Test 1
Define Statistics
Statistics is the science of collecting, organizing, summarizing, and analyzing information to draw a conclusion and answer questions. In addition, statistics is about providing a measure of confidence in any conclusions.
There are three outfield positions (left field, center field, right field). Given this, how might the graph be misleading when it combines them all in the representation? In comparison to other single positions.
The chart seems to show that one position has many more MVPs because three positions are combined into one. They should be separated.
A(n) ??? is a person or object that is a member of the population being studied.
individual
Which histogram depicts a higher standard deviation?
Histogram b depicts the higher standard deviation, because the distribution has more dispersion. Note: While A's range is only 40 to 60, B's is 30 to 80.
Explain the difference between a single-blind and a double-blind experiment.
In a single-blind experiment, the subject does not know which treatment is received. In a double-blind experiment, neither the subject nor the researcher in contact with the subject knows which treatment is received.
The accompanying histogram represents the total tax collected by a federal tax collection service for fifty regions of varying sizes and populations in a certain country. Explain why the graph is misleading. Although countries can be varied sizes, why are they represented the same?
It does not take into account the size and population of each region.
Match the histograms on the right to the summary statistics given.
Just look at them my dude and guess, if you have time, do math or whateva
Distinguish between nonsampling error and sampling error.
Nonsampling error is the error that results from undercoverage, nonresponse bias, response bias, or data-entry errors. Sampling error is the error that results because a sample is being used to estimate information about a population.
A newspaper article claimed that the afternoon hours were the worst in terms of robberies and provided the graph to the right in support of this claim. Explain how this graph is misleading. The graph has times such as: 6am - 9am 9am - 12pm 12pm - 6pm
Not all of the time intervals are the same size. Redistributing the time interval so they are all the same size may lead to a different shape.
What does it mean when sampling is done without replacement?
Once an individual is selected, the individual cannot be selected again.
Find the population mean or sample mean as indicated. Population: 3, 2, 15, 14, 16
Population mean is represented by μ μ = 10 because (3 + 2 + 15 + 14 + 16)/5 = 10
Find the population variance and standard deviation. 6, 15, 27, 33, 39 a. population variance? b. standard deviation?
Population variance is represented by σ^2 a. σ^2 = 144 1. find the mean = 24 2. in numerator, mean by all the values individually and square that value, and then add those all together 3. denominator is n which is 4. divide numerator by denominator = 144 b. σ = 12 Find the square root of the population variance
What is replication in an experiment?
Replication is applying each treatment to more than one experimental unit.
Find the population mean or sample mean as indicated. Sample: 22, 10, 6, 13, 24
Sample mean is represented as an "x" with a line over it x = 15 because (22 + 10 + 6 + 13 + 24)/5 = 15
Find the sample variance and standard deviation. 8, 53, 11, 50, 37, 22, 34, 27, 28, 29 a. sample variance? b. standard deviation?
Sample variation is represented by s^2 a. s^2 = 212.99 1. find the mean = 29.9 2. in numerator, mean by all the values individually and square that value, and then add those all together 3. denominator is n - 1 which is 10 - 1 = 9 4. divide numerator by denominator = 212.99 b. s = 14.6 Fine the square root of the standard variation
Find the sample variance and standard deviation. 18, 14, 2, 7, 8 a. sample variance? b. standard deviation?
Sample variation is represented by s^2 a. s^2 = 39.2 1. find the mean = 9.8 2. in numerator, mean by all the values individually and square that value, and then add those all together 3. denominator is n - 1 which is 5 - 1 = 4 4. divide numerator by denominator = 39.2 b. s = 6.3 Fine the square root of the standard variation
Nissan wants to administer a satisfaction survey to its current customers. Using their customerdatabase, the company randomly selects 80 customers and asks them about their level of satisfaction with the company.
Simple Random
Determine whether 45% is a parameter or a statistic in this: A study of 42 out of hundreds of people in a dining room showed that 45% of people enjoyed their meal.
Statistic because the data set of 42 people in a dining hall is a sample. Note: A parameter is a numerical summary of a population, which is an entire group to be studied. A statistic is a numerical summary of a sample, which is a subset of the population that is being studied.
The median for the given set of six ordered data values is 32.5. 7, 12, 25, X, 41, 48 What is the missing value?
