Review Sheet Exam 1

Try to identify each of the four major flaws in the following. A daily newspaper ran a survey by asking readers to call in their response to this question: "Do you support the development of atomic weapons that could kill millions of innocent people?" It was reported that 20 readers responded and 87% said "no" while 13% said "yes."

1) Biased wording = loaded question 2) Sample too small 3) Voluntary response 4) impossible to have those increments of %.

A researcher at the Sloan-Kettering Cancer Research Center was once criticized for falsifying data. Among his data were figures obtained from 6 groups of mice, with 20 individual mice in each group. These values were given for the percentage of successes in each group: 53%, 58%, 63%, 46%, 48%, 67%. What is the major flaw?

100%/20 = 5% If there is 20 mice then the results should be in 5% incriminates.

A thief steals an ATM card and must randomly guess the correct pin code that consist of four digits (each 0 to 9) that must be entered correctly in the correct order. Repetition is allowed. What is the probability of a correct guess on the first try?

10^4 = 10,000 P(Right) = 1/10,000

When four golfers are about to begin a game, they often toss a tee to randomly select the order in which they tee off. What is the probability that they tee off in alphabetical order by last name?

4! = 24 P(alphabetical) = 1/24

When Mendel conducted his famous genetics experiments with peas, one sample of offspring consisted of 428 green peas and 152 yellow peas. Based on those results, estimate the probability of getting an offspring pea that is green. Is the result reasonably close to the expected value of ¾, as Mendel claimed?

428 green + 152 Yellow = 580 total. P(Green) = 428/580 = .738. Yes .738 is reasonably close to .75 that Mendel expected.

When testing for current in a cable with five color-coded wires, the author used a meter to test two wires at a time. How many different tests are required for every possible pairing of two wires?

5C2 = 10

In "Ages of Oscar-winning Best Actor and Actresses" (Mathematics Teacher magazine) by Richard Brown and Gretchen Davis, the authors compare ages of actors and actresses at the time they won Oscars. a. Find the five number summaries. b. Find the IQR for each data set. c. Construct boxplots for the actors and actresses ages. d. Compare the two different sets of data. Actresses: 50 44 35 80 26 28 41 21 61 38 49 33 74 30 33 41 31 35 41 42 37 26 34 34 35 26 61 60 34 24 30 37 31 27 39 34 26 25 33 Actors: 32 37 36 32 51 53 33 61 35 45 55 39 76 37 42 40 32 60 38 56 48 48 40 43 62 43 42 44 41 56 39 46 31 47 45 60 46 40 36

A) Actors: min 31, Q1 37, med 43, Q3 51, Max 76 Actresses: min 21, Q1 30, med 34, Q3 41, Max 80 B) Actors: IQR = Q3 - Q1 IQR = 14 years Actresses: IQR = Q3 - Q1 IQR = 11 years C) See image: D)Both graphs are skewed to the right, showing Hollywood favors younger performers, we can see women are much younger than men when winning Oscars.

The Digital Pet Rock Company was recently successfully funded and must now appoint a president, CFO, CEO, and COO. It must also appoint a strategic planning committee with four different members. There are 10 qualified candidates, and officers can also serve on the committee. a. How many different ways can the four officers be appointed? b. How many different ways can a committee of four be appointed? c. What is the probability of randomly selecting the committee members and getting the four youngest of the qualified candidates?

A) 10P4 = 5,040 B) 10C4 = 210 C) P(4 youngest) = 1/210 = .00476

In a Gallup poll, 1038 adults were surveyed. a. 52% said that secondhand smoke is "very harmful." What is the actual number of adults who said that secondhand smoke is "very harmful"? b. Among the 1038 surveyed adults, 52 said that secondhand smoke is "not at all harmful." What is the percentage of people who chose "not at all harmful?"

