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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

The mean and standard deviation of the grades of a statistics course and an English course are \((\mu=80, \sigma=4.5)\) and \((\mu=85, \sigma=4.0),\) respectively. A student attends both the courses and scores 85 in statistics and 95 in English. Which grade is relatively better? Explain why.

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 \(p\).

Playing the lottery The state of Ohio has several statewide lottery options. One is the Pick 3 game in which you pick one of the 1000 three-digit numbers between 000 and 999\. The lottery selects a three-digit number at random. With a bet of \(\$ 1,\) you win \(\$ 500\) if your number is selected and nothing (\$0) otherwise. (Many states have a very similar type of lottery.) (Source: Background information from www.ohiolottery.com.) a. With a single \(\$ 1\) bet, what is the probability that you win \(\$ 500 ?\) b. Let \(X\) denote your winnings for a \(\$ 1\) bet, so \(x=\$ 0\) or \(x=\$ 500\). Construct the probability distribution for \(X\). c. Show that the mean of the distribution equals 0.50 , corresponding to an expected return of 50 cents for the dollar paid to play. Interpret the mean. d. In Ohio's Pick 4 lottery, you pick one of the 10,000 fourdigit numbers between 0000 and 9999 and (with a \(\$ 1\) bet) win \(\$ 5000\) if you get it correct. In terms of your expected winnings, with which game are you better off \(-\) playing Pick \(4,\) or playing Pick 3 ? Justify your answer.

The Mental Development Index (MDI) of the Bayley Scales of Infant Development is a standardized measure used in observing infants over time. It is approximately normal with a mean of 100 and a standard deviation of 16 a. What proportion of children has an MDI of (i) at least \(120 ?\) (ii) at least \(80 ?\) b. Find the MDI score that is the 99 th percentile. c. Find the MDI score such that only \(1 \%\) of the population has MDI below it.

Profit and the weather From past experience, a wheat farmer living in Manitoba, Canada, finds that his annual profit (in Canadian dollars) is 80,000 if the summer weather is typical, 50,000 if the weather is unusually dry, and 20,000 if there is a severe storm that destroys much of his crop. Weather bureau records indicate that the probability is 0.70 of typical weather, 0.20 of unusually dry weather, and 0.10 of a severe storm. Let \(X\) denote the farmer's profit next year. a. Construct a table with the probability distribution of \(X\) b. What is the probability that the profit is 50,000 or less? c. Find the mean of the probability distribution of \(X\). Interpret. d. Suppose the farmer buys insurance for 3000 that pays him 20,000 in the event of a severe storm that destroys much of the crop and pays nothing otherwise. Find the probability distribution of his profit. Find the mean and summarize the effect of buying this insurance.
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