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Chapter 4: Problem 45

A sample of concrete specimens of a certain type is selected, and the compressive strength of each specimen is determined. The mean and standard deviation are calculated as \(\bar{x}=3000\) and \(s=500\), and the sample histogram is found to be well approximated by a normal curve. a. Approximately what percentage of the sample observations are between 2500 and 3500 ? b. Approximately what percentage of sample observations are outside the interval from 2000 to 4000 ? c. What can be said about the approximate percentage of observations between 2000 and 2500 ? d. Why would you not use Chebyshev's Rule to answer the questions posed in Parts (a)-(c)?

Short Answer

Expert verified
a. Approximately 68% of the sample observations are between 2500 and 3500. b. Approximately 5% of the sample observations lie outside the interval from 2000 to 4000. c. Approximately 20% of the observations fall between 2000 and 2500. d. Chebyshev's Rule is not applicable here because the distribution form is already known to be normal.

Step by step solution

01

Find the Percentage of Observations Between 2500 and 3500

The value of 2500 is one standard deviation below the mean and 3500 one standard deviation above the mean. For a normal distribution, approximately 68% of observations fall within one standard deviation of the mean.
02

Determine the Percentage of Observations Outside the Interval from 2000 to 4000

The value of 2000 is two standard deviations below the mean, and 4000 is two standard deviations above the mean. In a normal distribution, about 95% of observations are within two standard deviations of the mean. Thus, about 100% - 95% = 5% of observations are outside this range.
03

Establish the Percentage of Observations Between 2000 and 2500

The value of 2500 is one standard deviation below the mean while 2000 is two standard deviations below the mean. In a standard normal distribution, about 34% of observations fall between the mean and one standard deviation below the mean, and about 14% fall between one and two standard deviations below the mean. Hence, the approximate percentage of observations between 2000 and 2500 is 34% - 14% = 20%.
04

Discuss the Inappropriateness of Chebyshev's Rule

Chebyshev's Rule is typically used when a specific distribution form is not known. Here, it has been explicitly stated that the histogram is well approximated by a normal curve. Hence, the properties of a normal distribution, which are more precise, were used.

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Key Concepts

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

Normal Distribution
The normal distribution is a foundational concept in statistics, often referred to as a bell curve due to its shape. It describes how data points are spread over a range of values, with most observations falling close to the mean.
This type of distribution is symmetric around its mean, conveying that:
  • The mean, median, and mode of a normal distribution are equal.
  • The curve is bell-shaped and extends infinitely in both directions.
Understanding normal distribution helps us make predictions about the data and assess probabilities for different outcomes.
Standard Deviation
Standard deviation is a measure that quantifies the amount of variation or dispersion in a set of data values. A low standard deviation indicates that data points are close to the mean, while a high standard deviation indicates widespread data points.
It can be calculated using the formula: \[s = \sqrt{\frac{1}{n-1} \sum_{i=1}^{n} (x_i - \bar{x})^2}\] Where:
  • \(s\) is the standard deviation.
  • \(n\) is the number of observations.
  • \(x_i\) is each individual data point.
In the context of a normal distribution, standard deviation defines the spread of the data along the bell curve.
Chebyshev's Rule
Chebyshev鈥檚 Rule provides a conservative estimate for the proportion of data within a certain number of standard deviations from the mean, applicable to any dataset, regardless of distribution.
The rule states that at least \[\left(1 - \frac{1}{k^2}\right)\] of data values lie within \(k\) standard deviations of the mean where \(k > 1\).
This rule isn't used with normal distributions when more precise estimates are available, as seen here.
Since the histogram resembles a normal distribution, using properties specific to normal distributions yields much more precise predictions.
Normal Curve
The normal curve is the graphical representation of a normal distribution, characterized by its bell shape. The "peak" of the curve represents the mean of the dataset.
Some key properties include:
  • The sides of the curve asymptotically approach the horizontal axis.
  • It's symmetric about the mean, showing equal spread of data on either side.
  • The area under the curve corresponds to probability and total area equals 1.
Understanding the normal curve is crucial for interpreting statistical data effectively, especially when predicting standard deviation ranges.
Percentages in Standard Deviation Ranges
In a normal distribution, specific percentages of observations fall within certain standard deviation ranges from the mean. These benchmarks include:
  • About 68% within 卤1 standard deviation.
  • Approximately 95% within 卤2 standard deviations.
  • Almost 99.7% within 卤3 standard deviations.
These percentages help determine the probability of observed data values, making them essential for statistical analysis.
Knowing these percentages allows us to answer questions about data distribution thoroughly, as demonstrated in the sample exercise where the percentages were used to find various observation intervals.

