### Understanding Normal DistributionWhen dealing with normal distribution, certain key parameters define the shape and characteristics of the distribution. The mean (average) provides the central point, and the standard deviation measures how spread out the values are around the mean. Using these parameters, we can determine the percentage of the area under the curve for specific ranges of values.#### Example Problem: Normal Distribution AnalysisGiven the problem:**For a normal distribution where the mean is 50 and the standard deviation is 10, what percentage of the area is:**a. above a score of 47? b. below a score of 53? c. between the scores of 40 and 47? d. between the scores of 35 and 65? e. above a score of 72? f. below a score of 31 and above a score of 69? g. between the scores of 55 and 62? h. between the scores of 32 and 47? **Let's break these down:**1. **Above a score of 47** - We determine the Z-score and find the percentage of area to the right of this value.2. **Below a score of 53** - Z-score calculation gives us the percentage of area to the left of this value.3. **Between the scores of 40 and 47** - Calculate Z-scores for both values and find the area between these two points.4. **Between the scores of 35 and 65** - Similar process, calculate Z-scores and find the area between these scores.5. **Above a score of 72** - Determine the Z-score for 72 and find the area to the right.6. **Below a score of 31 and above a score of 69** - Calculate and sum up the areas for Z-scores of 31 (to the left) and 69 (to the right).7. **Between the scores of 55 and 62** - Use Z-scores to find the area between these two scores.8. **Between the scores of 32 and 47** - Calculate Z-scores and determine the area between these ranges.### Detailed Step-by-Step SolutionsTo solve these problems, follow these detailed steps:1. Calculate the Z-score: \[ Z = \frac{X - \mu}{\sigma

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5.4. For a normal distribution where the mean is 50 and the standard deviation is 10, what percentage of the area is a. above a score of 47? b. below a score of 53? c. between the scores of 40 and 47?

5.4. For a normal distribution where the mean is 50 and the standard deviation is 10, what percentage of the area is a. above a score of 47? b. below a score of 53? c. between the scores of 40 and 47?

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Estimation

A new framework of data analysis where the new statistical methods are used for interpreting the results and analyzing the data is known as estimation in statistics.

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### Understanding Normal Distribution

When dealing with normal distribution, certain key parameters define the shape and characteristics of the distribution. The mean (average) provides the central point, and the standard deviation measures how spread out the values are around the mean. Using these parameters, we can determine the percentage of the area under the curve for specific ranges of values.

#### Example Problem: Normal Distribution Analysis

Given the problem:
**For a normal distribution where the mean is 50 and the standard deviation is 10, what percentage of the area is:**

a. above a score of 47?
b. below a score of 53?
c. between the scores of 40 and 47?
d. between the scores of 35 and 65?
e. above a score of 72?
f. below a score of 31 and above a score of 69?
g. between the scores of 55 and 62?
h. between the scores of 32 and 47?

**Let's break these down:**

1. **Above a score of 47**
- We determine the Z-score and find the percentage of area to the right of this value.

2. **Below a score of 53**
- Z-score calculation gives us the percentage of area to the left of this value.

3. **Between the scores of 40 and 47**
- Calculate Z-scores for both values and find the area between these two points.

4. **Between the scores of 35 and 65**
- Similar process, calculate Z-scores and find the area between these scores.

5. **Above a score of 72**
- Determine the Z-score for 72 and find the area to the right.

6. **Below a score of 31 and above a score of 69**
- Calculate and sum up the areas for Z-scores of 31 (to the left) and 69 (to the right).

7. **Between the scores of 55 and 62**
- Use Z-scores to find the area between these two scores.

8. **Between the scores of 32 and 47**
- Calculate Z-scores and determine the area between these ranges.

### Detailed Step-by-Step Solutions

To solve these problems, follow these detailed steps:

1. Calculate the Z-score:
\[ Z = \frac{X - \mu}{\sigma

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