Determine the smallest sample size required to estimate the population mean under the following specifications:
a. e = 2.4 confidence level = 80% data between 50 and 150
d. e = 1.2 confidence level = 90%, data between 25 and 175
To determine the smallest sample size required to estimate the population mean, we can use the formula for sample size calculation:
n = (Z^2 * σ^2) / (e^2)
Where:
n = sample size
Z = Z value corresponding to the desired confidence level (from the standard normal distribution table)
σ = population standard deviation (if known)
e = maximum error margin or margin of error
a. For the first specification (e = 2.4, confidence level = 80%), we need to determine the Z value for an 80% confidence level. To find this value, we need to find the Z value for the upper tail of 90% and divide it by 2.
Z = (1 - 0.80) / 2
Z = 0.10 / 2
Z = 0.05 (look for Z value closest to 0.05 in the standard normal distribution table, which is approximately 1.28)
Next, we need to determine the population standard deviation (σ). Since it is not provided in the question, we will assume it is unknown. In such cases, we can use a conservative estimate of σ based on the range of the data.
Range = maximum value - minimum value
Range = 150 - 50
Range = 100
Assuming a conservative estimate, we can use σ = Range / 4 (which is equivalent to assuming the data is uniformly distributed).
σ = 100 / 4
σ = 25
Now we can substitute the values into the sample size formula:
n = (Z^2 * σ^2) / (e^2)
n = (1.28^2 * 25^2) / (2.4^2)
n = (1.6384 * 625) / 5.76
n = 1022.5 / 5.76
n ≈ 177.84
Therefore, the smallest sample size required to estimate the population mean within a 2.4 error margin and 80% confidence level, assuming a uniform distribution between 50 and 150, is approximately 178.
d. For the second specification (e = 1.2, confidence level = 90%), we can follow the same steps as above. First, we need to determine the Z value for a 90% confidence level.
Z = (1 - 0.90) / 2
Z = 0.05 (look for Z value closest to 0.05 in the standard normal distribution table, which is approximately 1.645)
Next, we need to determine the population standard deviation (σ). Again, assuming it is unknown, we can use a conservative estimate based on the range of the data.
Range = maximum value - minimum value
Range = 175 - 25
Range = 150
Conservative estimate: σ = Range / 4
σ = 150 / 4
σ = 37.5
Now we can substitute the values into the sample size formula:
n = (Z^2 * σ^2) / (e^2)
n = (1.645^2 * 37.5^2) / (1.2^2)
n ≈ 108.2
Therefore, the smallest sample size required to estimate the population mean within a 1.2 error margin and 90% confidence level, assuming a uniform distribution between 25 and 175, is approximately 109.