X = 40 (x3 + x4)/ 2 = 32.5 --> (25 + X)/2 = 32.5 ---> 25 + X = 65 --> X = 40
An insurance company crashed four cars in succession at 5 miles per hour. The cost of repair for each of the four crashes was $429, $469, $402, $228. a. range? b. sample variance? c. standard deviation?
a. $241 because ($469 - $228) = $241 b. 11298 c. $106.29
One year Perry had the lowest ERA (earned-run average, mean number of runs yielded per nine innings pitched) of any male pitcher at his school, with an ERA of 3.13. Also, Sally had the lowest ERA of any female pitcher at the school with an ERA of 2.74. For the males, the mean ERA was 3.966 and the standard deviation was 0.727. For the females, the mean ERA was 4.706 and the standard deviation was 0.812. Find their respective z-scores. Which player had the better year relative to their peers, Perry or Sally? (Note: In general, the lower the ERA, the better the pitcher.) a. Perry had an ERA with a z-score of ??? Sally had an ERA with a z-score ??? b. Which player had a better year in comparison with their peers?
a. - 1.15 and - 2.42 Use the formula: z = (x − μ) / σ Perry: (3.13 - 3.966) / .727 = - 1.15 Sally: (2.74 - 4.706) / .812 = - 2.42 b. Sally had a better year because of a lower z-score.
The following data represent the dividend yields (in percent) of a random sample of 28 publicly traded stocks. 1.36, 0.4, 0.23, 1.19, 0.48, 2.59, 1.71, 1.11, 0.33, 0.66, 0, 0, 3.17, 0.34, 0.38, 3.52, 0.4, 0.51, 2.6, 3.06, 0, 0.48, 0, 0.05, 1.45, 1.91, 2.08, 0.36 a. What is the five number summary? b. Determine the shape
a. 0, .34, .5, 1.81, 3.52 (round 2 decimal places) 1. Reorder: 0, 0, 0, 0, 0.05, 0.23, 0.33, 0.34, 0.36, 0.38, 0.4, 0.4, 0.48, 0.48, 0.51, 0.66, 1.11, 1.19, 1.36, 1.45, 1.71, 1.91, 2.08, 2.59, 2.6, 3.06, 3.17, 3.52 2. Identify, Lowest: 0, Highest: 3.52 and then Median: .495, and Q1: .335, and Q3: 1.81 b. The distribution is skewed to the right. Note: this means the right whisker is longer than the left, and the median is more to the left
a. The traditional course has a range of ???, while the "flipped" course has a range of ???. The ??? course has more dispersion. b. The traditional course has a standard deviation of ???, while the "flipped" course has a standard deviation of ???. The ??? course has more dispersion. c. Suppose the score of 59.0 in the traditional course was incorrectly recorded as 590. How does this affect the range? Standard deviation? d. What property does this illustrate?
a. 29.8 and 29.5 and traditional b. 8.977 and 7.988 and traditional 1. find mean = 2. repeat ( Xi - X)^2 + ... for numerator 3. denominator = n - 1 4. Divide them 5. Square root that value c. 533.3 and 143.651 1. find mean = (change 59 for 590) 2. repeat ( Xi - X)^2 + ... for numerator (change value) 3. denominator = n - 1 4. Divide them 5. Square root that value d. Neither the range nor the standard deviation is resistant. (due to the noticeable value change)
The data represent the age of world leaders on their day of inauguration. Find the five-number summary, and construct a boxplot for the data. Comment on the shape of the distribution. 50, 69, 62, 65, 41, 55, 55, 52, 44, 61, 48, 48, 58, 56 a. What is the five-number summary? b. Shape of the graph?
a. 41, 48, 55, 61, 69 1. 41, 44, 48, 48, 50, 52, 55, 55, 56, 58, 61, 62, 65, 69 2. Identify Lowest: 41 and Highest: 69 3. Find the Median: 55 and then cut in in half to find the median of the upper half (Q1) and lower half (Q3) b. The distribution is roughly symmetric.
In a certain city, the average 20- to 29-year old man is 69.8 inches tall, with a standard deviation of 3.0 inches, while the average 20- to 29-year old woman is 64.1 inches tall, with a standard deviation of 3.8 inches. Who is relatively taller, a 75-inch man or a 70-inch woman? Find the corresponding z-scores. Who is relatively taller, a 75-inch man or a 70-inch woman? The z-score for the man, ???, is larger than the z-score for the woman, ???, so he is relatively taller.