A) 52% of 1038 = 540 who said it was harmful. B) .050096 or 5% said it wasn't all harmful

2. Determine whether the given values are from a discrete or continuous data set. a. George Washington's presidential salary was $25,000 per year, and the current annual presidential salary is $400,000. b. A statistics student obtains sample data and finds that the mean weight of cars in the sample is 3126 lb. c. In a survey of 1059 adults, it is found that 39% of them have guns in their homes (based on a Gallup poll). d. When 19,218 gas masks from branches of the U.S. military were tested, it was found that 10,322 of them were defective (based on data from Time magazine).

A) Discrete B) Continuous C) Discrete D) Discrete

Determine whether the events are disjoint for a single trial: a. Receiving a phone call from a volunteer survey subject who opposes all government taxation Receiving a phone call from a volunteer survey subject who approves of government taxation b. Randomly selecting a United States Senator currently holding office Randomly selecting an African American elected official

A) Disjoint B) Not disjoint

Determine whether the given description corresponds to an observational study or an experiment. a. Patients are given Lipitor to determine whether this drug has the effect of lowering high levels of cholesterol. b. Much controversy arose over a study of patients with syphilis who were not given a treatment that could have cured them. Their health was followed for years after they were found to have syphilis. c. The Duchess County Bureau of Weights and Measures randomly selects gas stations and obtains 1 gallon of gas from each pump. The amount pumped is measured for accuracy. d. Cruise ship passengers are given magnetic bracelets, which they agree to wear in an attempt to eliminate or diminish the effects of motion sickness.

A) Experiment B) Observational Study C) Observational study D) Experiment

Determine which of the four levels of measurement (nominal, ordinal, interval, ratio) is most appropriate. a. Heights of women basketball players in the WNBA. b. Ratings of fantastic, good, average, poor, or unacceptable for blind dates. c. Current temperatures of the classrooms at your college. d. Numbers on the jerseys of women basketball players in the WNBA. e. Consumer Reports magazine ratings of "best buy, recommended, not recommended" f. Social Security Numbers g. The number of "yes" responses received when 1250 drivers are asked if they have ever used a cell phone while driving. h. Zip codes

A) Ratio B) Ordinal C) Interval D) Nominal E) Ordinal F) Nominal G) Ratio H) Nominal

Identify the type of observational study: cross-sectional, retrospective, or prospective. a. A researcher from the New York University School of Medicine obtains data about head injuries by examining the hospital records from the past five years. b. A researcher from Mt. Sinai Hospital in New York City plans to obtain data by following (to the year 2014) siblings of victims of a plane crash. c. The US Labor Department obtains current unemployment data by polling 50,000 people this month. d. An economist collects data by interviewing people who won the lottery between the years 1995 and 2000.

A) Retrospective B) Prospective C) Cross-Sectional D) Retrospective

Identify the (a) sample and (b) population. Also determine whether the sample is likely to be representative of the population. a. A reporter for Newsweek stands on a street corner and asks 10 adults if they feel that the current president is doing a good job. b. Nielson Media Research surveys 5000 randomly selected households and finds that among the TV sets in use, 19% are tuned to 60 Minutes (based on data from USA Today). c. In a Gallup poll of 1059 randomly selected adults, 39% answered "yes" when asked "Do you have a gun in your home?" d. A graduate student at the University of Newport conducts a research project about how adult Americans communicate. She begins with a survey mailed to 500 of the adults that she knows. She asks them to mail back a response to this question: "Do you prefer to use e-mail or snail mail (the US Postal Service)?" She gets back 65 responses, with 42 of them indicating a preference for snail mail.

A) Sample: 10 selected Adults, Pop: All Adults in America. - Not representative (Convince sample) B)Sample: 5000 randomly selected Households. Pop: All households in the U.S. - Likely to be Representative C) Sample: The 1059 Selected Adults. Pop: All adults in the U.S. - This is Representative since it is random D) Sample: the 65 respondents. Pop: All adult Americans. - This is not representative since it was sent via snail mail which was one of the options.

