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Chapter 7: Problem 1

Describe the procedure for finding the area under any normal curve.

Short Answer

Expert verified
Convert to Z-scores, use the Z-score table, and subtract areas under the curve for the desired range.

Step by step solution

01

Understand the Normal Distribution

Before finding the area under a normal curve, understand that a normal distribution is a bell-shaped curve that is symmetric about the mean. The total area under the curve is equal to 1.
02

Standardize the Variable

Convert the variable to a standard normal variable (Z) using the formula: \[ Z = \frac{X - \mu}{\sigma} \] Here, \(X\) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation of the distribution.
03

Use Z-Scores Table

Find the corresponding Z-score using standard normal distribution tables. These tables provide the area to the left of a given Z-score under the standard normal curve.
04

Calculate the Required Area

To find the area under the curve for a specific range of values, determine the Z-scores for the range endpoints and use the Z-score table to find the corresponding areas. Subtract the smaller area from the larger area to get the required area.

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Key Concepts

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

normal distribution
Normal distribution is a fundamental concept in statistics. It represents a continuous probability distribution that is symmetrical around its mean, giving it a bell-shaped curve. The normal distribution is crucial because it describes many natural phenomena, such as heights, test scores, and measurement errors.

Its key features include:
  • Symmetry: The curve is perfectly symmetric about the mean.
  • Single Peak: The highest point on the curve is at the mean, which is also the median and mode.
  • Asymptotic Nature: The tails of the curve approach but never touch the horizontal axis.
Understanding the normal distribution helps in various applications, like calculating probabilities and finding the area under the curve, which we'll explore further.
Z-scores
A Z-score, also known as a standard score, is a measurement that describes a value's position in relation to the mean of a group of values. It is expressed in terms of standard deviations from the mean. The formula to calculate a Z-score is:
\[ Z = \frac{X - \mu}{\sigma} \]
Here:
  • X: The value being standardized.
  • \( \mu \): The mean of the distribution.
  • \( \sigma \): The standard deviation of the distribution.
Z-scores are crucial in finding probabilities and areas under the normal curve because they standardize different distributions to a common scale. By converting a value to a Z-score, you can easily find corresponding probabilities using the standard normal distribution table.
standard deviation
Standard deviation (\( \sigma \)) is a measure of the amount of variation or dispersion in a set of values. It tells us how spread out the values are around the mean (\( \mu \)). A low standard deviation indicates that the values are close to the mean, while a high standard deviation means that the values are spread out over a wider range.

It is calculated using the formula:
\[ \sigma = \sqrt{ \frac{\sum (X_i - \mu)^2}{N} } \]
where:
  • \( X_i \): Each individual value.
  • \( \mu \): The mean of the values.
  • N: The number of values.
Knowing the standard deviation is important because it helps in understanding the distribution's spread and in converting values to Z-scores, which we use to find areas under the normal curve.
mean
The mean (\( \mu \)) is one of the most commonly used statistical measures, representing the average of a set of values. It is obtained by summing all the individual values and then dividing by the number of values. The formula for the mean is:
\[ \mu = \frac{\sum X_i}{N} \]
where:
  • \( \sum X_i \): The sum of all individual values.
  • N: The number of values.
The mean serves as the central point for the normal distribution curve, and it is crucial for converting values to Z-scores. Understanding the mean is essential for assessing the data's central tendency and for further calculations involving Z-scores and standard deviations.

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Most popular questions from this chapter

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 standard deviation 118 chips according to a study by cadets of the U.S. Air Force Academy. (Source: Brad Warner and Jim Rutledge, Chance, Vol. 12, No. \(1,1999,\) pp. \(10-14 .\) ) (a) What is the probability that a randomly selected 18 ounce bag of Chips Ahoy! cookies contains between 1000 and 1400 chocolate chips? (b) What is the probability that a randomly selected 18 ounce bag of Chips Ahoy! cookies contains fewer than 1000 chocolate chips? (c) What proportion of 18 -ounce bags of Chip Ahoy! cookies contains more than 1200 chocolate chips? (d) What proportion of 18 -ounce bags of Chip Ahoy! cookies contains fewer than 1125 chocolate chips? (e) What is the percentile rank of an 18 -ounce bag of Chip Ahoy! cookies that contains 1475 chocolate chips? (f) What is the percentile rank of an 18-ounce bag of Chip Ahoy! cookies that contains 1050 chocolate chips?

Assume the random variable \(X\) is normally distributed with mean \(\mu=50\) and standard deviation \(\sigma=7 .\) Compute the following probabilities. Be sure to draw a normal curve with the area corresponding to the probability shaded. $$P(40 \leq X \leq 49)$$

In clinical trials of a medication whose purpose is to reduce the pain associated with migraine headaches, \(2 \%\) of the patients in the study experienced weight gain as a side effect. Suppose a random sample of 600 users of this medication is obtained. Use the normal approximation to the binomial to (a) approximate the probability that exactly 20 will experience weight gain as a side effect. (b) approximate the probability that 20 or fewer will experience weight gain as a side effect. (c) approximate the probability that 22 or more patients will experience weight gain as a side effect. (d) approximate the probability that between 20 and 30 patients, inclusive, will experience weight gain as a side effect.

True or False: Suppose \(X\) is a binomial random variable. To approximate \(P(3 \leq X<7)\) using the normal probability distribution, we compute \(P(3.5 \leq X<7.5)\).

List the conditions required for a binomial experiment.
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