A large bank knows the average credit card debt owed by their card holders is 1155 with a standard deviation of 349. In an investigation into the extent of credit card debt, the bank collects random sample of 44 credit card accounts. Calculate what is the probability that the mean credit card debt for the same of accounts is more than 1008?

To calculate the probability that the mean credit card debt for a sample of 44 accounts is more than 1008, we can use the **Central Limit Theorem** (CLT), which states that for sufficiently large sample sizes, the sampling distribution of the sample mean is approximately normal.

We are given:

- Population mean \( \mu = 1155 \)
- Population standard deviation \( \sigma = 349 \)
- Sample size \( n = 44 \)
- We want to find \( P(\bar{x} > 1008) \), where \( \bar{x} \) is the sample mean.

### Step 1: Compute the Standard Error (SE) of the mean

The **Standard Error** of the mean is calculated as:

\[
SE = \frac{\sigma}{\sqrt{n}} = \frac{349}{\sqrt{44}} = \frac{349}{6.6332} \approx 52.63
\]

### Step 2: Compute the Z-score

Next, we calculate the **Z-score**, which represents how many standard errors the value 1008 is away from the mean 1155:

\[
Z = \frac{\bar{x} - \mu}{SE} = \frac{1008 - 1155}{52.63} = \frac{-147}{52.63} \approx -2.79
\]

### Step 3: Find the corresponding probability

Using the Z-table, we find the probability corresponding to \( Z = -2.79 \). The Z-table gives the probability to the left of a given Z-value.

- \( P(Z < -2.79) \approx 0.0026 \)

Since we want the probability that the sample mean is **greater than** 1008, we need to find \( P(Z > -2.79) \):

\[
P(\bar{x} > 1008) = 1 - P(Z < -2.79) = 1 - 0.0026 = 0.9974
\]

### Final Answer:

The probability that the mean credit card debt for the sample is more than 1008 is approximately **0.9974**, or 99.74%.

Would you like more details or further explanation?

Here are 5 related questions you might explore:
1. How does the sample size affect the probability calculation in the Central Limit Theorem?
2. What happens if the standard deviation of the population changes?
3. How can the Central Limit Theorem be applied to non-normal data distributions?
4. What is the probability that the sample mean is less than 1155?
5. How do we handle cases where the population standard deviation is unknown?

**Tip**: When working with sample means, remember that larger sample sizes make the sampling distribution of the mean closer to a normal distribution, regardless of the shape of the population distribution.

Calculate the probability that the mean credit card debt for a random sample of 44 accounts is more than 1008. Using the Central Limit Theorem, this problem covers key concepts like Z-scores, standard error, and probability calculation with a population mean of 1155 and standard deviation of 349.

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