Identify which of these types of sampling is used: random, systematic, convenience, stratified, or cluster. a. Satellites are used to collect sample data for estimating deforestation rates. The Forest Resources Assessment of the United Nations (UN) Food and Agriculture Organization uses a method of selecting a sample of a 10-km-wide square at every 1º intersection of latitude and longitude. b. We randomly select 50 full-time college students in each of the 50 states. c. We randomly select four colleges and survey all of their full-time students. d. A pollster stops each person passing her office door and asks the person to rate the last movie that he or she saw (on a scale of 1 star to 4 stars). e. In a Kelton Research poll, 1114 Americans 18 years of age or older were called after their telephone numbers were randomly generated by a computer, and 36% of the respondents said that they believe in the existence of UFOs.

A) Systematic B) Stratified C) Cluster D) Convenience E) Random

a. A data set has a distribution that is uniform. If all of the values are converted to z scores, what is the shape of the distribution of the z scores? b. A data set has a distribution that is bell shaped. If all the values are converted to z scores, what is the shape of the distribution of the z scores? c. In general, how is the shape of a distribution affected if all values are converted to z scores?

A) Uniform B) Bell-shaped C) It is not affected, the distribution remains the same after turning into z-scores.

A woman wrote to Dear Abby and claimed that she gave birth 308 days after a visit from her husband, who was in the Navy. Lengths of pregnancies have a mean of 268 days and a standard deviation of 15 days. Find the z score for 308 days. Is such a length significant? What do you conclude?

Her pregnancy length is significantly high since it is above a z-score of 2. Either a rare event occurred or she cheated on her husband.

The 20 brain volumes (cm^3) from a data set in your textbook, "IQ and Brain Size" vary from a low of 963 cm^3 to a high of 1439 cm^3. Use the range rule of thumb to estimate the standard deviation s and compare the result to the exact standard deviation of 124.9 cm^3.

High: 1439 cm^3 Low: 963 cm^3 S = Range/4 = 119 cm^3

In 1986, the New York City subway fare cost $1, and as of this writing, the current cost is $2.50, so the 1986 price was multiplied by 2.5. In the accompanying graph, the large bill is 2.5 times as tall and 2.5 times as wide as the small bill. How is the graph deceptive?

Increasing by a factor of 2.5 is fine but the length & width is increased by 2.5 the area becomes larger by a factor 6.25. Greatly exaggerating the the increase.

Find the mean, median, mode and midrange for the given sample data. Given that these ratings are based on 12 randomly selected pages, is the mean of this sample likely to be a reasonable estimate of the mean reading level of the book? Find the range, variance, and standard deviation for the given sample data. Given that these ratings are based on 12 randomly selected pages, is the standard deviation of this sample likely to be a reasonable estimate of the standard deviation of the reading levels for all pages in the whole book? 85.3 84.3 79.5 82.5 80.2 84.6 79.2 70.9 78.6 86.2 74.0 83.7

Mean: X-bar = 80.75 Median: 81.35 Mode: none Mid-range: (70.9+86.2)/2 = 78.55 Range: (86.2-70.9) = 15.3 Variance: S^2 = 21.92 Std. Dev: S = 4.68

Stanford Binet IQ scores have a mean of 100 and a standard deviation of 16. Albert Einstein had an IQ of 160. a. What is the difference between Einstein's IQ and the mean? b. How many standard deviations is that? c. Convert Einstein's IQ score to a z score. d. If we consider IQ scores that convert to z scores between -2 and 2 to be neither significantly low nor significantly high, is Einstein's IQ significant?

Mue (Mean): 100 Stand Dev: 16 X: 160 A) X - Mean = 60 Iq points. B) 60/stand dev = 3.75 stand deviations. C) Z = (160-100)/2.8 = 5.71 D) Einstein's IQ is way above 2 standard deviations above the mean, his IQ is significantly high.

