Statistically Significant Weiss's Introductory Statistics, Tenth Edition, is the ideal textbook for introductory statistics classes that emphasize statistical reasoning and critical thinking. Comprehensive in its coverage, Weiss's meticulous style offers careful, detailed explanations to ease the learning process. With more than 1, data sets and over 3, exercises, this text takes a data-driven approach that encourages students to apply their knowledge and develop statistical understanding.
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The new edition incorporates the most up-to-date methods and applications to present the latest information in the field. Having identical frequency distributions implies that the total number of observations and the numbers of observations in each class are identical.
Thus, the relative frequencies will also be identical. Having identical relative frequency distributions means that the ratio of the count in each class to the total is the same for both frequency distributions.
However, one distribution may have twice or some other multiple the total number of observations as the other.
Chapter 2, Organizing Data would be different, but would have the same relative frequency distribution.
If, however, the two data sets have different numbers of observations, using relative-frequency distributions is more appropriate because the total of each set of relative frequencies is 1, putting both distributions on the same basis for comparison.
The classes are the days of the week and are presented in column 1. The frequency distribution of the networks is presented in column 2. The relative frequency distribution is presented in column 3. The classes are the NCAA wrestling champions and are presented in column 1.
The frequency distribution of the champions is presented in column 2. Minnesota Arizona St. Iowa St. The frequency distribution of the colleges is presented in column 2. The classes are the class levels and are presented in column 1. The frequency distribution of the class levels is presented in column 2.
The classes are the regions and are presented in column 1. The frequency distribution of the regions is presented in column 2. Class Level NE. Chapter 2, Organizing Data c We multiply each of the relative frequencies by degrees to obtain the portion of the pie represented by each region. The classes are the days and are presented in column 1. The frequency distribution of the days is presented in column 2. Class Level Su M.
The result is DAY The result is. Political View Liberal. The result is VIEW. Rank Professor. The result is RANK. Payer Medicare Medicaid. Color Red. Click Pie Options, check decreasing volume, click OK. Click OK twice. The result is TYPE The result is TYPE. Column 2 contains the marital status and column 3 contains the number of drinks. Chapter 2, Organizing Data divorced. Column 2 contains the preference for how the members want to receive the ballots and column 3 contains the highest degree obtained by the members.
From the tool bar, select Stat. The results are. One important reason for grouping data is that grouping often makes a large and complicated set of data more compact and easier to understand. For class limits, marks, cutpoints and midpoints to make sense, data must be numerical. They do not make sense for qualitative data classes because such data are nonnumerical.
The most important guidelines in choosing the classes for grouping a data set are: 1 the number of classes should be small enough to provide an effective summary, but large enough to display the relevant characteristics of the data; 2 each observation must belong to one, and only one, class; and 3 whenever feasible, all classes should have the same width. In the alternate method called limit grouping, we used the notation a-b to indicate a class that extends from a to b, including both.
For example, is a class that includes both 30 and The alternate method is especially appropriate when all of the data values are integers. If the data include values like For limit grouping, we find the class mark, which is the average of the lower and upper class limit. For cutpoint grouping, we find the class midpoint, which is the average of the two cutpoints. A frequency histogram shows the actual frequencies on the vertical axis; whereas, the relative frequency histogram always shows proportions between 0 and 1 or percentages between 0 and on the vertical axis.
An advantage of the frequency histogram over a frequency distribution is that it is possible to get an overall view of the data more easily. A disadvantage of the frequency histogram is that it may not be possible to determine exact frequencies for the classes when the number of observations is large. By showing the lower class limits or cutpoints on the horizontal axis, the range of possible data values in each class is immediately known and the class mark or midpoint can be quickly determined.
This is particularly helpful if it is not convenient to make all classes the same width. The use of the class mark or midpoint is appropriate when each class consists of a single value which is, of course, also the midpoint.
Use of the class marks or midpoints is not appropriate in other situations since it may be difficult to determine the location of the class limits or cutpoints from the values of the class marks or midpoints , particularly if the class marks or midpoints are not evenly spaced. Class Marks or midpoints cannot be used if there is an open class.
If the classes consist of single values, stem-and-leaf diagrams and frequency histograms are equally useful. If only one diagram is needed and the classes consist of more than one value, the stem-and-leaf diagram allows one to retrieve all of the original data values whereas the frequency histogram does not. Finally, stem-and-leaf diagrams are not very useful with very large data sets and may present problems with data having many digits in each number.
