  
n
=
130
,
σ
=
3.5
,
α
=
1
−
0.1
=
0.9
,
   
Z
   
α
2
  
=
Z
  
0.45
   
=
1.645
  (Round to the nearest hundredth)

To solve the problem, it appears you are looking to find the margin of error or the confidence interval for a population parameter given the sample size (\( n \)), standard deviation (\( \sigma \)), and a specified confidence level \( \alpha \).

Here's the given dataTo solve the problem, it appears you are looking to find the margin of error or the confidence interval for a population parameter given the sample size (\( n \)), standard deviation (\( \sigma \)), and a specified confidence level \( \alpha \).

Here's the given 

- \( n = 130 \) (sample size)
- \( \sigma = 3.5 \) (population standard deviation)
- \( \alpha = 1 - 0.1 = 0.9 \) (confidence level)
- \( Z_{\alpha/2} = Z_{0.45} = 1.645 \) (Z-score corresponding to the confidence level)

### Step 1: Margin of Error Calculation

The margin of error \( E \) for the confidence interval can be calculated using the formula:

\[
E = Z_{\alpha/2} \times \frac{\sigma}{\sqrt{n}}
\]

### Step 2: Substitute the Values

\[
E = 1.645 \times \frac{3.5}{\sqrt{130}}
\]

### Step 3: Compute the Margin of Error

First, compute the square root of the sample size:

\[
\sqrt{130} \approx 11.4018
\]

Next, divide the standard deviation by the square root of the sample size:

\[
\frac{3.5}{11.4018} \approx 0.307
\]

Now, multiply by the Z-score:

\[
E \approx 1.645 \times 0.307 \approx 0.5053
\]

### Step 4: Round the Margin of Error

Rounding to the nearest hundredth:

\[
E \approx 0.51
\]

### Final Answer:
The margin of error is approximately **0.51**.

Would you like a detailed explanation on how this relates to the confidence interval or any other questions?

Here are 5 related questions:

1. How does increasing the sample size \( n \) affect the margin of error?
2. What happens to the margin of error if the confidence level \( \alpha \) is increased to 0.95?
3. Can you explain the difference between population standard deviation \( \sigma \) and sample standard deviation \( s \)?
4. What role does the Z-score play in constructing confidence intervals?
5. How do you interpret a 90% confidence interval in practical terms?

**Tip:** Always remember that the margin of error decreases as the sample size increases, leading to a more precise estimate.

n = 130, σ = 3.5, α = 1 − 0.1 = 0.9, Zα/2 = Z0.45 = 1.645 (Round to the nearest hundredth)

Learn how to calculate the margin of error for a sample size of 130, a standard deviation of 3.5, and a 90% confidence level. This problem involves statistics concepts such as confidence intervals, the Z-score, and the formula for margin of error. Suitable for Grades 11-12 and college students.

Math Problem Statement

n

130 , σ

3.5 , α

1 − 0.1

Solution

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