1.73 and 1.55 Use the formula: z = (x − μ) / σ Man: (75 - 69.8) / 3 = 1.73 Woman: (70 - 64.1) / 3.8 = 1.55
For a large sporting event the broadcasters sold 66 ad slots for a total revenue of $154 million. What was the mean price per ad slot?
2.3 because (154/66) = 2.333
The percent change in the stock price of the company from January 2015 to December 2015 was ??? Jan: 28.54 and Dec: 37.43
31.1% because (28.54 x 37.43)/28.54
Suppose babies born after a gestation period of 32 to 35 weeks have a mean weight of 2700 grams and a standard deviation of 900 grams while babies born after a gestation period of 40 weeks have a mean weight of 3200 grams and a standard deviation of 470 grams. If a 34-week gestation period baby weighs 2475 grams and a 40-week gestation period baby weighs 2975 grams, find the corresponding z-scores. Which baby weighs less relative to the gestation period? Find the corresponding z-scores. Which baby weighs relatively less? The baby born in week ??? weighs relatively less since its z-score, ??? is smaller than the z-score of ??? for the baby born in week 34.
40 and -0.48 and -0.25 Use the formula: z = (x − μ) / σ 34-week: (2475 - 2700) / 900 = - 0.25 40 week: (2975 - 3200) / 470 = - 0.48 - As the magnitude of the z-score increases, the relative difference of the observation from the mean increases. - The baby with the larger z-score weighs relatively more, and the baby with the smaller z-score weighs relatively less.
What does it mean when a part of the population is under-represented?
A part of the population is under-represented when it is proportionally smaller in a sample than in its population.
Explain the difference between a population and a sample
A population is the entire group that is being studied while a sample is a subset of the population that is being studied.
Define simple random sampling
A sample of size n from a population of size N is obtained through simple random sampling if every possible sample of size n has an equally likely chance of occurring. The sample is then called a simple random sample.
For IQ scores of students, state whether you would expect a histogram of the data to be bell-shaped, uniform, skewed left, or skewed right.
Bell Shaped
To determine customer opinion of their pricing, Amtrak randomly selects 80 trains during a certain week and surveys all passengers on the trains.
Cluster Note: A cluster sample is obtained by selecting individuals within a randomly selected group of individuals.
The ??? is the difference between consecutive lower class limits.
class width
True or False? The shape of the distribution shown is best classified as skewed left. Description: Peak is at the left and it decreases as you move to the right.
False, it is skewed right because the tail decreases in that direction.
Determine whether the following statement is true or false. Explain: When obtaining a stratified sample, the number of individuals included within each stratum must be equal.
False. Within stratified samples, the number of individuals sampled from each stratum should be proportional to the size of the strata in the population.
After giving a statistics exam, Professor Dang determined the following five-number summary for her class results: 58, 66, 75, 87, 96 Use this information to draw a boxplot of the exam scores.
The correct graph is the one which has the line beginning at 58, a verticle line at 66, then 75, and again at 87, and the line ends at 96. (66, 75, 87 make up the box, while 58 is the left whisker and 96 is the right one)
The safety manager at Bumbler Enterprises provides the graph shown on the right to the plant manager and claims that the rate of worker injuries has been reduced by 67% over a 12-year period. Does the graph support his claim? Explain.
The graph does not support his claim. The vertical scale does not start at 0 which distorts the percent of change.
Comment on any trends
The percentage of high school graduates who enrolled in college has generally increased, though there have been some down years. Note: Although there are many falls in the graph, overall it increases
In a relative frequency distribution, what should the relative frequencies add up to?
The relative frequencies should add up to 1
Is the statement below true or false? There is not one particular frequency distribution that is correct, but there are frequency distributions that are less desirable than others.
The statement is true. Any correctly constructed frequency distribution is valid. However, some choices for the categories or classes give more information about the shape of the distribution.
Determine whether the study depicts an observational study or an experiment: Fourth-grade students are randomly divided into two groups. One group is taught English using traditional techniques. The other is taught English using a reform method. After 1 year, each group is given an achievement test to compare its proficiency with that of the other group.
The study is an experiment because the researchers control one variable to determine the effect on the response variable.
??? are the categories by which data are grouped.
classes
Determine whether the study depicts an observational study or an experiment: Office workers are randomly divided into two groups. One group takes meditation breaks throughout the day; the other takes 10-minute walks every 2 hours. After 1 month, each group is given a stress test to compare stress levels.
