https://file.aiquickdraw.com/mathbot/file/1729157112047eqha7njv.jpg

The problem states that a national bank analyzed a random sample of 365 cheque accounts and found that 78 were overdrawn. It asks to estimate, with 90% confidence, the percentage of all bank accounts at the GK branch that were **not overdrawn**, and interpret the findings.

### Step-by-Step Solution:

1. **Determine the sample proportion of overdrawn accounts:**
   - Let \( p \) represent the sample proportion of overdrawn accounts:
   \[
   p = \frac{78}{365} \approx 0.2137
   \]
   So, about 21.37% of the accounts were overdrawn.

2. **Proportion of accounts that were not overdrawn:**
   - The percentage of accounts that were not overdrawn is:
   \[
   1 - p = 1 - 0.2137 = 0.7863
   \]
   So, about 78.63% of the accounts were not overdrawn in the sample.

3. **Calculate the standard error (SE):**
   The standard error for a proportion is calculated as:
   \[
   SE = \sqrt{\frac{p(1 - p)}{n}}
   \]
   Where:
   - \( p = 0.2137 \)
   - \( 1 - p = 0.7863 \)
   - \( n = 365 \)

   Substituting the values:
   \[
   SE = \sqrt{\frac{0.2137 \times 0.7863}{365}} \approx 0.0212
   \]

4. **Find the z-value for 90% confidence level:**
   For a 90% confidence interval, the critical value \( z \) is approximately 1.645.

5. **Calculate the margin of error (ME):**
   The margin of error is given by:
   \[
   ME = z \times SE = 1.645 \times 0.0212 \approx 0.0349
   \]

6. **Construct the confidence interval:**
   The confidence interval for the proportion of accounts that were not overdrawn is:
   \[
   \text{Lower bound} = 0.7863 - 0.0349 = 0.7514
   \]
   \[
   \text{Upper bound} = 0.7863 + 0.0349 = 0.8212
   \]
   Therefore, the 90% confidence interval for the proportion of accounts that were **not** overdrawn is \( [0.7514, 0.8212] \), or between 75.14% and 82.12%.

### Interpretation:
We are 90% confident that the true percentage of accounts at the GK branch of the bank that were not overdrawn is between 75.14% and 82.12%.

---

Would you like further details or explanations on any specific step?

Here are 5 related questions you might find interesting:
1. How would the confidence interval change if we wanted a 95% confidence level?
2. What is the significance of using a random sample in this context?
3. How does increasing the sample size affect the margin of error?
4. Why is it important to interpret confidence intervals cautiously in statistical analysis?
5. Could we use a different method to estimate the confidence interval, and how would that affect the result?

**Tip**: Confidence intervals provide a range of plausible values for a population parameter, but they are not absolute guarantees. Always interpret them as estimations based on the sample data.

A national bank analysed a random sample of 365 cheque accounts at their GK branch and found that 78 of them were overdrawn. Estimate, with 90% confidence, the percentage of all bank accounts at the GK branch of the bank those were not overdrawn. Interpret the findings.

Calculate the 90% confidence interval for the percentage of non-overdrawn bank accounts at the GK branch using a sample of 365 accounts. This problem involves key statistical concepts such as sample proportion, standard error, and margin of error, suitable for students in Grades 10-12.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation