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Chapter 6: Problem 2

A normal distribution is informally described as a probability distribution that is "bell-shaped" when graphed. Draw a rough sketch of a curve having the bell shape that is characteristic of a normal distribution.

Short Answer

Expert verified
Draw a symmetric bell curve with its peak at the mean and descending symmetrically on either side.

Step by step solution

01

Understand the Characteristics of a Normal Distribution

A normal distribution shows data that are symmetrically distributed, forming a 'bell curve'. This means the highest point in the curve is the mean, median, and mode of the dataset.
02

Draw the Horizontal Axis

Draw a horizontal line to represent the x-axis. Label it to represent the range of values in your dataset.
03

Mark the Mean in the Center

Mark the center of your x-axis with the mean value. This is where the peak of the bell curve will be.
04

Draw the Bell Curve

Start at a low point on the left side of the mean. The curve should increase, peak at the mean, and then symmetrically decrease to a low point on the right side of the mean.
05

Label the Axes

Label the y-axis to represent the probability density and indicate that it increases upwards from the x-axis.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Bell Curve
A normal distribution is often referred to as a 'bell curve' because of its distinctive, bell-like shape when graphed. The curve starts low, rises smoothly to a peak, and then descends symmetrically.
The peak represents the highest probability or the most frequent occurrences of the dataset values. This shape is essential in understanding how data spreads.
Why is it called a bell curve? Because, like a bell, it starts at a low point, rises to a peak, and then falls again.
  • Starts low on both sides
  • Peaks at the center
  • Smooth and symmetrical descent
Mean, Median, and Mode
In a normal distribution, the mean, median, and mode are all located at the same point, right at the peak of the bell curve. This property is important because it indicates perfect symmetry in the data.
  • Mean: The average of all data points
  • Median: The middle value when data points are arranged in order
  • Mode: The most frequently occurring value
Since they all coincide in a normal distribution, it simplifies further analysis and interpretation of the data. This also helps in indicating that data is evenly distributed around this central point.
Symmetrical Distribution
One of the key characteristics of a normal distribution is its symmetrical nature. If you were to fold the distribution graph in half at its center (the mean), the two halves would match perfectly.
Symmetrical distribution means:
  • Equal spread on both sides of the mean
  • Same shape and size on the left and right of the mean
  • No skewness
This symmetry implies that values are equally likely to occur on either side of the mean, helping to predict outcomes and understand probabilities more effectively.
Probability Density
In the context of a normal distribution, the y-axis of the bell curve represents probability density. This shows how the probability is distributed across different values of the dataset.
Key points to understand probability density:
  • Higher points indicate greater probability
  • The total area under the curve is equal to 1
  • Helps in understanding the likelihood of different outcomes
Probability density functions (PDFs) are essential in statistics for representing the distribution of continuous variables. The area under the curve between any two points represents the probability that the value will fall within that range.

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Most popular questions from this chapter

Sampling with Replacement The Orangetown Medical Research Center randomly selects 100 births in the United States each day, and the proportion of boys is recorded for each sample. a. Do you think the births are randomly selected with replacement or without replacement? b. Give two reasons why statistical methods tend to be based on the assumption that sampling is conducted with replacement, instead of without replacement.

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When women were finally allowed to become pilots of fighter jets, engineers needed to redesign the ejection seats because they had been originally designed for men only. The ACES-II ejection seats were designed for men weighing between 140 lb and 211 lb. Weights of women are now normally distributed with a mean of 171 lb and a standard deviation of 46 lb (based on Data Set 1 "Body Data" in Appendix B). a. If I woman is randomly selected, find the probability that her weight is between 140 lb and 211 lb. b. If 25 different women are randomly selected, find the probability that their mean weight is between 140 lb and 211 lb. c. When redesigning the fighter jet ejection seats to better accommodate women, which probability is more relevant: the result from part (a) or the result from part (b)? Why?

Use the data in the table below for sitting adult males and females (based on anthropometric survey data from Gordon, Churchill, et al.). These data are used often in the design of different seats, including aircraft seats, train seats, theater seats, and classroom seats. (Hint: Draw a graph in each case.) $$\begin{array}{|l|l|l|l|} \hline & \text { Mean } & \text { St. Dev. } & \text { Distribution } \\ \hline \text { Males } & 23.5 \mathrm{in} . & 1.1 \mathrm{in} . & \text { Normal } \\ \hline \text { Females } & 22.7 \mathrm{in} . & 1.0 \mathrm{in} . & \text { Normal } \\ \hline \end{array}$$ Significance Instead of using 0.05 for identifying significant values, use the criteria that a value \(x\) is significantly high if \(P(x \text { or greater) } \leq 0.01\) and a value is significantly low if \(P(x \text { or less }) \leq 0.01 .\) Find the back-to-knee lengths for males, separating significant values from those that are not significant. Using these criteria, is a male back-to- knee length of 26 in. significantly high?
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