An IQ test is designed so that the mean is 100 and the standard deviation is 

1313 

for the population of normal adults. Find the sample size necessary to estimate the mean IQ score of nurses such that it can be said with 

9595​% 

confidence that the sample mean is within 

55 

IQ points of the true mean. Assume that 

sigmaσequals=1313 

and determine the required sample size using technology. Then determine if this is a reasonable sample size for a real world calculation.
The required sample size is 

2626. 

​(Round up to the nearest​ integer.)

Part 2

Would it be reasonable to sample this number of​ nurses?

NoNo. 

This number of IQ test scores is a fairly 

smallsmall 

number.

YesYes. 

This number of IQ test scores is a fairly 

smallsmall 

number.

NoNo. 

This number of IQ test scores is a fairly 

largelarge 

number.

YesYes. 

This number of IQ test scores is a fairly 

largelarge 

number.

To solve this problem, we need to find the sample size necessary to estimate the mean IQ score of nurses within a certain margin of error, using the provided confidence level and standard deviation. Let's break it down step by step.

### Part 1: Calculate the Required Sample Size

**Given:**
- Mean (\(\mu\)) = 100 (this is for context, not needed for sample size calculation)
- Standard Deviation (\(\sigma\)) = 13
- Margin of Error (E) = 5
- Confidence Level = 95%

#### Step 1: Find the Z-Score for a 95% Confidence Level
For a 95% confidence level, the Z-score is approximately 1.96.

#### Step 2: Use the Formula for Sample Size
The formula to calculate the sample size \(n\) is:
\[
n = \left(\frac{Z \times \sigma}{E}\right)^2
\]
Where:
- \(Z\) is the Z-score (1.96 for 95% confidence)
- \(\sigma\) is the standard deviation (13)
- \(E\) is the margin of error (5)

Substituting the values:
\[
n = \left(\frac{1.96 \times 13}{5}\right)^2
\]

#### Step 3: Calculate \(n\)
\[
n = \left(\frac{25.48}{5}\right)^2 = \left(5.096\right)^2 \approx 25.97
\]
Since the sample size must be an integer, we round up to the nearest whole number:
\[
n \approx 26
\]

### Part 2: Determine if the Sample Size is Reasonable

Given that the required sample size is 26, we now consider whether this is reasonable:

- **Yes**: This number of IQ test scores is a fairly **small** number.
  - 26 is a manageable and reasonable sample size for a study, especially in fields like psychology or nursing where getting access to 26 individuals is feasible.

**Answer:**
- **The required sample size is 26.**
- **Yes.** This number of IQ test scores is a fairly **small** number.

---

Do you want further explanation on any of these steps or more questions on related topics?

Here are some related questions you might find helpful:
1. How do confidence levels affect the required sample size?
2. What factors should be considered when determining a sample size for a study?
3. How do you calculate the margin of error in a confidence interval?
4. Why is it important to round up when calculating sample size?
5. How does the Z-score change with different confidence levels?

**Tip:** When estimating a population parameter, always consider the feasibility of collecting the required sample size in the context of the study's resources and timeframe.

Learn how to calculate the sample size required to estimate the mean IQ score of nurses with a 95% confidence level and within 5 IQ points of the true mean. This problem involves using statistics and confidence interval concepts, including Z-scores and standard deviation. Suitable for advanced college-level statistics.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation