the question.
Step-by-step explanation:
The question asks for the minimum sample size needed to estimate a population mean within 2 units with a 95% confidence level when the population standard deviation is known to be 8. This can be calculated using the formula for the margin of error in a confidence interval for a population mean, which includes the Z-score associated with the 95% confidence level, the population standard deviation, and the desired margin of error (E).
The formula is given by:
E = Z*(σ/√n)
Where E is the margin of error, Z is the Z-score for 95% confidence (which is approximately 1.96), σ is the population standard deviation, and n is the sample size.
To find the minimum sample size:
- Identify the given values: E = 2, Z = 1.96, σ = 8.
- Plug these values into the formula: 2 = 1.96*(8/√n).
- Solve for n: n = (1.96*8/2)^2.
- Calculate the value: n = (7.84/2)^2 = (3.92)^2 = 15.3664.
- Since we always round up in sample size calculations to ensure the margin of error is not exceeded, we round to the next whole number which is 16.
Thus, the minimum sample size needed is 16. This is not one of the choices provided in the question, which suggests there may have been a typographical error in the options given. The closest provided option that is greater than the calculated sample size is Option C, 61, which would certainly meet the required margin of error but exceeds the minimum required sample size.