population has a mean of 200 and a standard deviation of .50 Suppose a simple random sample of size  is selected and  is used to estimate . Use *z*-table.

**a.** What is the probability that the sample mean will be within plus or minus 6 of the population mean (to  decimals)?

To solve this problem, we need to calculate the probability that the sample mean will be within plus or minus 6 of the population mean. This means we want to find the probability that the sample mean, \(\bar{x}\), is between 194 and 206, given that the population mean is 200 and the population standard deviation is 0.50.

### Steps:

1. **Determine the standard error of the mean** (\(\text{SE}\)):
   The standard error (SE) is given by:
   \[
   \text{SE} = \frac{\sigma}{\sqrt{n}}
   \]
   where \(\sigma\) is the population standard deviation and \(n\) is the sample size (which was not provided in the problem; I'll leave it as \(n\) for now).

2. **Find the z-scores for 194 and 206**:
   The z-score is calculated as:
   \[
   z = \frac{x - \mu}{\text{SE}}
   \]
   where \(x\) is the value for which you are calculating the z-score, and \(\mu\) is the population mean.

   So, for \(x = 194\) and \(x = 206\), the z-scores are:
   \[
   z_{194} = \frac{194 - 200}{\text{SE}} = \frac{-6}{\text{SE}}
   \]
   \[
   z_{206} = \frac{206 - 200}{\text{SE}} = \frac{6}{\text{SE}}
   \]

3. **Use the z-table to find the probabilities**:
   Once we have the z-scores, we can use the z-table to find the probabilities corresponding to these z-scores.

If you can provide the sample size \(n\), I can continue with the calculations and give you the final probability.

Would you like to provide \(n\) or have more details on any of these steps? 

---

Here are 5 related questions to expand on the problem:
1. How does the sample size \(n\) affect the standard error and the overall probability?
2. How would the probability change if the standard deviation was higher or lower?
3. What does it mean if the z-score is positive or negative?
4. How do you interpret a z-score in the context of a probability distribution?
5. What is the significance of using a z-table in statistical calculations?

**Tip:** The larger the sample size, the smaller the standard error, which leads to a more accurate estimate of the population mean.

A population has a mean of 200 and a standard deviation of 0.50. Suppose a simple random sample is selected and the sample mean is used to estimate the population mean. What is the probability that the sample mean will be within plus or minus 6 of the population mean? Use the z-table.

Learn how to calculate the probability that a sample mean will be within plus or minus 6 of a population mean of 200, given a standard deviation of 0.50. This solution uses z-scores and the z-table, covering key concepts such as the standard error and normal distribution. Suitable for high school and undergraduate statistics students.

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