The study is an experiment because the researchers control one variable to determine the effect on the response variable.
Determine whether the study depicts an observational study or an experiment: A study is conducted to determine if there is a relationship between Parkinson's disease and childhood head trauma. Doctors look at the hospital records for patients with
The study is an observational study because the study examines individuals in a sample, but does not try to influence the response variable
Determine whether the value is a parameter or a statistic. The average age of men who had walked on the moon was 39 years, 11 months, 15 days.
The value is a parameter because the men who had walked on the moon are a population.
Determine whether 23% is a parameter or a statistic in this: A study of 6,076 adults in public rest rooms found that 23% did not wash their hands before exiting.
The value is a statistic because the 6,076 adults in public rest rooms are a sample. Note: A parameter is a numerical summary of a population, which is an entire group to be studied. A statistic is a numerical summary of a sample, which is a subset of the population that is being studied.
Determine whether the quantitative variable is discrete or continuous: Distance an athlete can jump
The variable is continuous because it is not countable. Note: the number can change and vary from person to person
Determine whether the quantitative variable is discrete or continuous: goals scored by a player in a soccer season
The variable is discrete because it is countable. Note: A discrete variable is a quantitative variable that has either a finite number of possible values or a countable number of possible values. A continuous variable is a quantitative variable that has an infinite number of possible values that are not countable.
Determine whether the quantitative variable is discrete or continuous: Passes made by a defenseman in a hockey game
The variable is discrete because it is countable. Note: In a single hockey game, it is a fixed and countable number.
Determine whether the variable is qualitative or quantitative: eye color
The variable is qualitative because it is an attribute characteristic.
Determine whether the variable is qualitative or quantitative: weight
The variable is quantitative because it is a numerical measure.
The graph is as describes: Depends on the Situation: 10% Morally Wrong: 20% Morally Acceptable: 70% a. If there are 211 million adults in the country, how many believe that the action is morally wrong? b. If a polling organization claimed that the results of the survey indicate that 9% of adults in the country believe that the action is acceptable in certain situations, would you say this statement is descriptive or inferential? Why? The statement is ??? because it ???
a. 42 million because (211 x .20) = 42 b. inferential and makes a prediction Note: Descriptive statements give information that is known. Inferential statements use known information to make predictions about unknown things that are related.
Speed (km/hr):Number of Players 10-14.9: 5 15-19.9: 10 20-24.9: 38 25-29.9: 308 30-34.9: 225 a. There are ??? classes. b. The lower class limit for the second class is ??? c. The upper class limit for the second class is ??? d. The class width is ???
a. 5 b. 15 c. 19.9 d. 5
a. The median of variable x is ??? b. The third quartile of variable y is ??? c. Which variable has more dispersion? Why? d. Describe the shape of the variable x. Support your position. e. Describe the shape of the variable y. Support your position. Note: In a boxplot, vertical lines are drawn at Q1, M, and Q3. The central vertical line is drawn at the value of the median.
a. 80 because that is where the line in the middle of the box is b. 100 because the third quartile variable is where the box ends c. Variable y—the interquartile range of variable y is larger than that of variable x. d. Symmetric—the median is the center of the box and the left and right whiskers are about the same length. e. Skewed left—the median is right of center in the box and the left whisker is longer than the right whisker.
a. What was the most frequent and least frequent outcome? b. Determine the percentage of time an 8 was observed.
a. 9 and 12 Note: look at the highest and lowest bar b. 17% Note: count up the frequency of 8 and divide that by the total frequency times 100.
a. What is a bar graph? b. What is a Pareto chart?
a. A bar graph is a horizontal or vertical representation of the frequency or relative frequency of the categories. The height of each rectangle represents the category's frequency or relative frequency. b. A Pareto chart is a bar graph whose bars are drawn in decreasing order of frequency or relative frequency.
a. What does it mean when an observational study is retrospective? b. What does it mean when an observational study is prospective?
a. A retrospective study requires that individuals look back in time or require the researcher to look at existing records. b. A prospective study collects the data over time.