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Most popular questions from this chapter

Fiber content (in grams per serving) and sugar content (in grams per serving) for 18 high fiber cereals (www.consumerreports.com) are shown below. Fiber Content \(\begin{array}{rrrrrrrrr}7 & 10 & 10 & 7 & 8 & 7 & 12 & 12 & 8 \\ 13 & 10 & 8 & 12 & 7 & 14 & 7 & 8 & 8\end{array}\) Sugar Content \(\begin{array}{llllllll}11 & 6 & 14 & 13 & 0 & 18 & 9 & 10\end{array}\) $$ \begin{array}{rrrrrrrrr} 11 & 6 & 14 & 15 & 0 & 18 & 9 & 10 \\ 6 & 10 & 17 & 10 & 10 & 0 & 9 & 5 & 11 \end{array} $$ a. Find the median, quartiles, and interquartile range for the fiber content data set. b. Find the median, quartiles, and interquartile range for the sugar content data set. C. Are there any outliers in the sugar content data set? d. Explain why the minimum value for the fiber content data set and the lower quartile for the fiber content data set are equal. e. Construct a comparative boxplot and use it to comment on the differences and similarities in the fiber and sugar distributions.

The article "Taxable Wealth and Alcoholic Beverage Consumption in the United States" (Psychological Reports [1994]: \(813-814\) ) reported that the mean annual adult consumption of wine was 3.15 gallons and that the standard deviation was 6.09 gallons. Would you use the Empirical Rule to approximate the proportion of adults who consume more than 9.24 gallons (i.e., the proportion of adults whose consumption value exceeds the mean by more than 1 standard deviation)? Explain your reasoning.

An experiment to study the lifetime (in hours) for a certain brand of light bulb involved putting 10 light bulbs into operation and observing them for 1000 hours. Eight of the light bulbs failed during that period, and those lifetimes were recorded. The lifetimes of the two light bulbs still functioning after 1000 hours are recorded as \(1000+.\) The resulting sample observations were \(\begin{array}{llllllll}480 & 790 & 1000+ & 350 & 920 & 860 & 570 & 1000+\end{array}\) \(\begin{array}{ll}170 & 290\end{array}\) Which of the measures of center discussed in this section can be calculated, and what are the values of those measures?

The accompanying table gives the mean and standard deviation of reaction times (in seconds) for each of two different stimuli: $$ \begin{array}{lcc} & \text { Stimulus } & \text { Stimulus } \\ & 1 & 2 \\ \hline \text { Mean } & 6.0 & 3.6 \\ \text { Standard deviation } & 1.2 & 0.8 \\ \hline \end{array} $$ If your reaction time is 4.2 seconds for the first stimulus and 1.8 seconds for the second stimulus, to which stimulus are you reacting (compared to other individuals) relatively more quickly?

The report "Who Moves? Who Stays Put? Where's Home?" (Pew Social and Demographic Trends, December 17, 2008 ) gave the accompanying data for the 50 U.S. states on the percentage of the population that was born in the state and is still living there. The data values have been arranged in order from largest to smallest. \(\begin{array}{lllllllllll}75.8 & 71.4 & 69.6 & 69.0 & 68.6 & 67.5 & 66.7 & 66.3 & 66.1 & 66.0 & 66.0 \\ 65.1 & 64.4 & 64.3 & 63.8 & 63.7 & 62.8 & 62.6 & 61.9 & 61.9 & 61.5 & 61.1\end{array}\) \(\begin{array}{lllllllllll}59.2 & 59.0 & 58.7 & 57.3 & 57.1 & 55.6 & 55.6 & 55.5 & 55.3 & 54.9 & 54.7 \\ 54.5 & 54.0 & 54.0 & 53.9 & 53.5 & 52.8 & 52.5 & 50.2 & 50.2 & 48.9 & 48.7\end{array}\) \(\begin{array}{llllll}48.6 & 47.1 & 43.4 & 40.4 & 35.7 & 28.2\end{array}\) a. Find the values of the median, the lower quartile, and the upper quartile. b. The two smallest values in the data set are 28.2 (Alaska) and 35.7 (Wyoming). Are these two states outliers? c. Construct a boxplot for this data set and comment on the interesting features of the plot.
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