suppose x has a mound-shaped distribution with sigma=3. Find the minimal sample size required so that for a 95% confidence interval, the maximal margin of errors is E=0.4
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To find the minimal sample size required, we need to use the formula for the margin of error:
Margin of Error (E) = Z * (σ / √n)
Where:
- E is the desired maximal margin of error
- Z is the Z-score corresponding to the desired level of confidence (95% confidence interval in this case)
- σ is the population standard deviation
- n is the sample size
In this case, the population standard deviation (σ) is given as 3, and the desired maximal margin of error (E) is 0.4.
First, we need to find the Z-score corresponding to the 95% confidence level. The Z-score can be found using a Z-table or a statistical software. For a 95% confidence level, the Z-score is approximately 1.96.
Substituting the values into the formula, we have:
0.4 = 1.96 * (3 / √n)
Now, let's solve the equation for n:
0.4 / 1.96 = √n / 3
(0.4 / 1.96)^2 = n / 3^2
(0.4^2 / 1.96^2) * 9 = n
n = (0.4^2 / 1.96^2) * 9
n = 0.04 / 0.038416
n ≈ 1.039
Since the sample size (n) should be an integer (we can't have a fraction of a participant), we need to round up to the nearest whole number. Therefore, the minimal sample size required is 2 participants.
Note: It is unusual to have such a small sample size, so it's important to consider whether this sample size is suitable for the specific research or study.