To solve this problem, we can apply the Central Limit Theorem (CLT) and the properties of the normal distribution.

Given Information:

• The population mean ($\mu$) = 500
• The population standard deviation ($\sigma$) = 100
• The sample size ($n$) = 20
• We are asked to find the probability that a random sample of 20 students received scores at or below 400.

Steps to solve:

1. Calculate the mean and standard deviation of the sampling distribution: Since we're working with a sample, we need to adjust the standard deviation to account for the sample size using the formula for the standard error of the mean:

   $\text{Standard Error (SE)} = \frac{\sigma}{\sqrt{n}}$ Where: $\sigma = 100, \quad n = 20$

   Substituting the values:

   $SE = \frac{100}{\sqrt{20}} \approx \frac{100}{4.472} \approx 22.36$

2. Standardize the score using the Z-score formula: The Z-score formula is:

   $Z = \frac{X - \mu}{SE}$ Where:

   • $X = 400$ (the score we're interested in)
   • $\mu = 500$ (population mean)
   • $SE \approx 22.36$ (standard error)

   Substituting the values:

   $Z = \frac{400 - 500}{22.36} = \frac{-100}{22.36} \approx -4.47$

3. Find the probability associated with this Z-score: A Z-score of -4.47 is extremely low, so we look up the corresponding probability in the standard normal distribution table or use a Z-table calculator.

   The probability of getting a Z-score of -4.47 or lower is essentially 0. Since it is so far in the lower tail of the distribution, the probability is practically negligible.

Conclusion:

The probability that a random sample of 20 students scored at or below 400 is extremely close to 0.

Let me know if you would like further clarification or details on any of these steps!

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5 Related Questions:

1. How does the Central Limit Theorem apply to sample means and sample sizes?
2. What is the impact of increasing the sample size on the standard error and the Z-score?
3. How do we interpret extreme Z-scores in terms of probability?
4. What would happen if the mean of the population were lower than 500? How would the Z-score change?
5. How can the probability be calculated for scores above a certain threshold instead of below?

Tip: The standard error of the mean decreases as the sample size increases, which makes it easier to detect significant deviations from the population mean in larger samples.