A quality-control manager randomly selects 30 bottles of motor oil that were filled on April 25 to assess the calibration of the filling machine. a. What is the population in the study? b. What is the sample in the study?
a. All bottles of motor oil produced in the plant on April 25. b. The 30 bottles of motor oil selected in the plant on April 25.
a. What is an observational study? b. What is an designed experiment? c. ??? allows the researcher to claim causation between an explanatory variable and a response variable.
a. An observational study measures the value of the response variable without attempting to influence the value of either the response or explanatory variables. b. A designed experiment is when a researcher assigns individuals to a certain group, intentionally changing the value of an explanatory variable, and then recording the value of the response variable for each group. c. A designed experiment
Class: Frequencies 61-64: 34 65-67: 68 68-69: 200 70: 198 71-72: 121 73-76: 87 77-80: 48 a. Find the Class Midpoints b. Sample Mean? c. Standard Deviation?
a. Class Midpoints 61-64: 63 65-67: 66.5 68-69: 69 70: 70.5 71-72: 72 73-76: 75 77-80: 79 Note: Find the midpoint of each value by adding the two consecutive lower class levels and dividing it by 2 (ie. 70 + 71 / 2 = 70.5 or 77 + 81 / 2 = 79) b. sample mean: 70.7 c. standard deviation: 3.5 Note: repeat ( Xi - X)^2(frequency)+ ... for numerator and denominator is still n - 1, divide, and square root
a. What is meant by confounding? b. What is a lurking variable? c. What is a confounding variable?
a. Confounding in a study occurs when the effects of two or more explanatory variables are not separated. Therefore, any relation that may exist between an explanatory variable and the response variable may be due to some other variable or variables not accounted for in the study. b. A lurking variable is an explanatory variable that was not considered in a study, but that affects the value of the response variable in the study. In addition, lurking variables are typically related to explanatory variables in the study. c. A confounding variable is an explanatory variable that was considered in a study whose effect cannot be distinguished from a second explanatory variable in the study
a. What is a cross-sectional study? b. What is a case-control study? c. Which is the superior observational study?
a. Cross-sectional studies are observational studies that collect information about individuals at a specific point in time or over a very short period of time. b. Case-control studies are observational studies that are retrospective, meaning that they require individuals to look back in time or require the researcher to look at existing records. c. Neither study is always the superior to the other. Both have advantages and disadvantages that depend on the situation.
Number: Frequency 0: 2 1: 13 2: 14 3: 7 4: 3 5: 1 a. Frequencies? b. What percentage of households in the survey have three televisions? c. What percentage of households in the survey have four or more televisions? d. The distribution is skewed ???
a. Frequencies: 0: .05 1: .325 2: .35 3: .175 4: .075 5: .025 b. 17.5% because (.175 x 100) c. 10% because (.075 + .025)(100) d. right because it decreases as you move right
Number of Children Under 5: Number of Households 0: 16 1: 13 2: 14 3: 5 4: 2 a. Relative Frequencies? b. What percentage of households has 2 children under the age of 5? c. What percentage of households has one or two children under the age of 5?
a. Frequencies: 0: .32 1: .26 2: .28 3: .1 4: .04 b. 28% Note: 14/50 = .28(100) = 28% c. 54% Note: 13 + 14/50 = .54(100) = 54%
Las Vegas: 1157 Orlando: 1064 New York: 889 Chicago: 682 San Diego: 917 a. Construct a relative frequency distribution of the data. b. What proportion of the tickets were for New York? c. A local news broadcast reported that 19.5% of tickets purchased from the airline are for flights to San Diego. What is wrong with this statement?
a. Frequencies: Las Vegas: .246 Orlando: .226 New York: .189 Chicago: .145 San Diego: .195 b. For New York: .189 c. No level of confidence is provided along with the estimate.
Never: 280 Rarely: 506 Sometimes: 920 Most of the time: 471 Always: 62 a. Relative Frequency of the Data? b. What percentage of respondents answered "Always"? c. What percentage of respondents answered "Never" or "Rarely"? d. Suppose a person claims that, "2.8% of all people in the nation always eat out." Is this a descriptive or inferential statement?
a. Frequencies: Never: .125 Rarely: .226 Sometimes: .411 Most of the time: .21 Always: .028 b. 2.8% c. 35.1% d. Inferential
a. How is the graph misleading? b. What could be done to improve the graphic?
a. It looks like twice as much money is spent on wages than products. Note: Although products is 24% and wages is 31%, the bar for wages is double the products bar. b. The bars should be more proportional to their percentages.
