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 to work this out. How do I put this information into an equation or a formula. If I could figure that out im pretty sure I could get the answer
please help :(
Find table in the back of your statistics text labeled something like "areas under normal distribution" to find the proportion/probability (±.025) and its Z score.
For 99%, use ±.005.
95% = mean ± Z SEm
SEm = SD/√n
To calculate the margin of error for a confidence interval, you can use the following formula:
Margin of Error = Critical Value × (Standard Deviation / Square Root of Sample Size)
1. For a 95% confidence interval with a sample size of 400:
a) To find the critical value, you can use a standard normal distribution table or use a calculator. For a 95% confidence level, the critical value is approximately 1.96.
Margin of Error = 1.96 × (100 / √400)
Margin of Error = 1.96 × (100 / 20)
Margin of Error = 1.96 × 5
Margin of Error = 9.8
b) For a 95% confidence level, the margin of error is 9.8.
2. For a 95% confidence interval with a sample size of 1600:
Margin of Error = 1.96 × (100 / √1600)
Margin of Error = 1.96 × (100 / 40)
Margin of Error = 1.96 × 2.5
Margin of Error = 4.9
3. For a 99% confidence interval with a sample size of 400:
a) To find the critical value, for a 99% confidence level, the critical value is approximately 2.58.
Margin of Error = 2.58 × (100 / √400)
Margin of Error = 2.58 × (100 / 20)
Margin of Error = 2.58 × 5
Margin of Error = 12.9
b) For a 99% confidence level, the margin of error is 12.9.
The effect of the sample size:
As the sample size increases, the margin of error becomes smaller. This means that with a larger sample size, the confidence interval becomes narrower and provides a more precise estimate of the population mean.
The effect of the choice of confidence level:
Choosing a higher confidence level, such as 99%, increases the margin of error. This means that the confidence interval becomes wider and provides a less precise estimate of the population mean. The higher confidence level requires a larger margin of error to account for the increased level of confidence.
To find the margin of error for a confidence interval for estimating the population mean, you can use the formula:
Margin of Error = Z * (Standard Deviation / √n),
where Z is the z-score corresponding to the desired confidence level, Standard Deviation is the population standard deviation, and n is the sample size.
(i) For a sample size of 400:
1. Find the z-score corresponding to a 95% confidence level. This can be obtained from a standard normal distribution table or using a statistical calculator. For a 95% confidence level, the z-score is approximately 1.96.
2. Substitute the values into the formula:
Margin of Error = 1.96 * (100 / √400) = 1.96 * (100 / 20) = 1.96 * 5 = 9.8.
Therefore, the margin of error is 9.8.
(ii) For a sample size of 1600:
1. Find the z-score corresponding to a 95% confidence level (same as in part (i)). The z-score is approximately 1.96.
2. Substitute the values into the formula:
Margin of Error = 1.96 * (100 / √1600) = 1.96 * (100 / 40) = 1.96 * 2.5 = 4.9.
Therefore, the margin of error is 4.9.
The effect of sample size: As the sample size increases, the margin of error decreases. This means that larger sample sizes typically result in more precise estimates of the population mean.
(b) To find the margin of error for different confidence levels:
(i) For a confidence level of 95% (same as in part (i)):
Margin of Error = 1.96 * (100 / √400) = 9.8 (as determined previously).
(ii) For a confidence level of 99%:
1. Find the z-score corresponding to a 99% confidence level. For a 99% confidence level, the z-score is approximately 2.576.
2. Substitute the values into the formula:
Margin of Error = 2.576 * (100 / √400) = 2.576 * (100 / 20) = 2.576 * 5 = 12.88.
Therefore, the margin of error is 12.88.
The effect of the choice of confidence level: As the confidence level increases, the margin of error also increases. This is because a higher confidence level requires capturing a wider range of potential values, leading to a larger margin of error.