If a score is selected at random from the distribution above, what is the probability that it will be between 30 and 49

To answer your question about the probability that a score selected at random will be between 30 and 49, we need more information about the type of distribution you are referring to (e.g., normal distribution, uniform distribution, etc.), and some parameters like the mean, standard deviation, or frequency of the scores.

For now, I'll explain how to calculate this for two common cases:

### 1. **Normal Distribution**
If the distribution is normal (bell-shaped), the probability of a score falling between 30 and 49 can be calculated by using the **Z-scores** or a cumulative distribution function (CDF). The steps are:
- Standardize the scores (convert them to Z-scores) using:
  \[
  Z = \frac{X - \mu}{\sigma}
  \]
  where:
  - \(X\) is the score,
  - \(\mu\) is the mean,
  - \(\sigma\) is the standard deviation.
- Use the Z-tables or CDF to find the cumulative probability for 30 and 49.
- Subtract the cumulative probability at 30 from the cumulative probability at 49.

### 2. **Uniform Distribution**
If the distribution is uniform (all scores are equally likely within a given range), the probability of a score being between 30 and 49 is simply the fraction of the total range that falls within this interval:
\[
P(30 \leq X \leq 49) = \frac{\text{Interval Length}}{\text{Total Range of Scores}}
\]
For example, if the total range is from 0 to 100, the probability would be:
\[
P(30 \leq X \leq 49) = \frac{49 - 30}{100 - 0} = \frac{19}{100} = 0.19 \text{ or } 19\%
\]

Could you provide more details on the type of distribution or any specific parameters?

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Would you like further explanation on any of these? Here are 5 related questions to deepen your understanding:

1. How do you find the Z-score for a value in a normal distribution?
2. What is a cumulative distribution function (CDF) and how is it used in probability?
3. How do you calculate probabilities for a discrete probability distribution?
4. What are the differences between discrete and continuous distributions?
5. How do you calculate probabilities in a binomial distribution?

**Tip:** In a normal distribution, most values cluster around the mean, and the probability of extreme values (far from the mean) decreases quickly.

If a score is selected at random from the distribution above, what is the probability that it will be between 30 and 49?

Learn how to calculate the probability of a score between 30 and 49 in both normal and uniform distributions. This guide covers key concepts like Z-scores, CDF, and uniform probability, suitable for high school students in Grades 10-12.

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