We are working with a normal distribution here, where the mean (μ) is 21.5 and the standard deviation (σ) is 5.3. To solve these parts, we'll use the Z-score formula to convert the raw scores into Z-scores, and then we can find the corresponding probabilities using standard normal distribution tables or a calculator:

$Z = \frac{X - \mu}{\sigma}$

Part (a) - Probability that a score is less than 17:

1. Calculate the Z-score for $X = 17$:

$Z = \frac{17 - 21.5}{5.3} = \frac{-4.5}{5.3} \approx -0.8491$

2. Find the probability corresponding to this Z-score. Using the standard normal distribution table or a calculator, the probability for $Z = -0.8491$ is approximately:

$P(Z < -0.8491) \approx 0.1985$

Thus, the probability that a randomly selected student scores less than 17 is approximately 0.1985.

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Part (b) - Probability that a score is between 14.1 and 28.9:

1. Calculate the Z-score for 14.1:

$Z = \frac{14.1 - 21.5}{5.3} = \frac{-7.4}{5.3} \approx -1.3962$

2. Calculate the Z-score for 28.9:

$Z = \frac{28.9 - 21.5}{5.3} = \frac{7.4}{5.3} \approx 1.3962$

3. Find the probabilities for these Z-scores:

   • $P(Z < -1.3962) \approx 0.0816$
   • $P(Z < 1.3962) \approx 0.9184$

4. Subtract the two probabilities to find the probability that the score is between 14.1 and 28.9:

$P(14.1 < X < 28.9) = P(Z < 1.3962) - P(Z < -1.3962) = 0.9184 - 0.0816 = 0.8368$

Thus, the probability that a randomly selected student scores between 14.1 and 28.9 is approximately 0.8368.

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Would you like more details on this solution or further clarification?

Related Questions:

1. What is the probability of scoring above 25 on the test?
2. How would you find the probability for a score less than 10?
3. What Z-score corresponds to the top 10% of the distribution?
4. If a student scores 30, what is their percentile rank?
5. What is the probability of a score between 20 and 22?

Tip:

For normal distributions, knowing how to use Z-scores is crucial for finding probabilities and percentiles, especially when data doesn't match typical values from tables.