Body temperature of adults have a mean of 98.20º F and a standard deviation of 0.62º F. Is an adult body temperature of 100º F significantly low or significantly high?

Mue (mean): 98.20 F Stand Dev: 0.62 F Sig low: 96.96 F Sig High: 99.44 F

You need to conduct a study of longevity for people who were born after the end of World War II in 1945. If you were to visit graveyards and use the birth/death dates listed on the tombstones, would you get good results? Why or why not?

No! most/many who are born after 1945 are still alive! the tombstones would primarily show those who died young

Consider the nicotine content of 25 different king size cigarette brands. The average (mean) of those amounts is 1.26 mg. Is this result likely to be a good estimate of the average (mean) of all cigarettes smoked in the United States? Why or why not?

Not good estimate since some Cigarettes are more popular and others more rare. So a weighted mean of all types is more fitting.

If someone is randomly selected, find the probability that his or her birthday is not in October. Ignore leap years.

P(Not in oct) = 1-P(is in oct) = 1-31/365 = 334/365 = .915

A study of 150 randomly selected American Airlines flights showed that 108 arrived on time. a. What is the estimated probability of an American Airlines flight arriving late? b. Is it unlikely for an American Airlines flight to arrive late?

P(time on) = 108/150 A) P(late) = 1-P(on time) = 1 - 108/150 = .28 B) No it is not unlikely since it will occur 28% of the time.

You have a 5% probability of winning a raffle your friend is trying to sell you on. You purchase four raffle tickets for $20. What is the probability that all four raffle tickets will be losers?

P(win) = 0.05 P(lose) = 1 - 0.05 = 0.95 P(4 lose) = (0.95) ^4 = 0.815

A survey with 12 questions is designed so that 3 of the questions are identical and 4 other questions are identical (except for minor changes in wording). How many different ways can the 12 questions be arranged?

Permutation when some are identical. 12!/3!4! = 3,326,400

A die was drilled with a hole and filled with a lead weight, then proceeded to roll it 200 times. The results are given the frequency distribution to the right. Find the standard deviation. (see packet #20 for table)

Put outcomes into L1, freq into L2 -> S = 11.6

Which is relatively better: A score of 85 on a psychology test or a score of 45 on an economics test? Scores on the psychology test have a mean of 90 and a standard deviation of 10. Scores on the economics test have a mean of 55 and a standard deviation of 5.

Relatively we did better in Psychology test since it had a higher z-score.

List the statistics that are resistant, and list the statistics that are not resistant.

Resistant: Median & Mode Not Resistant: Mean, Range, Mid-range, Standard Deviation, Variance

In "Ages of Oscar-winning Best Actor and Actresses" (Mathematics Teacher magazine) by Richard Brown and Gretchen Davis, frequency distributions and stem-and-leaf plots are used to compare ages of actors and actresses at the time they won Oscars. Here are the results for recent winners. a. Construct a frequency distribution for the actors' ages. b. Construct a stem-and-leaf plot for the actresses' ages. c. Looking at the stem-and-leaf plot, explain what this shows about actresses who win. What is the shape of the distribution? (Use packet #15 for list)

See image for A) & B). C) Skewed to to the right. Most actresses win win when they are younger.

Adult males have heights with a mean 69.0 in. and a standard deviation of 2.8 in. Find the z scores corresponding to the following: a. Actor Danny DeVito, who is 5 ft. tall b. NBA basketball player Shaquille O'Neal, who is 7 ft. 1 in. tall c. A typical person who is 69.72 in. tall

See image:

Given the table, find a. P(woman or doesn't smoke) b. P(man or smokes) (See packet #36 for table)

See image:

Identify the class width, class midpoints, and class boundaries for the given frequency distribution. Also find relative and cumulative frequency. (use packet #12 for table)

See image:

One couple attracted media attention when their three children, born in different years, were all born on July 4. a. Ignoring leap years, find the probability that three randomly selected people were all born on July 4. b. Is the probability low enough so that such an event is not likely to occur somewhere in the United States over the course of several years? c. Ignoring leap years, find the probability that three randomly selected people all have the same birthday.

See image:

The principle of redundancy is used when system reliability is improved through redundant or backup components. Assume your alarm clock has a 0.975 probability of working on any given morning. a. What is the probability that your alarm clock will not work on the morning of an important final exam? b. If you had two such alarm clocks what is the probability that they both fail on the morning of an important final exam? c. With one alarm clock, we have a 0.975 probability of being awakened. What is the probability of being awakened if we are using two alarm clocks?

See image:

Using the data below of cell phone airport data speeds (Mbps) from sprint, find the following: a. The percentile corresponding to 11.7 Mbps b. The percentile corresponding to 0.3 Mbps c. Find the 85th percentile d. Find the 40th percentile 0.2 0.2 0.2 0.2 0.3 0.4 0.4 0.4 0.5 0.6 0.6 0.6 0.6 0.8 0.9 0.9 1.1 1.2 1.3 1.7 1.7 2.1 2.2 2.3 2.3 2.5 2.5 2.8 3.2 3.3 4.4 5.9 6.4 6.6 7.3 7.5 7.8 9.2 9.2 11.4 11.7 11.8 12.1 12.3 12.5 12.9 13.3 13.4 15.5 26.4

See image:

Using the seven items below, find the probability of randomly selecting three items and getting one that is red on the first selection, one that is green on the second selection, and an item that is blue on the third selection. a. Assume that each item is replaced before the next one is selected. b. Assume that none of the selected items is replaced before the others are selected. Blue Green Red White Green Red Green

See image:

a. Find P(negative test result | subject did not lie). b. Find P(subject did not lie | negative test result). c. Compare the results from parts (a) and (b). Are they equal?

See image: C) cutoff = They are not equal.

Three students take equivalent tests of a sense of humor and, after the laughter dies down, their scores are calculated. Which is the highest relative score? a. A score of 144 on a test with a mean of 128 and a standard deviation of 34. b. A score of 90 on a test with a mean of 86 and a standard deviation of 18. c. A score of 18 on a test with a mean of 15 and a standard deviation of 5.

Since .60 is highest z-score, the score of 18 is the best.

The given frequency distribution describes the speed of drivers ticketed by the Town of Poughkeepsie police. These drivers were traveling through a 30 mi/h speed zone on Creek Road. Construct a histogram corresponding to the given frequency distribution. What does the distribution suggest about the enforced speed limit compared to the posted speed limit? (Use packet #13 for table)

Since the smallest class is 42 - 45, already has the driver speeding by more than 10 mph this suggests they only ticket drivers who are excessively speeding. (skewed to the right) (see image for graphs/histogram)

Listed below are the measured weights (in milligrams) of a sample of Bufferin aspirin tablets. Given that this medication should be manufactured in a consistent way so that dosage amounts can be controlled, do the measures of variation seem to indicate that the variation is at an acceptable level? 672.2 679.2 669.8 672.6 672.2 662.2 662.7 661.3 654.2 667.4 667.0 670.7

Stand Dev: S = 6.67 mg Variance: S^2 = 44.47 mg^2 Range: 25.00 mg The range is high for something that should be very consistent and regulated. The variation is at an unacceptable level.

Using the frequency distribution of body temperatures given, construct the corresponding histogram. Using class midpoints, find the mean of the frequency distribution. What does the distribution suggest about the common belief that the average body temperature is 98.6º F? If the subjects are randomly selected, the temperatures should have a distribution that is approximately bell-shaped. Is it? (Use packet #14 for table)

The common belief is the most commonly occuring body temp. But this distribution has a mean a full class lower at 98.2 F. This is skewed to the left, it is not bell-shaped. See image for graphs/tables

Why do we need to know how to find variance?

Variance is an unbiased estimator.