As part of a college literature course, students must select three classic works of literature from the provided list and complete critical book reviews for each selected work. Write a short description of the processes that can be used to generate a simple random sample of three books. Obtain a simple random sample of size 3 from this list. 1. Huckleberry Finn 2. As I Lay Dying 3. Death of a Salesman 4. The Scarlet Letter 5. The Jungle 6. A Tale of Two Cities 7. Pride and Prejudice 8. The Sun Also Rises 9. Crime and Punishment a. Which of the following would produce a simple random sample? b. Use the portion of the random number table provided below to obtain a simple random sample of size 3 from this list. If you start on the left and take the first three numbers between 1 and 9, what three books would be selected from the numbered list? 86886 30695 66517
a. List each book on a separate piece of paper, place them all in a hat, and pick three. Number the books from 1 to 9 and use a random number table to produce 3 different one digit numbers corresponding to the books selected. b. The Sun Also Rises, A Tale of Two Cities, and Death of a Salesman Note: only count numbers which do not repeat
Speed (km/hr): Number of Players 10-13.9: 4 14-17.9: 6 18-21.9: 15 22-25.9: 75 26-29.9: 279 30-33.9: 235 a. Construct a relative frequency distribution. b. The percentage of players that had a top speed between 14 and 17.9 km/h was ??? c. The percentage of players that had a top speed less than 13.9 km/h was ???
a. Relative Frequencies: 10-13.9: .0065 14-17.9: .0098 18-21.9: .0244 22-25.9: .1221 26-29.9: .4544 30-33.9: .3827 b. .98% because (.0098 x 100) c. .65% because (.0065 x 100)
The owner of a shopping mall wishes to expand the number of shops available in the food court. He has a market researcher survey the first 110 customers who come into the food court during weekday evenings to determine what types of food the shoppers would like to see added to the food court. Complete parts (a) and (b) below. a. The survey has bias. Determine whether the flaw is due to the sampling method or the survey itself. For biased surveys, identify the cause of the error. What is the cause of the bias? b. Suggest a remedy to the problem. Which of the following is the best way to remedy this problem?
a. Sampling Bias b. Ask customers throughout the day on both weekdays and weekends.
A polling organization contacts 2294 teenagers who are 13 to 17 years of age and live in the United States and asks whether or not they had attended a concert this past year. a. What is the population in the study? b. What is the sample in the study?
a. Teenagers who are 13 to 17 years of age and live in the United b. The 2294 teenagers who are 13 to 17 years of age and live in the United
Description: low is 0, high is 20, middle is 10 and the blue box starts at 6 and goes to 14. a. Identify the shape of the distribution b. determine the five-number summary.
a. The distribution is roughly symmetric b. 0, 6, 10, 14, 20 Note: 0 represents the start, 6 represents where the box starts, 10 is the line in the middle of the box, 14 represents where the box ends, and 20 is where the line ends.
The accompanying data represent health care expenditures per capita as a percentage of the gross domestic product of a country from 2009 to 2015. Gross domestic product is the total value of all goods and services created during the course of the year. a. Construct a time-series plot that a politician would create to support the position that health care expenditures are increasing and must be slowed. b. Construct a time-series plot that the health care industry would create to refute the opinion of the politician. c. Explain how different measures may be used to support two completely different positions.
a. The graph chosen has a y-axis that starts at 10,000 and obviously increases at a steady rate b. This graph starts at 0 and remains pretty consistent/flat. c. The scales used in the graph can significantly affect the message. Also, the variable used to convey the message on the graph can make a large difference as well.
Determine whether the variable is qualitative or quantitative: a. Address b. Distance in miles from one city to another
a. The variable is qualitative because it is an attribute characteristic. b. The variable is quantitative because it is a numerical measure. Note: Qualitative variables allow for classification of individuals based on some attribute or characteristic. Quantitative variables provide numerical measures of individuals, and can be added or subtracted and provide meaningful results.
a. How is the bar graph misleading? What does the graph seem to convey? b. What does the new graph seem to convey?
a. The vertical axis starts at 34,500 instead of 0. This tends to indicate that the median earnings for females changed at a faster rate than it actually did. b. This graph indicates that median earnings for females have remained fairly constant over the given time period.
