A credit manager has chosen a sample of size 2,164 to estimate the mean balance of credit accounts to within $0.50. A pilot study of 50 accounts had a mean of $1,219 and a variance of 100. The level of confidence of the interval is:
90% ?
95% ?
98% ?
99% ?
To find the level of confidence for the interval, we need to use the formula for the confidence interval:
Sample Mean ± (Z * (Standard Deviation / sqrt(Sample Size)))
In this case, the sample mean is $1,219, the sample size is 50, the standard deviation is sqrt(100) = 10, and we want the interval to be within $0.50.
Let's calculate the Z-value for different levels of confidence using Z-table:
For 90% confidence level, Z = 1.645
For 95% confidence level, Z = 1.96
For 98% confidence level, Z = 2.33
For 99% confidence level, Z = 2.58
Now, let's calculate the confidence intervals for each level of confidence:
For 90% confidence level:
Lower bound = $1,219 - (1.645 * (10 / sqrt(50)))
Upper bound = $1,219 + (1.645 * (10 / sqrt(50)))
For 95% confidence level:
Lower bound = $1,219 - (1.96 * (10 / sqrt(50)))
Upper bound = $1,219 + (1.96 * (10 / sqrt(50)))
For 98% confidence level:
Lower bound = $1,219 - (2.33 * (10 / sqrt(50)))
Upper bound = $1,219 + (2.33 * (10 / sqrt(50)))
For 99% confidence level:
Lower bound = $1,219 - (2.58 * (10 / sqrt(50)))
Upper bound = $1,219 + (2.58 * (10 / sqrt(50)))
Now, let's compare the intervals with $0.50 to determine the level of confidence:
- If the interval is within $0.50, then that level of confidence is correct.
- If the interval is more than $0.50, then the level of confidence is too low.
- If the interval is less than $0.50, then the level of confidence is too high.
By comparing the intervals, we find:
- For 90% confidence level, the interval is wider than $0.50, so it is too low.
- For 95% confidence level, the interval is wider than $0.50, so it is too low.
- For 98% confidence level, the interval is narrower than $0.50, so it is too high.
- For 99% confidence level, the interval is narrower than $0.50, so it is too high.
Therefore, the correct level of confidence for the interval is 98%.