A. upposc statewide Standardizd test are normally distributed of G5 and a standard deviation of 4. Estimate the percentage of scores that were that the scores on a, Wth a mean 35
Continuous Probability Distributions
Probability distributions are of two types, which are continuous probability distributions and discrete probability distributions. A continuous probability distribution contains an infinite number of values. For example, if time is infinite: you could count from 0 to a trillion seconds, billion seconds, so on indefinitely. A discrete probability distribution consists of only a countable set of possible values.
Normal Distribution
Suppose we had to design a bathroom weighing scale, how would we decide what should be the range of the weighing machine? Would we take the highest recorded human weight in history and use that as the upper limit for our weighing scale? This may not be a great idea as the sensitivity of the scale would get reduced if the range is too large. At the same time, if we keep the upper limit too low, it may not be usable for a large percentage of the population!
![### Understanding Normal Distribution in Standardized Test Scores
Suppose that the scores on a statewide standardized test are normally distributed with a mean of 65 and a standard deviation of 4. Estimate the percentage of scores that were:
A. Above 73.
- **[ ] %**
B. Below 53.
- **[ ] %**
C. Between 61 and 73.
- **[ ] %**
When interpreting normal distributions, consider how the percentages align with the Z-scores and the empirical rule (68-95-99.7 rule) to help estimate high-level percentages within these ranges.
**Explanation of the approach:**
1. **Calculate Z-scores:** Convert raw scores to Z-scores to find the relative position on a standard normal distribution.
2. **Use Z-tables or normal distribution calculators:** Find the corresponding percentages for the Z-scores.
3. **Apply the Empirical Rule:** Use this rule to quickly estimate:
- 68% of the data falls within one standard deviation (σ) from the mean.
- 95% within two standard deviations.
- 99.7% within three standard deviations.
**Example Estimations:**
- For scores above 73:
- Z = (73 - 65) / 4 = 2
- Look up Z = 2 in the Z-table to find the area to the left, then subtract from 100% for the area to the right.
- For scores below 53:
- Z = (53 - 65) / 4 = -3
- Look up Z = -3 in the Z-table.
- For scores between 61 and 73:
- Calculate Z for 61 and 73, then find the area between these Z-scores.
For complete understanding and precise calculations, refer to Z-tables or technology-enabled normal distribution tools.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Ff5ede190-124d-4580-a2ef-b75915abe6b9%2Ff3e8d97d-8339-48e1-b37e-a1db2ce35be2%2F9my4w2.jpeg&w=3840&q=75)
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