Normal Curve Chapter 5 Directions Write The Answer To Each Q

Normal Curvechapter 5directionswrite The Answer To Each Question In T

Directions: Write the answer to each question in the space below. 1. If you go out 1 standard deviation unit on both sides of the mean in a normal distribution, what percentage of the cases will be captured? 2. If you go out 3 standard deviation units on both sides of the mean in a normal distribution, what percentage of the cases will be captured? 3. For X bar = 100.00 and S = 10.00, approximately what percentage of the cases lie between 80.00 and 120.00? 4. For X bar = 55.00 and S = 7.00, between what two values do approximately 95% of the cases lie in a normal distribution? 5. For X bar = 90.00 and S = 15.00, between what two values do 99.7% of the cases lie in a normal distribution? 6. What is the multiplier or the approximate 95% rule? 7. What is the multiplier for the 99.7% rule? 8. For all practice purposes, the normal curve has how many standard deviation units?

Paper For Above instruction

The normal distribution, often called the bell curve, is fundamental in statistics due to its widespread applications and properties. It describes how data points are distributed around a mean, with most values clustering near the mean and fewer occurring as you move away. Understanding the percentages of data within certain deviations from the mean is crucial for interpretation, prediction, and decision-making in various fields such as psychology, finance, and natural sciences.

1. Using the empirical rule, approximately 68% of the data falls within one standard deviation of the mean. When we extend one standard deviation on either side of the mean, about 68% of the cases are captured in a normal distribution. This rule is intuitive because it demonstrates that most data points lie close to the average, with probabilities decreasing as we move further away.

2. Extending to three standard deviations encompasses about 99.7% of the data points. Thus, almost all cases are within three standard deviations from the mean in a normal distribution. This fact underpins why the empirical rule is sometimes called the 68-95-99.7 rule, where 99.7% of the data falls within ±3 standard deviations.

3. Calculating the percentage of cases between 80.00 and 120.00 when X̄ = 100.00 and S = 10.00. The interval from 80 to 120 corresponds to the mean ± 2 standard deviations (100 ± 2×10). Since approximately 95% of data in a normal distribution lies within two standard deviations of the mean, about 95% of the cases are between these values.

4. When X̄ = 55.00 and S = 7.00, approximately 95% of cases lie between 55.00 ± 2×7.00, which is from 41.00 to 69.00. This interval captures about 95% of the data, as per the empirical rule.

5. For X̄ = 90.00 and S = 15.00, 99.7% of the cases lie within ± 3 standard deviations, i.e., between 90 - 3×15 = 45 and 90 + 3×15 = 135. This range captures almost all data points in the distribution.

The approximate multipliers for these ranges are:

- 95% rule multiplier: 2 (since about 95% of data falls within two standard deviations of the mean)

- 99.7% rule multiplier: 3 (since about 99.7% of data falls within three standard deviations)

The normal curve in practice spans approximately 6 standard deviations. To summarize, the standard approaches and empirical rules enable quick estimations of data spread, which are essential for data analysis and interpretation. These principles allow statisticians and researchers to understand the likelihood of data points falling within specific ranges without needing to analyze every individual data point.

References

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