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Chapter 7: Problem 3

True or False: The normal curve is symmetric about its mean, \(\mu .\)

Short Answer

Expert verified
True

Step by step solution

01

Understanding the Normal Curve

The normal curve, also known as the Gaussian distribution or bell curve, is a continuous probability distribution characterized by its symmetrical bell shape.
02

Symmetry of the Curve

A curve is symmetric if one half of it is a mirror image of the other half. For the normal distribution, the peak of the curve, which is the mean ( 渭 ), divides the curve into two identical halves.
03

Implications of Symmetry

Since the normal distribution is symmetric about the mean, any value below the mean has a corresponding value above the mean with the same probability.
04

Conclusion

Given that the normal curve is symmetric about its mean, the statement 'The normal curve is symmetric about its mean, 渭 ' is true.

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Key Concepts

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

Gaussian distribution
The Gaussian distribution is another name for the normal distribution. It describes how values are distributed in many real-world situations. This distribution is continuous, meaning it includes every possible value within a certain range rather than just specific points.
Gaussian distribution is defined by two main parameters: the mean (渭) and the standard deviation (蟽). The mean is the center of the distribution, where most values cluster, while the standard deviation measures how spread out the values are around the mean.
Understanding this distribution helps in predicting probabilities and outcomes within a data set. It's widely used in fields like statistics, finance, and natural sciences due to its natural occurrence in varied phenomena.
bell curve
The bell curve is a name given to the shape of the normal distribution, which looks like a bell. This visual representation helps in understanding the distribution鈥檚 properties.
In a bell curve, most data points gather around the mean, creating a peak. As you move away from the mean, the frequency of data points gradually decreases on both sides, forming the downwards slopes of the bell.
Key characteristics:
  • Symmetry: The bell curve is symmetric around the mean. Each half is a mirror image of the other.
  • Unimodal: There is only one peak.
  • Asymptotic: The tails of the curve get closer to the horizontal axis but never touch it.
This symmetry is crucial as it implies consistent frequencies for values equally distant from the mean.
mean symmetry
Mean symmetry is a critical property of the normal distribution. It implies that the distribution is evenly balanced around the mean value.
If you were to fold the normal curve at the mean, each side would align perfectly with the other. This symmetry indicates that any deviation below the mean is mirrored by an equivalent deviation above the mean with equal probability.
Understanding mean symmetry can simplify the analysis of data. Since both sides are identical, you often only need to study one side to make inferences about the entire distribution. Recognizing this can help in predicting behaviors and outcomes based on the given data.
This property is the basis for many statistical methods and concepts, making it essential to grasp fully.

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Most popular questions from this chapter

The mean incubation time of fertilized chicken eggs kept at \(100.5^{\circ} \mathrm{F}\) in a still-air incubator is 21 days. Suppose that the incubation times are approximately normally distributed with a standard deviation of 1 day. Source: University of Illinois Extension (a) Draw a normal model that describes egg incubation times of fertilized chicken eggs. (b) Find and interpret the probability that a randomly selected fertilized chicken egg hatches in less than 20 days. (c) Find and interpret the probability that a randomly selected fertilized chicken egg takes over 22 days to hatch. (d) Find and interpret the probability that a randomly selected fertilized chicken egg hatches between 19 and 21 days. (e) Would it be unusual for an egg to hatch in less than 18 days? Why?

The lives of refrigerators are normally distributed with mean \(\mu=14\) years and standard deviation \(\sigma=2.5\) years Source: Based on information from Consumer Reports (a) Draw a normal curve with the parameters labeled. (b) Shade the region that represents the proportion of refrigerators that last for more than 17 years. (c) Suppose the area under the normal curve to the right of \(x=17\) is 0.1151 . Provide two interpretations of this result.

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 90th percentile

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 \(5.03 \mathrm{~mm} ?\)

Draw a normal curve and label the mean and inflection points. $$ \mu=50 \text { and } \sigma=5 $$
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