The distribution of cash withdrawals from the automatic teller machine at a
certain bank has a mean of $500 with a standard deviation of $70. To reduce the
incentives for robbery, the bank puts money into the machine every 12 hours and
it keeps the amount deposited fairly close to the expected total withdrawals for a
12-hour period. If 100 withdrawals were expected in each 12-hour period and
each withdrawal was independent, how much should the bank put into the
machine so that the probability of running out of money was 0.05?
To find the amount the bank should put into the machine so that the probability of running out of money is 0.05, we can use the concept of the Central Limit Theorem and the cumulative distribution function (CDF) of a normal distribution.
Here's how you can calculate the required amount:
Step 1: Find the average amount withdrawn per 12-hour period.
The average amount withdrawn per 12-hour period is given as $500, and there are 100 withdrawals expected in each period. Therefore, the average amount withdrawn per withdrawal is $500 / 100 = $5.
Step 2: Calculate the standard deviation of the amount withdrawn per withdrawal.
The standard deviation of the amount withdrawn per withdrawal can be calculated by dividing the standard deviation of total withdrawals in a 12-hour period by the square root of the number of withdrawals. In this case, the standard deviation of total withdrawals is $70, and there are 100 withdrawals. Thus, the standard deviation per withdrawal is $70 / sqrt(100) = $7.
Step 3: Calculate the standard deviation of the sum of 100 withdrawals.
The standard deviation of the sum of 100 withdrawals can be calculated by multiplying the standard deviation per withdrawal by the square root of the number of withdrawals. In this case, the number of withdrawals is 100, and the standard deviation per withdrawal is $7. Thus, the standard deviation of the sum of 100 withdrawals is $7 * sqrt(100) = $70.
Step 4: Find the z-score corresponding to a probability of 0.05.
The z-score represents the number of standard deviations an observation is away from the mean in a normal distribution. To find the z-score corresponding to a probability of 0.05 (or 5%), we can look it up in a standard normal distribution table or use statistical software. In this case, the z-score is approximately -1.645.
Step 5: Calculate the amount the bank should put into the machine.
To calculate the amount the bank should put into the machine, we need to find the value of the sum of 100 withdrawals that corresponds to the z-score of -1.645. This can be done by multiplying the z-score by the standard deviation of the sum of 100 withdrawals and adding it to the average amount withdrawn per 12-hour period. In this case, the z-score is -1.645 and the standard deviation of the sum of 100 withdrawals is $70. Thus, the amount the bank should put into the machine is $5 + (-1.645 * $70) = $5 - $115.15 ≈ $384.85.
Therefore, the bank should put approximately $384.85 into the machine to ensure that the probability of running out of money is 0.05.