https://file.aiquickdraw.com/mathbot/file/1727742447669bmbgf9dg.jpg

We need to calculate the probability that the sample mean SAT score is less than 484, given that:

- Population mean (μ) = 495
- Population standard deviation (σ) = 111
- Sample size (n) = 61
- We are asked for the probability \( P(\bar{X} < 484) \).

### Step-by-step solution:
1. **Calculate the standard error (SE) of the sample mean**:
   \[
   SE = \frac{\sigma}{\sqrt{n}} = \frac{111}{\sqrt{61}} \approx 14.217
   \]

2. **Compute the z-score** for the sample mean:
   \[
   Z = \frac{\bar{X} - \mu}{SE} = \frac{484 - 495}{14.217} \approx -0.774
   \]

3. **Find the probability** associated with the z-score of -0.774 using the standard normal distribution table or a calculator.

   A z-score of -0.774 corresponds to a cumulative probability of approximately **0.2194**.

Thus, the probability that the sample mean score is less than 484 is approximately **0.2194**.

---

Would you like more details on any of these steps or have any other questions?

### Related questions:
1. What is the z-score formula, and how does it apply in different contexts?
2. How does increasing the sample size affect the standard error?
3. What is the relationship between standard deviation and standard error?
4. How do we use a z-table to find probabilities for different z-scores?
5. What is the probability that the sample mean score is greater than 484?

### Tip:
When using a sample to estimate a population parameter, remember that increasing the sample size reduces the variability of the sample mean, making the estimate more reliable.

What is the probability that the sample mean score is less than 484? Given that the mean mathematics SAT score was 495, the standard deviation was 111, and a sample of 61 scores is chosen.

Find the probability that the sample mean SAT score is less than 484, using inferential statistics concepts such as the standard error, z-score, and normal distribution. The population mean SAT score is 495, and the standard deviation is 111 with a sample size of 61. This detailed step-by-step guide is suitable for statistics students.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation