Math 108 Test 1

Commercial dredging operations in ancient rivers occasionally uncover archaeological artifacts of great importance. One such artifact is Bronze Age spearheads recovered from ancient rivers in Ireland. A recent study gave the following information regarding discoveries of ancient bronze spearheads in Irish rivers. River Bann Blackwater Erne Shannon Barrow No. of spearheads 22 8 15 32 14

(a) Make a Pareto chart for these data. No. of Spearheads: 0, 10, 20, 30, 40: Shannon 32, Bann 22, Erne 15, Barrow 14, Blackwater 8 (b) Make a circle graph for these data Shannon 35.2% Blkwtr 8.8% Barr 15.4% Erne 16/5% Bann 24.2%

In a statistical study what is the difference between an individual and a variable?

An individual is a member of the population of interest. A variable is an aspect of an individual subject or object being measured.

What is the average miles per gallon (mpg) for all new cars? Using Consumer Reports, a random sample of 35 new cars gave an average of 21.1 mpg. (a) Identify the variable.

miles per gallon (b) Is the variable quantitative or qualitative? - quantitative (c) What is the implied population? all new cars

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

Find the class width. 25 (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 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 Relative Frequency 0.07 0.16 0.44 0.28 0.05 Cumulative Frequency 4 13 38 54 57 C. Draw a histogram Frequency: 0,5,10,15,20,15 Finish Times: 235.5, 260.5, 285.5, etc. D. Draw a relative-frequency histogram. Relative Frequency: 0, 0.1, 0.2, 0.3., etc Finish Times: 235.5, 260.5, 285.5, 310.5, etc (e) Categorize the basic distribution shape. mound-shaped symmetrical

For a set population, does a parameter ever change?

Never If there are three different samples of the same size from a set population, is it possible to get three different values for the same statistic? - Data from samples may vary from sample to sample, and so corresponding sample statistics may vary from sample to sample.

Consider these types of graphs: histogram, bar graph, Pareto chart, pie chart, stem-and-leaf display. (a) Which are suitable for qualitative data? (Select all that apply.)

Pareto chart pie chart bar graph (b) Which are suitable for quantitative data? (Select all that apply.) stem-and-leaf display Pareto chart bar graph pie chart histogram

Which technique for gathering data (observational study or experiment) do you think was used in the following studies? Explain your answer. (a) The U.S. Census Bureau tracks population age. In 1900, the percentage of the population that was 19 years old or younger was 44.4%. In 1930, the percentage was 38.8%; in 1970, the percentage was 37.9%; and in 2000, the percentage in the age group was down to 28.5% (The First Measured Century, T. Caplow, L. Hicks, B. J. Wattenberg).

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) After receiving the same lessons, a class of 100 students was randomly divided into two groups of 50 each. One group was given a multiple-choice exam covering the material in the lessons. The other group was given an essay exam. The average test scores for the two groups were then compared. -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.

A die is a cube with dots on each face. The faces have 1, 2, 3, 4, 5, or 6 dots. The table below is a computer simulation (from the software package Minitab) of the results of rolling a fair die 20 times. DATA DISPLAY ROW C1 C2 C3 C4 C5 C6 C7 C8 C9 C10 1 5 2 2 2 5 3 2 3 1 4 2 3 2 4 5 4 5 3 5 3 4 (a) Assume that each number in the table corresponds to the number of dots on the upward face of the die. Is it appropriate that the same number appears more than once? Why?

Yes, the outcome of the die roll can repeat. What is the outcome of the second roll? - 2 (b) If we simulate more rolls of the die, do you expect to get the same sequence of outcomes? Why or why not? No, because the process is random.

Categorize these measurements associated with student life according to level: nominal, ordinal, interval, or ratio. (a) Length of time to complete an exam

ratio (b) Time of first class - interval (c) Major field of study - nominal (d) Course evaluation scale: poor, acceptable, good - ordinal (e) Score on last exam (based on 100 possible points) - ratio (f) Age of student -ratio

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.

The time plot in the figure below gives the number of state and federal prisoners per 100,000 population (Source: Statistical Abstract of the United States, 120th Edition).

(a) Estimate the number of prisoners per 100,000 people for 1980 and for 1997. 1980: 140 1997: 440 (b) During the time period shown, there was increased prosecution of drug offenses, longer sentences for common crimes, and reduced access to parole. What does the time-series graph say about the prison population change per 100,000 people? The number of inmates per 100,000 increased (c) In 1997, the U.S. population was approximately 266,528,000 people. At the rate of 440 prisoners per 100,000 population, about how many prisoners were in the system? (Round answer to whole number.) 1,172,723 prisoners The projected U.S. population for the year 2020 is 323,612,000. If the rate of prisoners per 100,000 stays the same as in 1997, about how many prisoners do you expect will be in the system in 2020? (Round answer to whole number.) 1,423,893 prisoners

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.) - 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.

Driving under the influence of alcohol (DUI) is a serious offense. The following data give the ages of a random sample of 50 drivers arrested while driving under the influence of alcohol. This distribution is based on the age distribution of DUI arrests given in the Statistical Abstract of the United States (112th Edition). 46 16 41 26 22 33 30 22 36 34 63 21 26 18 27 24 31 38 26 55 31 47 27 43 35 22 64 40 58 20 49 37 53 25 29 32 23 49 39 40 24 56 30 51 21 45 27 34 47 35

(a) Make a stem-and-leaf display of the age distribution. Age of DUI Arrests 1 I 68 2 I 01122234456667779 3 I 00112344556789 4 I 0013567799 5 I 13568 6 I 34 (b) Make a frequency table using seven classes. (Give relative frequencies to 2 decimal places.) Class Limits 16 − 22 23 − 29 30 − 36 37 − 43 44 − 50 51 − 57 58 − 64 Class Boundaries 15.5 − 22.5 22.5 − 29.5 29.5 − 36.5 36.5 − 43.5 43.5 − 50.5 50.5 − 57.5 57.5 − 64.5 Midpoint 19 26 33 40 47 54 61 Frequency 8 11 11 7 6 4 3 Relative Frequency 0.16 0.22 0.22 0.14 0.12 0.08 0.06 Cumulative Frequency 8 19 30 37 43 47 50 (c) Make a histogram showing class boundaries. Frequency: 0, 2, 4, 6, etc. Age: 15.5, 22.5, 29.5, 36.5, etc. (d) Identify the shape of the distribution. skewed right

You are conducting a study of students doing work-study jobs on your campus. Among the questions on the survey instrument are the following. A. How many hours are you scheduled to work each week? Answer to the nearest hour. B. How applicable is this work experience to your future employment goals? Respond using the following scale: 1= not at all, 2 = somewhat, 3 = very

(a) Suppose you take random samples from the following groups: freshmen, sophomores, juniors, and seniors. What kind of sampling technique are you using (simple random, stratified, systematic, cluster, multistage, convenience)? -stratified sample (b) Describe the individuals of this study. - Students on your campus with work-study jobs. (c) What is the variable for question A? hours scheduled Classify the variable as qualitative or quantitative. -quantitative What is the level of measurement? ratio (d) What is the variable for question B? -rating of applicability of work experience to future employment Classify the variable as qualitative or quantitative. -qualitative What is the level of measurement? -ordinal (e) Is the proportion of responses "3 = very" to question B a statistic or a parameter? - statistic (f) Suppose only 40% of the students you selected for the sample respond. What is the nonresponse rate? -60% Do you think the nonresponse rate might introduce bias into the study? Explain. - Yes, the people choosing not to respond may have some characteristics that would bias the study. (g) Would it be appropriate to generalize the results of your study to all work-study students in the nation? Explain. -No, the sample frame is restricted to one campus.

Categorize the type of sampling (simple random, stratified, systematic, cluster, or convenience) used in each of the following situations. (a) To conduct a preelection opinion poll on a proposed amendment to the state constitution, a random sample of 10 telephone prefixes (first three digits of the phone number) was selected, and all households from the phone prefixes selected were called.

cluster sample (b) To conduct a study on depression among the elderly, a sample of 30 patients in one nursing home was used. - convenience sample (c) To maintain quality control in a brewery, every 20th bottle of beer coming off the production line was opened and tested. - systematic sample (d) Subscribers to the magazine Sound Alive were assigned numbers. Then a sample of 30 subscribers was selected by using a random-number table. The subscribers in the sample were invited to rate new compact disc players for a "What the Subscribers Think" column. - simple random sample (e) To judge the appeal of a proposed television sitcom, a random sample of 10 people from each of three different age categories was selected and those chosen were asked to rate a pilot show. - stratified sample