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Chapter 6: Problem 18

Verify the empirical rule by using Table A, software, or a calculator to show that for a normal distribution, the probability (rounded to two decimal places) within a. 1 standard deviation of the mean equals 0.68 . b. 2 standard deviations of the mean equals 0.95 . c. 3 standard deviations of the mean is very close to 1.00 .

Short Answer

Expert verified
a. 0.68, b. 0.95, c. Approximately 1.00.

Step by step solution

01

Understanding the Empirical Rule

The Empirical Rule states that for a normal distribution: 68% of data falls within 1 standard deviation (蟽) of the mean (渭), 95% falls within 2蟽, and 99.7% falls within 3蟽.
02

Using the Standard Normal Table for 1 Standard Deviation

For 1蟽, we calculate the probability using the standard normal distribution. The z-scores for -1蟽 and 1蟽 are -1 and 1, respectively. The probability, P(-1 鈮� Z 鈮� 1), is calculated as: P(Z 鈮� 1) - P(Z 鈮� -1). Using the standard normal table, P(Z 鈮� 1) 鈮� 0.8413 and P(Z 鈮� -1) 鈮� 0.1587. So, P(-1 鈮� Z 鈮� 1) = 0.8413 - 0.1587 = 0.6826, which rounds to 0.68.
03

Calculating for 2 Standard Deviations

For 2蟽, find the probability from -2 to 2 using z-scores. P(-2 鈮� Z 鈮� 2) is determined by P(Z 鈮� 2) - P(Z 鈮� -2). The values from the standard normal table are P(Z 鈮� 2) 鈮� 0.9772 and P(Z 鈮� -2) 鈮� 0.0228. Therefore, P(-2 鈮� Z 鈮� 2) = 0.9772 - 0.0228 = 0.9544, which rounds to 0.95.
04

Checking for 3 Standard Deviations

For 3蟽, calculate the probability from -3 to 3. P(-3 鈮� Z 鈮� 3) is P(Z 鈮� 3) - P(Z 鈮� -3). From the standard normal table, P(Z 鈮� 3) 鈮� 0.9987 and P(Z 鈮� -3) 鈮� 0.0013. Thus, P(-3 鈮� Z 鈮� 3) = 0.9987 - 0.0013 = 0.9974, which rounds to approximately 1.00.

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Key Concepts

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

Understanding Normal Distribution
A normal distribution is a type of continuous probability distribution that is symmetric around its mean. It is also known as a Gaussian distribution or bell curve due to its characteristic bell shape.
The mean, median, and mode of a normal distribution are all equal, located at the highest point, or center of the bell.
The further away from the mean, the lower the probability of finding data points there.
  • The curve is described by its mean (渭) and standard deviation (蟽).
  • Its total area sums up to 1, representing whole probability (100%).
  • In a real-world context, many variables are approximately normally distributed, such as intelligence scores and heights of people.
Understanding the distribution helps in predicting probabilities and the likelihood of certain outcomes within a dataset.
Exploring Standard Deviation
Standard deviation is a measure of how spread out the numbers in a data set are. A smaller standard deviation means the data points are close to the mean, while a larger one indicates they are more spread out.
This concept is essential to comprehend the variability within the data.
  • Standard deviation is always a non-negative number.
  • It is denoted by \( \sigma \) for a population and \( s \) for a sample.
  • In a normal distribution, the standard deviation dictates the width and spread of the bell curve.
The Empirical Rule uses standard deviation as a key component to provide a quick estimate of the distribution of data within one, two, and three standard deviations from the mean.
Breaking Down Z-scores
Z-scores are a way of standardizing scores in a dataset by representing them in terms of standard deviations away from the mean. They are a valuable tool for understanding how far away a particular value is from the mean and for comparing scores from different distributions.
  • A Z-score is calculated by the formula: \( Z = \frac{(X - \mu)}{\sigma} \), where \( X \) is the value, \( \mu \) is the mean, and \( \sigma \) is the standard deviation.
  • Z-scores can be positive, negative, or zero, depending on whether \( X \) is above, below, or at the mean, respectively.
  • Using Z-scores, one can determine the probability of a score occurring within a normal distribution.
By comparing Z-scores to a standard normal distribution table, we can find a data point's position relative to the mean of the dataset, helping to apply the Empirical Rule effectively.

