In an election, uppose that 50% of voters support the incumbent candidate. If we poll 115 of these voters at random, the probability distribution for the proportion of the polled voters that support the incumbent candidate can be modeled by the normal distribution pictured below.
In an election, uppose that 50% of voters support the incumbent candidate. If we poll 115 of these voters at random, the probability distribution for the proportion of the polled voters that support the incumbent candidate can be modeled by the normal distribution pictured below.
In an election, uppose that 50% of voters support the incumbent candidate. If we poll 115 of these voters at random, the probability distribution for the proportion of the polled voters that support the incumbent candidate can be modeled by the normal distribution pictured below.
In an election, uppose that 50% of voters support the incumbent candidate. If we poll 115 of these voters at random, the probability distribution for the proportion of the polled voters that support the incumbent candidate can be modeled by the normal distribution pictured below. Complete the boxes accurate to two decimal places.
Transcribed Image Text:This graph represents a bell curve, also known as a normal distribution. The bell curve is symmetrical and depicts the distribution of data points where most values cluster around a central peak and probabilities for values taper off equally on both sides.
### Key Features of the Bell Curve:
- **Symmetrical Shape**: The curve is symmetrical around the mean. This means that if you fold the graph in half at the center, the two sides would match perfectly.
- **Central Peak**: The highest point on the curve represents the mean, median, and mode of the data set.
- **Standard Deviations**: The horizontal axis marks the standard deviations from the mean. The sections highlighted in red rectangles are spaces where standard deviation values would normally be indicated, showing how data is spread out.
### Understanding the Distribution:
- **68-95-99.7 Rule**: This rule applies to the normal distribution and states that approximately 68% of the data falls within one standard deviation, 95% within two, and 99.7% within three standard deviations from the mean.
- **Use in Statistics**: The bell curve is crucial in fields such as statistics, research, and business, where it helps to predict outcomes and understand variability within data sets.
This visualization provides an essential foundation for understanding many statistical analyses and hypothesis testing scenarios.
Features Features Normal distribution is characterized by two parameters, mean (µ) and standard deviation (σ). When graphed, the mean represents the center of the bell curve and the graph is perfectly symmetric about the center. The mean, median, and mode are all equal for a normal distribution. The standard deviation measures the data's spread from the center. The higher the standard deviation, the more the data is spread out and the flatter the bell curve looks. Variance is another commonly used measure of the spread of the distribution and is equal to the square of the standard deviation.
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