The following data represent the high-temperature distribution for a summer month in a city for some of the last 130 years. Treat the data as a population. Temperature: Days 50-59: 2 60-69: 313 70-79: 1413 80-89: 1530 90-99: 478 100-109: 14 a. Mean? b. Standard Deviation? c. According to the empirical rule, 95% of days in the month will be between what two temperatures?
a. mean (μ): 80.9 1. Find the midpoints 2. Numerator = midpoint x frequencies added up 3. Denominator = frequencies added up b. Standard Deviation (σ): 8.3 1. Numberator = midpoint - mean (frequency) added up 2. Denominator = frequencies added up 3. divided and square rooted c. 64.3 and 97.5 If a distribution is roughly bell shaped than the empirical rule states the following. 68% of the data lie between μ−1σ and μ+1σ. 95% of the data lie between μ−2σ and μ+2σ. 99.7% of the data lie between μ−3σ and μ−3σ. Therefore, use μ−2σ and μ+2σ. μ−2σ --> 80.9 - 2(8.3) = 64.3 μ+2σ --> 80.9 + 2(8.3) = 97.5
An insurance company crashed four cars of the same model at 5 miles per hour. The costs of repair for each of the four crashes were $419, $441, $483, and $230. a. mean? b. median? c. mode?
a. mean: $393.25 because (419 + 441 + 483 + 230)/4 = 393.25 b. median: $430 reorder 230, 419, 441, 483 --> (419 + 441)/2 = 430 c. mode: does not exist no numbers repeat
A concrete mix is designed to withstand 3000 pounds per square inch (psi) of pressure. The following data represent the strength of nine randomly selected casts (in psi). 3950, 4100, 3100, 3000, 2940, 3830, 4100, 4050, 3420 a. mean? b. median? c. mode
a. mean: 3610 b. median: 3830 reorder to fine 3830 in the middle 2940, 3000, 3100, 3420, 3830, 3950, 4050, 4100, 4100 c. mode: 4100 repeated twice
The following data represent the amount of time (in minutes) a random sample of eight students took to complete the online portion of an exam in a particular statistics course. 64.6, 71.5, 82.9, 111.8, 128.4, 93.1, 94.7, 124.2 a. mean? b. median? c. mode?
a. mean: 96.4 (64.6 + 71.5 + 82.9 + 111.8 + 128.4 + 93.1 + 94.7 + 124.2)/8 b. median: reorder 64.6, 71.5, 82.9, 93.1, 94.7, 111.8, 128.4, 124.2 --> (93.1 + 94.7)/2 = 93.9 c. mode: does not exist
The frequency distribution was obtained using a class width of 0.5 for data on cigarette tax rates. Use the frequency distribution to approximate the population mean and population standard deviation. Compare these results to the actual mean μ=$1.623 and standard deviation σ= $1.044. Tax Rate: Frequency 0.00-0.49: 6 0.50-0.99: 13 1.00-1.49: 7 1.50-1.99: 6 2.00-2.49: 6 2.50-2.99: 6 3.00-3.49: 3 3.50-3.99: 3 4.00-4.49: 1 a. Population Mean? b. Standard Deviation? c. Compare these results to the values found using the actual data.
a. population mean: 1.681 1. Numerator = midpoints x frequencies + ...added up 2. Denominator = frequencies added up b. standard deviation: 1.089 1. Numerator = (midpoint - mean)^2(frequency) +... 2. Denominator = frequencies added up 3. Square Root that c. The grouped values are both slightly larger.
Compute the range and sample standard deviation for strength of the concrete (in psi). 3960, 4080, 3200, 3200, 2960, 3870, 4080, 4060 a. range? b. standard deviation?
a. range: 1120 b. standard deviation: 471.8 1. find mean = 3676.25 2. repeat (3960 - 3676.25)^2 + ... for numerator 3. denominator = n - 1 which is 8 - 1 = 7 4. Divide = 222569.6429 5. Square root = 471.8
The following data represent the flight time (in minutes) of a random sample of seven flights from one city to another city. 285, 271, 261, 267, 258, 264, 270 a. range? b. standard deviation?
a. range: 27 b. standard deviation: 1. find mean = 268 2. repeat (285 - 268)^2 + ... for numerator 3. denominator = n - 1 which is 7 - 1 = 6 4. Divide = 77.667 5. Square root = 8.8
Recently, a random sample of 13-18 year olds was asked, "How much do you currently have in savings?" The data in the table represent the responses to the survey. Approximate the mean and standard deviation amount of savings. Savings: Frequency $0-$199: 337 $200-$399: 99 $400-$599: 53 $600-$799: 19 $800-$999: 12 $1000-$1199: 7 $1200-$1399: 4 a. Sample Mean? b. Sample Standard Deviation?
