PRACTICE PROBLEMS FOR SECTION 2.4
1 Prepare a pie chart and bar chart for the data in Problem 2 of Section 2.3.
2 Prepare a pie chart and bar chart for the data in Problem 3 of Section 2.3 and comment on the cars the senior citizens like to drive.
3 Prepare a line graph for the data in Problem 5 of Section 2.3 and state whether these data show any patterns. Read the data columnwise.
4 Use the data in Problem 6 of Section 2.3 to do the following:Construct a frequency histogram for these data.Construct a relative frequency histogram for these data.Construct a frequency polygon for these data.Construct an ogive curve for these data.
5 Construct two stem‐and‐leaf diagrams for the data in Problem 4 of Section 2.3, using increments of 10 and 5, and comment on which diagram is more informative.
6 Construct a stem‐and‐leaf diagram for the data in Problem 6 of Section 2.3. Then, reconstruct the stem‐and‐leaf diagram you just made by dividing each stem into two stems and comment on which diagram is more informative.
7 A manufacturing company is very training oriented. Every month the company sends some of its engineers for six‐sigma training. The following data give the number of engineers who were sent for six‐sigma training during the past 30 months:182016301416222416141619182423281812181517212225272319182026Using technology, prepare a complete frequency distribution table for these data.
8 A manufacturer of men's shirts is interested in finding the percentage of cotton in fabric used for shirts that are in greater demand. In order to achieve her goal, she took a random sample of 30 men who bought shirts from a targeted market. The following data shows the cotton content of shirts bought by these men (some men bought more than one shirt, so that here ):35256535503540506555255565352535455565553545354520354045356535503530356535253520356535303565353025356535653520352535303565356535303530653530352035653555353035653565353035653535Prepare a single‐valued frequency distribution table for these data.Prepare a pie chart for these data and comment on the cotton contents in these shirts.
9 The following data give the number of patients treated per day during the month of August at an outpatient clinic in a small California town:20302535324640384441373540414338373532402326272921232833392029Prepare a complete frequency distribution table for the data using six classes.On how many days during August were 36 or more patients treated in the clinic?
10 The following data give the number of parts that do not meet certain specifications in 50 consecutive batches manufactured in a given plant of a company:1619222527183630202429403031343621252428263024161921302420222432271824201733352932363928261718252729Construct a frequency histogram and a frequency polygon for these data.
11 A manufacturer of a part is interested in finding the life span of the part. A random sample of 30 parts gave the following life spans (in months):232530323642282421434648393034352421165425343723242826192737Construct a relative frequency histogram and a cumulative frequency histogram for these data. Comment on the life span of the part in question.
12 The following data give the number of accidents per week in a manufacturing plant during a period of 25 weeks:0412342103402132420535014Construct a single‐valued frequency distribution table for these data.Construct a frequency histogram for these data.During how many weeks was the number of accidents less than 2?During how many weeks was the number of accidents at least 3?What is the relative frequency of 0 accidents?
13 Compressive strengths were measured on 60 samples of a new metal that a car manufacturing company is considering for use in bumpers with better shock‐absorbent properties. The data are shown below:59.758.359.061.558.763.868.265.663.562.459.463.264.560.060.561.568.566.661.358.559.261.360.460.662.163.564.467.367.964.265.469.367.364.562.371.760.760.266.768.564.265.167.059.561.763.167.568.569.261.562.368.466.565.769.362.568.060.562.360.5Prepare a complete frequency distribution table.Construct a frequency histogram.Construct a relative frequency histogram.Construct a frequency and relative frequency polygon.Construct a cumulative frequency histogram and then draw the ogive curve for these data.
14 Refer to the data in Problem 13 above. Construct a stem‐and‐leaf diagram for these data.
15 The following data give the consumption of electricity in kilowatt‐hours during a given month in 30 rural households in Maine:260290280240250230310305264286262241209226278206217247268207226247250260264233213265206225Construct, using technology, a stem‐and‐leaf diagram for these data.Comment on what you learn from these data.
2.5 Numerical Measures of Quantitative Data
Methods used to derive numerical measures for sample data as well as population data are known as numerical methods.
Definition 2.5.1
Numerical measures computed by using data of the entire population are referred to as parameters.
Definition 2.5.2
Numerical measures computed by using sample data are referred to as statistics.
In the field of statistics, it is standard practice to denote parameters by letters of the Greek alphabet and statistics by letters of the Roman alphabet.
We divide numerical measures into three categories: (i) measures of centrality, (ii) measures of dispersion, and (iii) measures of relative position. Measures of centrality give us information about the center of the data, measures of dispersion give information about the variation around the center of the data, and measures of relative position tell us what percentage of the data falls below or above a given measure.
2.5.1 Measures of Centrality
Measures of centrality are also known as measures of central tendency. Whether referring to measures of centrality or central tendency, the following measures are of primary importance:
1 Mean
2 Median
3 Mode
The mean, also sometimes referred to as the arithmetic mean, is the most useful and most commonly used measure of centrality. The median is the second most used, and the mode is the least used measure of centrality.
Mean
The mean of a sample or a population is calculated by dividing the sum of the data measurements by the number of measurements in the data. The sample mean is also known as sample average and is denoted by
