A worldwide organization of academics claims that the mean IQ score of its members is 112 , with a standard deviation of 15 . A randomly selected group of 35 members of this organization is tested, and the results reveal that the mean IQ score in this sample is 114.2.
If the organization's claim is correct, what is the probability of having a sample mean of 114.2 or less for a random sample of this size?
Carry your intermediate computations to at least four decimal places. Round your answer to at least three decimal places.

To solve this problem, we'll use the concept of the sampling distribution of the sample mean. The sample mean (\(\bar{X}\)) for a sample of size \(n\) from a population with mean \(\mu\) and standard deviation \(\sigma\) is normally distributed with the following parameters:

- Mean of the sample mean: \(\mu_{\bar{X}} = \mu\)
- Standard deviation of the sample mean (Standard Error): \(\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}}\)

### Step 1: Define the parameters
Given:
- Population mean (\(\mu\)) = 112
- Population standard deviation (\(\sigma\)) = 15
- Sample size (\(n\)) = 35
- Sample mean (\(\bar{X}\)) = 114.2

### Step 2: Calculate the standard error of the mean
\[
\sigma_{\bar{X}} = \frac{\sigma}{\sqrt{n}} = \frac{15}{\sqrt{35}}
\]

### Step 3: Calculate the Z-score
The Z-score is calculated using the formula:
\[
Z = \frac{\bar{X} - \mu_{\bar{X}}}{\sigma_{\bar{X}}}
\]

### Step 4: Find the probability
Finally, we will use the Z-score to find the probability of obtaining a sample mean of 114.2 or less.

Let's carry out these calculations.The probability of obtaining a sample mean of 114.2 or less, given that the organization's claim is correct, is approximately **0.807**.

This means there's an 80.7% chance that a randomly selected sample of 35 members would have a mean IQ score of 114.2 or less.

Do you have any questions or would you like further details?

Here are five related questions you might find interesting:

1. What is the probability of getting a sample mean greater than 114.2 in this scenario?
2. How would the probability change if the sample size were increased to 50?
3. What is the probability of obtaining a sample mean between 110 and 115?
4. How would the probability change if the population standard deviation were 10 instead of 15?
5. What Z-score corresponds to the top 5% of sample means?

**Tip:** Understanding the concept of the standard error is crucial, as it shows how sample size influences the variability of the sample mean.

Calculate the probability of obtaining a sample mean IQ score of 114.2 or less from a sample of 35 members, given a population mean IQ score of 112 and a standard deviation of 15. Understand the concept of sampling distribution, standard error of the mean, and normal distribution in statistical analysis.

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