a. sample mean: $239 1. Find the midpoint of each value by adding the two consecutive lower class levels and dividing it by 2 (ie. 0 + 200 / 2 = 100 or 800 + 1000 / 2 = 900) 2. Multiply all those midpoints by their individual frequency and then add all those values 3. Divide it by all the frequency values added up 4. 14100/59 = $239 b. standard deviation: 236 1. find sample mean = $239 2. repeat ( Xi - X)^2(frequency)+ ... for numerator (ie. (300 - 239)^2(53) + (500 - 239)^2(19)...) 3. denominator = n - 1 which is 531 - 1 = 530 4. Divide the numerator and denominator 5. Square root that value
The accompanying frequency distribution represents the square footage of a random sample of 500 houses that are owner occupied year round. Approximate the mean and standard deviation square footage. Square footage: Frequency 0−499: 9 500−999: 13 1,000−1,499: 33 1,500−1,999: 121 2,000−2,499: 119 2,500−2,999: 83 3,000−3,499: 45 3,500−3,999: 41 4,000−4,499: 26 4,500−4,999: 10 a. Mean Square Footage? b.Standard Deviation?
a. sample mean: 2438 1. Find the midpoint of each value by adding the two consecutive lower class levels and dividing it by 2 (ie. 500 + 1000 / 2 = 750 and 2000 + 2500 / 2 = 2250) 2. Multiply all those midpoints by their individual frequency and then add all those values 3. Divide it by all the frequency values added up b. standard deviation: 934 1. find sample mean = 2438 2. repeat ( Xi - X)^2(frequency)+ ... for numerator (ie. (2500 - 2438)^2(9) + (7500 - 2438)^2(13)...) 3. denominator = n - 1 which is 500 - 1 = 499 4. Divide the numerator and denominator 5. Square root that value
The following data represent the number of people aged 25 to 64 years covered by health insurance (private or government) in 2018. Approximate the mean and standard deviation for age. Age: Number (millions) 25-34: 26.7 35-44: 31.2 45-54: 39.4 55-64: 25.1
a. sample mean: 45.14 1. Find the midpoint of each value by adding the two consecutive lower class levels and dividing it by 2 (ie. 25 + 35 / 2 = 30 and 45 + 55 / 2 = 50) 2. Multiply all those midpoints by their individual frequency and then add all those values 3. Divide it by all the frequency values added up b. standard deviation: 10.47 1. find sample mean = 45.14 2. repeat ( Xi - X)^2(frequency)+ ... for numerator (ie. (30 - 45.14)^2(26.7 + (40-45.14)^2(31.2)...) 3. denominator = n (added frequencies, do not subtract by 1 in populations!) 4. Divide the numerator and denominator 5. Square root that value
The ??? class limit is the smallest value within the class and the ??? class limit is the largest value within the class.
lower and upper
A frequency distribution lists the ??? of occurrences of each category of data, while a relative frequency distribution lists the ??? of occurrences of each category of data.
number proportion
In a boxplot, if the median is to the left of the center of the box and the right whisker is substantially longer than the left whisker, the distribution is skewed ???
right Note: the longer side in which the numbers dwindle is the direction it is skewed "right whisker is substantially longer"
To estimate the percentage of defects in a recent manufacturing batch, a quality control manager at Microsoft selects every 10th software CD that comes off the assembly line starting with the third until she obtains a sample of 50 software CDs.
systematic Note: A systematic sample is obtained by selecting every kth individual from the population (the first individual selected is a random number from 1 to k). There are many different types of sampling: A simple random sample has each possible sample of a given size is equally likely to occur. A stratified sample is obtained by separating the population into nonoverlapping groups and then obtaining a simple random sample from each group. A convenience sample is a sample in which the individuals are easily obtained.
Determine whether the following statement is true or false. Explain: Inferences based on voluntary response samples are generally not reliable.
True, because it is often the case that the individuals who volunteer do not accurately represent the population.
Determine whether the following statement is true or false. Explain: When conducting a cluster sample, it is better to have fewer clusters with more individuals when the clusters are heterogeneous.
True, because when the clusters are heterogeneous, they are scaled down versions of the population.
