A bank officer wants to determine the amount of the average total monthly deposits per customer at the bank. He believes an estimate of this average using a confidence interval is sufficient.
a) How large a sample should he take to be within $200 of the actual average with 99% confidence? He assumes the standard deviation of total monthly deposits for all customers is about $1000.
b) What about 95%? How do the two results compare? Explain your answer.
To answer this question, we need to use the formula for sample size calculation for determining a confidence interval.
a) To find the sample size needed to be within $200 of the actual average with 99% confidence, we'll use the following formula:
n = [(Z * σ) / E]^2
Where:
n = required sample size
Z = z-score corresponding to the desired confidence level (in this case 99%)
σ = standard deviation of total monthly deposits ($1000)
E = margin of error ($200)
First, let's find the value of the z-score corresponding to a 99% confidence level. You can either check the z-table or use a statistical software or calculator. The z-score corresponding to a 99% confidence level is approximately 2.576.
Substituting the values into the formula:
n = [(2.576 * 1000) / 200]^2
n = 13.464^2
n ≈ 181.148
Therefore, the bank officer should take a sample size of at least 182 customers to estimate the average total monthly deposits per customer within $200 with 99% confidence.
b) To find the sample size needed to be within $200 of the actual average with 95% confidence, we'll use the same formula as before but with a different z-score:
n = [(Z * σ) / E]^2
For a 95% confidence level, the z-score is approximately 1.96 (again, you can refer to the z-table or use a statistical software or calculator).
Substituting the values into the formula:
n = [(1.96 * 1000) / 200]^2
n ≈ 9.8^2
n ≈ 96.04
Therefore, the bank officer should take a sample size of at least 97 customers to estimate the average total monthly deposits per customer within $200 with 95% confidence.
Comparing the two results, we can observe that the required sample size for a higher confidence level (99%) is larger (182) compared to the lower confidence level (95%) which only requires a sample size of 97. This is because a higher confidence level results in a narrower confidence interval and, therefore, requires a larger sample size to achieve the desired level of precision. In other words, as the desired level of confidence increases, larger sample sizes are needed to ensure a more accurate estimation.