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Most popular questions from this chapter

Suppose your favorite coffee machine offers 12 ounce cups of coffee. The actual amount of coffee put in the cup by the machine varies according to a normal distribution, with mean equal to 13 ounces and standard deviation equal to 0.6 ounces. For each question below, sketch a graph, mark the mean, shade the area to which the answer refers and compute the percentage. a. What percentage of cups will be filled with less than 12 ounces? b. What percentage of cups will be filled with more than 12.5 ounces? c. What percentage of cups will have in between 12 and 13 ounces of coffee?

Ideal number of children Let \(X\) denote the response of a randomly selected person to the question, "What is the ideal number of children for a family to have?" The probability distribution of \(X\) in the United States is approximately as shown in the table, according to the gender of the person asked the question. $$ \begin{array}{lcc} \hline \begin{array}{l} \text { Probability Distribution of } \boldsymbol{X}=\text { Ideal Number } \\\ \text { of Children } \end{array} \\ \hline \boldsymbol{x} & \mathbf{P}(\boldsymbol{x}) \text { Females } & \mathbf{P}(\boldsymbol{x}) \text { Males } \\ \hline 0 & 0.01 & 0.02 \\ 1 & 0.03 & 0.03 \\ 2 & 0.55 & 0.60 \\ 3 & 0.31 & 0.28 \\ 4 & 0.11 & 0.08 \\ \hline \end{array} $$ a. Show that the means are similar, 2.50 for females and 2.39 for males. b. The standard deviation for the females is 0.770 and 0.758 for the males. Explain why a practical implication of the values for the standard deviations is that males hold slightly more consistent views than females about the ideal family size.

Binomial needs fixed \(n\) For the binomial distribution, the number of trials \(n\) is a fixed number. Let \(X\) denote the number of girls in a randomly selected family in Canada that has three children. Let \(Y\) denote the number of girls in a randomly selected family in Canada (that is, the number of children could be any number). A binomial distribution approximates well the probability distribution for one of \(X\) and \(Y\), but not for the other. a. Explain why. b. Identify the case for which the binomial applies and identify \(n\) and \(p\).

Most phones use lithium-ion (Li-ion) batteries. These batteries have a limited number of charge and discharge cycles, usually falling between 300 and \(500 .\) Beyond this lifespan, a battery gradually diminishes below \(50 \%\) of its original capacity. a. Suppose the distribution of the number of charge and discharge cycles was normal. What values for the mean and the standard deviation are most likely to meet the assumption of normality of this variable? b. Based on the mean and the standard deviation calculated in part a, find the 95 th percentile.

Selling at the right price An insurance company wants to examine the views of its clients about the prices of three car insurance plans launched last year. It conducts a survey with two sets of plans with different prices and finds that: \- If plan A is sold for \(\$ 150\), plan \(\mathrm{B}\) for \(\$ 250\), and plan \(\mathrm{C}\) for \(\$ 350,\) then \(45 \%\) of the customers would be interested in plan \(\mathrm{A}, 15 \%\) in plan \(\mathrm{B},\) and \(40 \%\) in plan \(\mathrm{C}\). \- If plans \(\mathrm{A}, \mathrm{B},\) and \(\mathrm{C}\) are sold for \(\$ 170, \$ 250\), and \(\$ 310\) respectively, then \(15 \%\) of the customers would be interested in plan \(\mathrm{A}, 40 \%\) in plan \(\mathrm{B},\) and \(45 \%\) in plan \(\mathrm{C}\). a. For the first pricing set, construct the probability distribution of \(X=\) selling price for the sale of a car insurance plan, find its mean, and interpret. b. For the second pricing set, construct the probability distribution of \(X,\) find its mean, and interpret. c. Which pricing set is more profitable to the company? Explain.
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