The histogram especially one using relative frequencies is generally preferable. Data sets with a large number of observations may result in a stem of the stem-and-leaf diagram having more leaves than will fit on the line. In that case, the histogram would be preferable. You can reconstruct the stem-and-leaf diagram using two lines per stem. If there are still two few stems, you can reconstruct the diagram using five lines per stem, recording 10 and 11 on the first line, 12 and 13 on the second, and so on.
For the number of bedrooms per single-family dwelling, single-value grouping is probably the best because the data is discrete with relatively few distinct observations. For the ages of householders, given as a whole number, limit grouping is probably the best because the data are given as whole numbers and there are probably too many distinct observations to list them as single-value grouping.
For additional sleep obtained by a sample of patients by using a particular brand of sleeping pill, cutpoint grouping is probably the best because the data is continuous and the data was recorded to the nearest tenth of an hour. For the number of automobiles per family, single-value grouping is probably the best because the data is discrete with relatively few distinct observations. For gas mileages, rounded to the nearest number of miles per gallon, limit grouping is probably the best because the data are given as whole numbers and there are probably too many distinct observations to list them as single-value grouping.
For carapace length for a sample of giant tarantulas, cutpoint grouping is probably the best because the data is continuous and the data was recorded to the nearest hundredth of a millimeter. The resulting table follows. Number of Siblings 0 1 2 3 4. Frequency 8 17 11 3 1 40 b To get the relative frequencies, divide each frequency by the sample size of Relative Frequency 0.
Column 1 demonstrates that the data are grouped using classes based on a single value. These single values in column 1 are used to label the horizontal axis of the frequency histogram.
Suitable candidates for vertical axis units in the frequency histogram are the integers within the range 0 through 17, since these are representative of the magnitude and spread of the frequencies presented in column 2.
When classes are based on a single value, the middle of each histogram bar is placed directly over the single numerical value represented by the class. Also, the height of each bar in the frequency histogram matches the respective frequency in column 2. Figure a. The relative-frequency histogram in Figure b is constructed using the relative-frequency distribution presented in part b of this exercise. It has the same horizontal axis as the frequency histogram.
We notice that the relative frequencies presented in column 2 range in size from 0. Thus, suitable candidates for vertical-axis units in the relative-frequency histogram are increments of 0.
The middle of each histogram bar is placed directly over the single numerical value represented by the class. Also, the height of each bar in the relative-frequency histogram matches the respective relative frequency in column 2. Number of Persons. The frequency histogram in Figure a is constructed using the frequency distribution presented in part a of this exercise. Suitable candidates for vertical axis units in the frequency histogram are the integers within the range 0 through 13, since these are representative of the magnitude and spread of the frequencies presented in column 2.
Litter Size. Litter Size 1. Suitable candidates for vertical axis units in the frequency histogram are the integers within the range 0 through 7, since these are representative of the magnitude and spread of the frequencies presented in column 2. Number of Radios. Suitable candidates for vertical axis units in the frequency histogram are the integers within the range 1 through 12, since these are representative of the magnitude and spread of the frequencies presented in column 2.
The last class to construct is , since the largest single data value is Having established the classes, we tally the energy consumption figures into their respective classes. These results are presented in column 2, which lists the frequencies. Consumption mil. BTU b. The relative frequencies for all classes are presented in column 2. BTU The lower class limits of column 1 are used to label the horizontal axis of the frequency histogram.
Suitable candidates for vertical-axis units in the frequency histogram are the even integers 0 through 10, since these are representative of the magnitude and spread of the frequency presented in column 2. The height of each bar in the frequency histogram matches the respective frequency in column 2. We notice that the relative frequencies presented in column 2 vary in size from 0. Thus, suitable candidates for vertical axis units in the relative-frequency histogram are increments of 0.
The height of each bar in the relative-frequency histogram matches the respective relative frequency in column 2. All of the classes are presented in column 1. Having established the classes, we tally the age figures into their respective classes. Age b. Age Suitable candidates for vertical-axis units in the frequency histogram are the even integers 2 through 8, since these are representative of the magnitude and spread of the frequency presented in column 2.
Having established the classes, we tally the cheese consumption figures into their respective classes. Cheese Consumption b. Cheese Consumption Chapter 2, Organizing Data lower class limits of column 1 are used to label the horizontal axis of the frequency histogram. Suitable candidates for vertical-axis units in the frequency histogram are the even integers 2 through 7, since these are representative of the magnitude and spread of the frequency presented in column 2.
