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Chapter 7: Problem 29

Draw a normal curve and label the mean and inflection points. $$ \mu=30 \text { and } \sigma=10 $$

Short Answer

Expert verified
The mean is at 30, and the inflection points are at 20 and 40.

Step by step solution

01

- Draw the Horizontal Axis

Start by drawing a horizontal axis on your graph. This axis will represent the variable for which we are plotting the normal distribution.
02

- Plot the Mean

Identify the mean of the distribution, denoted as \( \mu = 30 \). Mark this point on the horizontal axis.
03

- Draw the Bell Curve

Sketch a symmetric bell-shaped curve centered on the mean (\( \mu = 30 \)). Ensure the curve is smooth and approaches the horizontal axis asymptotically.
04

- Identify and Mark the Inflection Points

The inflection points occur at \( \mu - \sigma \) and \( \mu + \sigma \). Calculate these points: \( 30 - 10 = 20 \) and \( 30 + 10 = 40 \). Mark these points on the horizontal axis.
05

- Label the Graph

Label the mean (\( \mu = 30 \)) and the inflection points (\(20\) and \(40\)) clearly on the graph.

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Key Concepts

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

mean
The mean is the average value of a data set. In a normal distribution, it is represented by the symbol \(\mu\). The mean is the center point of a bell curve.
Here鈥檚 how you can find and plot the mean on a graph:
  • Start with a horizontal axis.
  • Identify the mean value; in this exercise, it is given as \(\mu = 30\).
  • Mark this point on the horizontal axis. This will be the center of your bell curve.
By plotting the mean, you provide a reference point that helps in identifying the shape and center of the normal distribution.
inflection points
Inflection points are where the curve changes concavity. For a normal distribution, these points are crucial because they indicate the spread of the data.
To locate inflection points:
  • Identify the mean \(\mu\) and standard deviation \(\sigma\). In this exercise, \(\mu = 30\) and \(\sigma = 10\).
  • Calculate the positions: \(\mu - \sigma = 20\) and \(\mu + \sigma = 40\).
  • Mark these points on the horizontal axis.
These points divide the bell curve into sections where the curve transitions from being concave upwards to concave downwards and vice versa. This visual change helps you understand the distribution's variability.
standard deviation
Standard deviation is a measure of spread in your dataset. Represented by \(\sigma\), it indicates how much individual data points deviate from the mean.
A few key aspects to understand about standard deviation in a normal distribution:
  • It determines the width of the bell curve.
  • The smaller the standard deviation, the steeper and narrower the curve.
  • Larger standard deviations result in flatter and wider curves.
In this exercise, the standard deviation is \(10\). It helps in finding the inflection points, which are crucial for understanding the spread and shape of the distribution.
bell curve
A bell curve, or normal distribution curve, is a graphical representation of data that shows a symmetric, bell-shaped appearance. Here are its key features:
  • The mean (\(\mu\)) is at the center of the curve.
  • Inflection points are located one standard deviation (\(\sigma\)) away from the mean.
  • The curve approaches, but never touches the horizontal axis.
In this exercise, the bell curve is centered at the mean, which is 30. The curve is smoothly drawn so that it has two equal halves about this central point. This distribution is used in statistics to represent real-world data that clusters around an average value. Understanding the bell curve allows you to grasp concepts like probability and data variability easily.

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Most popular questions from this chapter

The reading speed of sixth-grade students is approximately normal, with a mean speed of 125 words per minute and a standard deviation of 24 words per minute. (a) Draw a normal model that describes the reading speed of sixth-grade students. (b) Find and interpret the probability that a randomly selected sixth-grade student reads less than 100 words per minute. (c) Find and interpret the probability that a randomly selected sixth-grade student reads more than 140 words per minute. (d) Find and interpret the probability that a randomly selected sixth-grade student reads between 110 and 130 words per minute. 0.3189 (e) Would it be unusual for a sixth grader to read more than 200 words per minute? Why?

The reaction time \(X\) (in minutes) of a certain chemical process follows a uniform probability distribution with \(5 \leq X \leq 10 .\) (a) Draw the graph of the density curve. (b) What is the probability that the reaction time is between 6 and 8 minutes? (c) What is the probability that the reaction time is between 5 and 8 minutes? (d) What is the probability that the reaction time is less than 6 minutes?

Assume that the random variable \(X\) is normally distributed, with mean \(\mu=50\) and standard deviation \(\sigma=7\). Find each indicated percentile for \(X\) The 9 th percentile

The number of chocolate chips in an 18 -ounce bag of Chips Ahoy! chocolate chip cookies is approximately normally distributed, with a mean of 1262 chips and a standard deviation of 118 chips, according to a study by cadets of the U.S. Air Force Academy. Source: Brad Warner and Jim Rutledge, Chance \(12(1): 10-14,1999\) (a) Determine the 30 th percentile for the number of chocolate chips in an 18 -ounce bag of Chips Ahoy! cookies. (b) Determine the number of chocolate chips in a bag of Chips Ahoy! that make up the middle \(99 \%\) of bags. (c) What is the interquartile range of the number of chips in Chips Ahoy! cookies?

Ball bearings are manufactured with a mean diameter of 5 millimeters \((\mathrm{mm})\). Because of variability in the manufacturing process, the diameters of the ball bearings are approximately normally distributed, with a standard deviation of \(0.02 \mathrm{~mm}\) (a) What proportion of ball bearings has a diameter more than \(5.03 \mathrm{~mm} ?\) (b) Any ball bearings that have a diameter less than \(4.95 \mathrm{~mm}\) or greater than \(5.05 \mathrm{~mm}\) are discarded. What proportion of ball bearings will be discarded? (c) Using the results of part (b), if 30,000 ball bearings are manufactured in a day, how many should the plant manager expect to discard? (d) If an order comes in for 50,000 ball bearings, how many bearings should the plant manager manufacture if the order states that all ball bearings must be between \(4.97 \mathrm{~mm}\) and \(5.03 \mathrm{~mm} ?\)
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