https://file.aiquickdraw.com/mathbot/file/1728073600804n5a6ojdd.jpg

The problem involves calculating probabilities and confidence intervals based on a sample of teenagers who do not have checking accounts.

### (a) Solution
The problem states that 15% of all teenagers in the United States do not have checking accounts. TeenChex is interested in targeting these teenagers from a random sample of 4,000 teenagers. 

- The expected number of teenagers without checking accounts is calculated as:

\[
\text{Expected number} = \text{Total sample} \times \text{Proportion without checking accounts}
\]
\[
= 4000 \times 0.15 = 600 \text{ teenagers}
\]

However, the value given in the image is 3,400, which might be a typo or confusion.

- The standard deviation \(\sigma\) is already calculated as 22.6.

### (b) Solution: Confidence Interval Calculation
To find the 95% confidence interval, we need to use the formula:

\[
\text{Confidence Interval} = \text{Expected Number} \pm (Z \times \sigma)
\]

For a 95% confidence level, \( Z \approx 1.96 \).

\[
\text{Lower Bound} = 600 - (1.96 \times 22.6) \approx 600 - 44.296 \approx 556
\]
\[
\text{Upper Bound} = 600 + (1.96 \times 22.6) \approx 600 + 44.296 \approx 644
\]

So, the 95% confidence interval is approximately 556 to 644 teenagers without checking accounts.

Let me know if you'd like further details on this or if you have additional questions! Here are some related questions:

1. How do we calculate a confidence interval for proportions?
2. What is the significance of the \( Z \)-score in confidence intervals?
3. How does sample size affect the confidence interval?
4. What does the standard deviation indicate in this context?
5. How would the interval change if the confidence level were 90% instead of 95%?

**Tip:** Always check if the sample proportion or probability is consistent with the given conditions, and verify any outlier values or unusual inputs.

Teenage Marketing: In a certain year, 15% of all teenagers in the United States had checking accounts. TeenChex Inc. is interested in targeting teenagers who do not already have a checking account. (a) If TeenChex selects a random sample of 4,000 teenagers, what number of teenagers without checking accounts can it expect to find? (b) Fill in the missing quantities for a 95% confidence interval.

Learn how to calculate the expected number of teenagers without checking accounts from a random sample of 4,000 teenagers, and find the 95% confidence interval. This problem covers key statistical concepts like probability, confidence intervals, and standard deviation. Suitable for students in Grades 10-12.

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