Find the margin of error for a 95% confidence interval for estimating the population mean when the sample standard deviation equals 100, with a sample size of (i) 400, (ii) 1600. What is the effect of the sample size? (b) Find the margin of error for estimating the population mean when the sample standard deviation equals 100 for a sample size of 400, using confidence level (i) 95%, (ii) 99%. What is the effect of the choice of confidence level?
I just don't understand how do I put this into an equation?
To find the margin of error for estimating the population mean, you can use the following formula:
Margin of Error = (Z * Standard Deviation) / √(Sample Size)
where Z represents the Z-score corresponding to the desired level of confidence. The Z-score can be obtained from standard normal distribution tables or using a statistical calculator.
Now let's calculate the margin of error for the given scenarios:
(i) Sample Size = 400, Confidence Level = 95%
To find the Z-score for a 95% confidence level, we look up the corresponding value from the standard normal distribution. The Z-score for a 95% confidence level is approximately 1.96.
Margin of Error = (1.96 * 100) / √(400)
Margin of Error = 1.96 * 10 / √(20)
Margin of Error ≈ 19.6 / 20
Margin of Error ≈ 0.98
Therefore, the margin of error for estimating the population mean with a sample size of 400 and a 95% confidence level is approximately 0.98.
(ii) Sample Size = 1600, Confidence Level = 95%
Using the same formula, the Z-score for a 95% confidence level is still approximately 1.96.
Margin of Error = (1.96 * 100) / √(1600)
Margin of Error = 1.96 * 10 / √(40)
Margin of Error ≈ 19.6 / 40
Margin of Error ≈ 0.49
Therefore, the margin of error for estimating the population mean with a sample size of 1600 and a 95% confidence level is approximately 0.49.
The effect of the sample size:
As the sample size increases (from 400 to 1600 in this case), the margin of error decreases. A larger sample size provides more precise estimates because there is less variability in the data.
(i) Sample Size = 400, Confidence Level = 99%
To find the Z-score for a 99% confidence level, we look up the corresponding value from the standard normal distribution. The Z-score for a 99% confidence level is approximately 2.58.
Margin of Error = (2.58 * 100) / √(400)
Margin of Error = 2.58 * 10 / √(20)
Margin of Error ≈ 25.8 / 20
Margin of Error ≈ 1.29
Therefore, the margin of error for estimating the population mean with a sample size of 400 and a 99% confidence level is approximately 1.29.
(ii) Sample Size = 400, Confidence Level = 99%
Using the same formula, the Z-score for a 99% confidence level is still approximately 2.58.
Margin of Error = (2.58 * 100) / √(400)
Margin of Error = 2.58 * 10 / √(20)
Margin of Error ≈ 25.8 / 20
Margin of Error ≈ 1.29
Therefore, the margin of error for estimating the population mean with a sample size of 400 and a 99% confidence level is also approximately 1.29.
The choice of confidence level:
The confidence level determines the level of certainty you have in your estimate. A higher confidence level (99% in this case) requires a larger margin of error compared to a lower confidence level (95% in this case). This means that with a higher confidence level, the range around the estimated mean is wider, providing a more cautious estimate but also a less precise one.
To find the margin of error for a confidence interval, you can use the formula:
Margin of Error = Critical Value * (Standard Deviation / √n)
In this equation:
- Critical Value: This value is determined by the desired confidence level. It represents the number of standard deviations from the mean to the edge of the confidence interval and can be obtained from a Z-table or using a statistical software.
- Standard Deviation: This is the sample standard deviation.
- n: This represents the sample size.
Case (i):
For a 95% confidence level and a sample size of 400:
- Critical Value: The critical value for a 95% confidence level is approximately 1.96.
- Standard Deviation: Given as 100.
- Sample Size: n = 400.
Plug the values into the formula:
Margin of Error = 1.96 * (100 / √400)
Simplifying, we get:
Margin of Error = 1.96 * (100 / 20)
Margin of Error = 1.96 * 5
Margin of Error = 9.8
Therefore, the margin of error is 9.8.
Case (ii):
For a 95% confidence level and a sample size of 1600:
- Critical Value: The critical value for a 95% confidence level is still approximately 1.96.
- Standard Deviation: Given as 100.
- Sample Size: n = 1600.
Plug the values into the formula:
Margin of Error = 1.96 * (100 / √1600)
Simplifying, we get:
Margin of Error = 1.96 * (100 / 40)
Margin of Error = 1.96 * 2.5
Margin of Error = 4.9
Therefore, the margin of error is 4.9.
The effect of the sample size is that as the sample size increases, the margin of error decreases. This means that with a larger sample size (1600 vs. 400), you can estimate the population mean with a smaller margin of error, indicating higher precision.
Moving on to part b:
Case (i):
For a confidence level of 95% and a sample size of 400:
- Critical Value: The critical value for a 95% confidence level is still 1.96.
- Standard Deviation: Given as 100.
- Sample Size: n = 400.
Plug the values into the formula:
Margin of Error = 1.96 * (100 / √400)
Simplifying, we get:
Margin of Error = 1.96 * (100 / 20)
Margin of Error = 1.96 * 5
Margin of Error = 9.8
Therefore, the margin of error is 9.8.
Case (ii):
For a confidence level of 99% and a sample size of 400:
- Critical Value: The critical value for a 99% confidence level is approximately 2.58.
- Standard Deviation: Given as 100.
- Sample Size: n = 400.
Plug the values into the formula:
Margin of Error = 2.58 * (100 / √400)
Simplifying, we get:
Margin of Error = 2.58 * (100 / 20)
Margin of Error = 2.58 * 5
Margin of Error = 12.9
Therefore, the margin of error is 12.9.
The effect of the choice of confidence level is that as the confidence level increases, the margin of error also increases. This means that with a higher confidence level (99% vs. 95%), the margin of error becomes larger, indicating a wider interval around the estimated population mean to account for the increased confidence level.