Having established the classes, we tally the anxiety questionnaire score figures into their respective classes. Anxiety b. Section 2. Having established the classes, we tally the audience sizes into their respective classes. Suitable candidates for vertical axis units in the frequency histogram are the integers within the range 1 through 5, since these are representative of the magnitude and spread of the frequencies presented in column 2.
The relative-frequency histogram in Figure b is constructed using the relative-frequency distribution obtained in part b of this exercise. We notice that the relative frequencies presented in column 3 range in size from 0. The height of each bar in the relativefrequency histogram matches the respective relative frequency in column 2. Having established the classes, we tally the cheetah speeds into their respective classes. The relative frequencies for all classes are presented in column 2 52 54 56 58 60 62 64 66 68 70 72 74 c.
The frequency histogram in Figure a is constructed using the frequency distribution obtained in part a of this exercise Column 1 demonstrates that the data are grouped using classes with class widths of 2.
Suitable candidates for vertical axis units in the frequency histogram are the integers within the range 0 through 8, since these are representative of the magnitude and spread of the frequencies presented in column 2.
Chapter 2, Organizing Data frequency histogram matches the respective relative frequency in column 2. Having established the classes, we tally the fuel tank capacities into their respective classes. Fuel 12 14 16 18 20 22 24 Suitable candidates for vertical axis units in the frequency histogram are the integers within the range 2 through 7, since these are representative of the magnitude and spread of the frequencies presented in column 2. The last class to construct is 7 - under 8, since the largest single data value is 7.
Chapter 2, Organizing Data of 2. Suitable candidates for vertical axis units in the frequency histogram are the integers within the range 0 through 10, since these are representative of the magnitude and spread of the frequencies presented in column 2. The horizontal axis of this dotplot displays a range of possible exam scores. To complete the dotplot, we go through the data set and record each exam score by placing a dot over the appropriate value on the horizontal axis.
The horizontal axis of this dotplot displays a range of possible ages. To complete the dotplot, we go through the data set and record each age by placing a dot over the appropriate value on the horizontal axis. Dotplot of AGE.
The data values range from 52 to 84, so the scale must accommodate those values. We stack dots above each value on two different lines using the same scale for each line.
The two sets of pulse rates are both centered near 68, but the Intervention data are more concentrated around the center than are the Control data. The data values range from 7 to 18, so the scale must accommodate those values. The Dynamic system does seem to reduce acute postoperative days in the hospital on the average. Since each data value consists of 3 or 4 digits ranging from to The last digit becomes the leaf and the remaining digits are the stems, so we have stems of 91 to The resulting stem-and-leaf diagram is 91 92 93 94 95 96 97 98 99 Splitting into two lines per stem, leafs of belong in the first stem and leafs of belong in the second stem.
The result is 28 29 29 30 30 Since each data value consists of a 2 digit number with a one digit decimal, we will make the leaf the decimal digit and the stems the remaining two digit numbers of 28, 29, 30, and The result is 28 29 30 Since each data value consists of 2 digits, each beginning with 1, 2, 3, or 4, we will construct the stem-and-leaf diagram with these four values as the stems.
The result is 1 2 3 4. The stem-and-leaf diagram in part b is more useful because by splitting the stems into two lines per stem, you have created more lines. Part a had too few lines. The stem with one line per stem is more useful. One gets the same impression regarding the shape of the distribution, but the two lines per stem version has numerous lines with no data, making it take up more space than necessary to interpret the data and giving it too many lines.
Since we have two digit numbers, the last digit becomes the leaf and the first digit becomes the stem. For this data, we have stems of 6 and 7. Splitting the data into five lines per stem, we put the leaves in the first stem, in the second stem, in the third stem, in the fourth stem, and in the fifth stem. For this data, we have stems of 6, 7, and 8. The graph indicates that:. With 20 patients in total, the number having cholesterol levels between and is 7 i.
Column 1 contains the numbers of pups borne in a lifetime for each of 80 female. Tables Tally Great White Sharks. The result is PUPS 3 4. Histogram, choose Simple and click OK. The frequency histogram is. To change to a relative-frequency histogram, before clicking OK the second time, click on the Scale button and the Y-Scale type tab, and choose Percent and click OK. The graph will look like the frequency histogram, but will have relative frequencies on the vertical scale instead of counts.
The numbers of pups range from 1 to 12 per female with 7 and 8 pups occurring more frequently than any other values. However, you can use an option in creating your histogram that will report the frequencies in each of the classes, essentially creating a grouped frequency distribution. Retrieve the data from the Weiss-Stats-CD. Clients who use PayPal are going to receive a payment invoice into their email which they sign up with.
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