an IQ test is designed so that the mean is 100 and the standars deviation is 17 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 95% confidence that the sample mean is within 6 IQ points of the true mean. assume that o = 17 and determine the required sample size using technology. then determine if this is a reasonable sample size for a real world calculation. would it be reasonable to sample this number of nurses?

To determine the sample size necessary for estimating the mean IQ score of nurses with 95% confidence, we can use the following formula for sample size in estimating a population mean:

\[
n = \left( \frac{Z_{\alpha/2} \cdot \sigma}{E} \right)^2
\]

Where:
- \( Z_{\alpha/2} \) is the critical value corresponding to the desired confidence level (for 95% confidence, \( Z_{\alpha/2} = 1.96 \))
- \( \sigma = 17 \) is the population standard deviation
- \( E = 6 \) is the margin of error (how close we want the estimate to be to the true mean)
- \( n \) is the required sample size

### Step-by-Step Calculation

1. **Identify the known values:**
   \[
   Z_{\alpha/2} = 1.96, \quad \sigma = 17, \quad E = 6
   \]

2. **Substitute into the formula:**
   \[
   n = \left( \frac{1.96 \cdot 17}{6} \right)^2
   \]

Now, I will calculate the sample size.The required sample size is 31. 

### Reasonableness of this Sample Size
A sample size of 31 nurses seems reasonable for most real-world studies. It's a manageable number in terms of conducting surveys or assessments in healthcare settings, where such data collection could be feasible.

Do you need further details on the calculation, or have any questions?

Here are five related questions for expanding the topic:
1. How does increasing the confidence level affect the sample size?
2. What happens to the required sample size if we reduce the margin of error?
3. How does changing the standard deviation influence the sample size?
4. Why is a normal distribution assumption important for calculating the sample size?
5. What are some other methods to calculate confidence intervals for non-normal populations?

**Tip**: When designing a study, always check if the sample size is practical and ensure that it can yield reliable results without being too small or unnecessarily large.

An IQ test is designed so that the mean is 100 and the standard deviation is 17 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 95% confidence that the sample mean is within 6 IQ points of the true mean. Assume that σ = 17 and determine the required sample size using technology. Then determine if this is a reasonable sample size for a real world calculation. Would it be reasonable to sample this number of nurses?

Find the sample size needed to estimate the mean IQ score of nurses with 95% confidence. This problem involves calculating the required sample size based on a margin of error of 6 IQ points, a population standard deviation of 17, and a confidence level of 95%. Includes step-by-step calculations and analysis of real-world feasibility.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation