Probability and Statistics: Week 1 Exercise
An important part of employee compensation is a benefits package, which might include health insurance, life insurance, child care, vacation days, retirement plan, parental leave, bonuses, etc. Suppose you want to conduct a survey of benefits packages available in private businesses in Hawaii. You want a sample size of 100. Some sampling techniques are described below. Categorize each technique as simple random sample, stratified sample, systematic sample, cluster sample, or convenience sample. -(a) Assign each business in the Island Business Directory a number, and then use a random-number table to select the businesses to be included in the sample. -(b) Use postal ZIP Codes to divide the state into regions. Pick a random sample of 10 ZIP Code areas and then include all the businesses in each selected ZIP Code area. -(c) Send a team of five research assistants to Bishop Street in downtown Honolulu. Let each assistant select a block or building and interview an employee from each business found. Each researcher can have the rest of the day off after getting responses from 20 different businesses. -(d) Use the Island Business Directory. Number all the businesses. Select a starting place at random, and then use every 50th business listed until you have 100 businesses. -(e) Group the businesses according to type: medical, shipping, retail, manufacturing, financial, construction, restaurant, hotel, tourism, other. Then select a random sample of 10 businesses from each business type.
(a)simple random sample (b)cluster sample (c)convenience sample (d)systematic sample (e)stratified sample
(c) Draw a histogram. (d)Draw a relative-frequency histogram. (e) Categorize the basic distribution shape.
(c)Frequency: 0,5,10,15,20,15 Finish Times: 235.5, 260.5, 285.5, etc. (d)Relative Frequency: 0, 0.1, 0.2, 0.3., etc Finish Times: 235.5, 260.5, 285.5, 310.5, etc (e)mound-shaped symmetrical
What is the difference between a class boundary and a class limit? (Select all that apply.)
-Class limits specify the span of data values that fall within a class. -Class boundaries are not possible data values. -Class limits are possible data values. -Class boundaries are values halfway between the upper class limit of one class and the lower class limit of the next.
A data set with whole numbers has a low value of 20 and a high value of 117. Find the class width for a frequency table with seven classes.
14
What is the difference between a parameter and a statistic?
A parameter is a numerical measurement describing data from a population. A statistic is a numerical measurement describing data from a sample.
How long does it take to finish the 1161-mile Iditarod Dog Sled Race from Anchorage to Nome, Alaska? Finish times (to the nearest hour) for 57 dogsled teams are shown below. 261 271 236 244 279 296 284 299 288 288 247 256 338 360 341 333 261 266 287 296 313 311 307 307 299 303 277 283 304 305 288 290 288 289 297 299 332 330 309 328 307 328 285 291 295 298 306 315 310 318 318 320 333 321 323 324 327 For this problem, use five classes. (a) Find the class width. (b) Make a frequency table showing class limits, class boundaries, midpoints, frequencies, relative frequencies, and cumulative frequencies. (Give relative frequencies to 2 decimal places.)
Class width 25 Class Limits 236-260 261-285 286-310 311-335 336-360 Class Boundaries 235.5-260.5 260.5-285.5 285.5-310.5 310.5-335.5 335.5-360.5 Midpoint 248 273 298 323 348 Frequency 4 9 25 16 3 Relative Frequency 0.07 0.16 0.44 0.28 0.05 Cumulative Frequency 4 13 38 54 57
A data set has values ranging from a low of 10 to a high of 52. What's wrong with using the class limits 10-19, 20-29, 30-39, 40-49 for a frequency table?
Each data value must fall into one class. The data values of 50 and above do not have a class.
If you were going to apply statistical methods to analyze teacher evaluations, which question form, A or B, would be better? Form A: In your own words, tell how this teacher compares with other teachers you have had. Form B: Use the following scale to rank your teacher as compared with other teachers you have had.
Form B would be better because statistical methods can be applied to the ordinal data.
Explain the difference between a stratified sample and a cluster sample. (Select all that apply.)
In a cluster sample, the clusters to be included are selected at random and then all members of each selected cluster are included. In a stratified sample, random samples from each strata are included.
Explain the difference between a simple random sample and a systematic sample. (Select all that apply.)
In a simple random sample, every sample of size n has an equal chance of being included. In a systematic sample, the only samples possible are those including every kth item from the random starting position.
Find the class limits for a frequency table with seven classes.
LOWER LIMITS(in order) 20 34 48 62 76 90 104
Suppose you are looking at the 2006 results of how the Echo generation classified specified items as either luxuries or necessities. Do you expect the results to reflect how the Echo generation would classify items in 2020? Explain.
No, the generation will have aged by 14 years and their perception of items as necessities or luxuries might well have changed by then.
Consider these number assignments for category items describing electronic ways of expressing personal opinions. 1 = Twitter; 2 = e-mail; 3 = text message; 4 = Facebook; 5 = blog Are these numerical assignments at the ordinal data level or higher? Explain.
No, they are at the nominal level as there is no apparent ordering in the responses.
What about at the interval level or higher? Explain.
No, while the data has an ordering, and the data can be compared to each other, the differences don't mean anything.
A data set has values ranging from a low of 10 to a high of 50. The class width is to be 10. What's wrong with using the class limits 10-20, 21-31, 32-42, 43-53 for a frequency table with a class width of 10?
The classes listed have a class width of 11.
A data set has values ranging from a low of 10 to a high of 50. What's wrong with using the class limits 10-20, 20-30, 30-40, 40-50 for a frequency table?
The classes overlap so that some data values, such as 20, fall within two classes.
Find the class limits for a frequency table with seven classes.
UPPER LIMITS(in order) 33 47 61 75 89 103 117
Consider these number assignments for category items describing usefulness of customer service. 1 = not helpful; 2 = somewhat helpful; 3 = very helpful; 4 = extremely helpful Are these numerical assignments at the ordinal data level? Explain.
Yes, the data has an ordering to its categories.
Consider the students in your statistics class as the population and suppose they are seated in four rows of 10 students each. To select a sample, toss a coin. If it comes up heads, you use the 20 students sitting in the first two rows as your sample. If it comes up tails, you use the 20 students sitting in the last two rows as your sample. (a) Does every student have an equal chance of being selected for the sample? Explain. (b) Is it possible to include students sitting in row 3 with students sitting in row 2 in your sample? Is your sample a simple random sample? Explain. (c) Describe a process you could use to get a simple random sample of size 20 from a class of size 40.
a) Yes, your seating location and the randomized coin flip ensure equal chances of being selected. b) No, it is not possible with this described method of selection; No, this is not a simple random sample. It is a cluster sample. c) Assign each student a number 1, 2, . . . , 40 and use a computer or a random-number table to select 20 students.
Are data at the nominal level of measurement quantitative or qualitative?
qualitative
Suppose you are assigned the number 1, and the other students in your statistics class call out consecutive numbers until each person in the class has his or her own number. (a) Explain how you could get a random sample of four students from your statistics class. (Select all that apply.) (b) Explain why the first four students walking into the classroom would not necessarily form a random sample. (Select all that apply.) (c) Explain why four students coming in late would not necessarily form a random sample. (Select all that apply.) (d) Explain why four students sitting in the back row would not necessarily form a random sample. (Select all that apply.) (e) Explain why the four tallest students would not necessarily form a random sample. (Select all that apply.)
(a) Explain how you could get a random sample of four students from your statistics class. (Select all that apply.) - Use a computer or random-number table to randomly select four students after numbers are assigned. (b) Explain why the first four students walking into the classroom would not necessarily form a random sample. (Select all that apply.) - Perhaps they are students with lots of free time and nothing else to do. - Perhaps they are students that had a class immediately prior to this one. - Perhaps they are excellent students who make a special effort to get to class early. - Perhaps they are students that needed less time to get to class. (c) Explain why four students coming in late would not necessarily form a random sample. (Select all that apply.) - Perhaps they are students that need more time to get to class. - Perhaps they are students that had a prior class go past scheduled time. - Perhaps they are lazy students that don't want to attend class. - Perhaps they are busy students who are never on time to class. (d) Explain why four students sitting in the back row would not necessarily form a random sample. (Select all that apply.) - Perhaps students in the back row came to class early. - Perhaps students in the back row came to class late. - Perhaps students in the back row do not pay attention in class. - Perhaps students in the back row are introverted. (e) Explain why the four tallest students would not necessarily form a random sample. (Select all that apply.) - Perhaps tall students generally are healthier. - Perhaps tall students generally sit together. - Perhaps tall students generally are athletes. - Perhaps tall students generally attend more classes.
What is the average miles per gallon (mpg) for all new hybrid small cars? Using Consumer Reports, a random sample of such vehicles gave an average of 35.7 mpg. (a) Identify the variable. (b) Is the variable quantitative or qualitative? (c) What is the implied population?
(a) miles per gallon (b) quantitative (c) all new hybrid small cars
Categorize these measurements associated with student life according to level: nominal, ordinal, interval, or ratio. (a) Length of time to complete an exam (b) Time of first class (c) Major field of study (d) Course evaluation scale: poor, acceptable, good (e) Score on last exam (based on 100 possible points) (f) Age of student
(a) ratio (b) interval (c) nominal (d) ordinal (e) ratio (f) ratio
In each of the following situations, the sampling frame does not match the population, resulting in undercoverage. Give examples of population members that might have been omitted. (a) The population consists of all 250 students in your large statistics class. You plan to obtain a simple random sample of 30 students by using the sampling frame of students present next Monday. (Select all that apply.) (b) The population consists of all 15-year-olds living in the attendance district of a local high school. You plan to obtain a simple random sample of 200 such residents by using the student roster of the high school as the sampling frame. (Select all that apply.)
(a)-Students who are out sick cannot be sampled. -Students who are on a school trip cannot be sampled. -Students who are skipping class cannot be sampled. (b)-Home-schooled students cannot be sampled. -Dropouts cannot be sampled.
Another display technique that is somewhat similar to a histogram is a dotplot. In a dotplot, the data values are displayed along the horizontal axis. A dot is then plotted over each data value in the data set. For more details, view How to Make a Dotplot. The figure below shows a dotplot generated by Minitab for the number of licensed drivers per 1000 residents by state, including the District of Columbia (Source: U.S. Department of Transportation). (a) From the dotplot, how many states have 600 or fewer licensed drivers per 1000 residents? (b) About what percentage of the states (out of 51) seem to have close to 800 licensed drivers per 1000 residents? (Round your answer to one decimal place.) (c) Consider the intervals 550 to 650, 650 to 750, and 750 to 850 licensed drivers per 1000 residents. In which interval do most of these states fall?
(a)1 (b)9.8% (c)650 to 750
The ogives shown are based on U.S. Census data and show the average annual personal income per capita for each of the 50 states. The data are rounded to the nearest thousand dollars. (a) How were the percentages shown in graph (ii) computed? (b) How many states have average per capita income less than 37.5 thousand dollars? (c) How many states have average per capita income between 42.5 and 52.5 thousand dollars? (d) What percentage of the states have average per capita income more than 47.5 thousand dollars?
(a)The percentages in graph (ii) were computed by dividing each of the cumulative frequencies in graph (i) by 50 and then converting those values into percents. (b)34 (c) 7 (d) 2%
Which technique for gathering data (observational study or experiment) do you think was used in the following studies? (a) The Colorado Division of Wildlife netted and released 774 fish at Quincy Reservoir. There were 219 perch, 315 blue gill, 83 pike, and 157 rainbow trout. (b) The Colorado Division of Wildlife caught 41 bighorn sheep on Mt. Evans and gave each one an injection to prevent heartworm. A year later, 38 of these sheep did not have heartworm, while the other three did. (c) The Colorado Division of Wildlife imposed special fishing regulations on the Deckers section of the South Platte River. All trout under 15 inches had to be released. A study of trout before and after the regulation went into effect showed that the average length of a trout increased by 4.2 inches after the new regulation. (d) An ecology class used binoculars to watch 23 turtles at Lowell Ponds. It was found that 18 were box turtles and 5 were snapping turtles.
(a)This is an observational study because observations and measurements of individuals are conducted in a way that doesn't change the response or the variable being measured. (b)This is an experiment because a treatment was deliberately imposed on the individuals in order to observe a possible change in the response or variable being measured. (c)This is an experiment because a treatment was deliberately imposed on the individuals in order to observe a possible change in the response or variable being measured. (d)This is an observational study because observations and measurements of individuals are conducted in a way that doesn't change the response or the variable being measured.
You are interested in the weights of backpacks students carry to class and decide to conduct a study using the backpacks carried by 30 students. (a) Give some instructions for weighing the backpacks. Include unit of measure, accuracy of measure, and type of scale. (b) Do you think each student asked will allow you to weigh his or her backpack? (c) Do you think telling students ahead of time that you are going to weigh their backpacks will make a difference in the weights?
(a)Use kilograms as the standard unit of measurement. Round the weight to the nearest whole number. A platform would be the recommended scale to use in this scenario because it is more accurate than a spring scale and has a higher weight capacity than a digital scale. (b)Some students may refuse to allow the weighing. (c)Informing students before class may cause students to remove items before class.
Look at the histogram below, which shows mileage, in miles per gallon (mpg), for a random selection of older passenger cars. (a) Is the shape of the histogram essentially bimodal?
(a)Yes, because the histogram has two peaks.
Which technique for gathering data (sampling, experiment, simulation, or census) do you think was used in the following studies? (a) An analysis of a sample of 31,000 patients from New York hospitals suggests that the poor and the elderly sue for malpractice at one-fifth the rate of wealthier patients. (Journal of the American Medical Association). (b) The effects of wind shear on airplanes during both landing and takeoff were studied by using complex computer programs that mimic actual flight. (c) A study of all league football scores attained through touchdowns and field goals was conducted by the National Football League to determine whether field goals account for more scoring events than touchdowns (USA Today). (d) An Australian study included 588 men and women who already had some precancerous skin lesions. Half got skin cream containing a sunscreen with a sun protection factor of 17; half got an inactive cream. After 7 months, those using the sunscreen with the sun protection had fewer precancerous skin lesions (New England Journal of Medicine).
(a)sampling (b)simulation (c)census (d)experiment