statistics exam #1
How many ways can a committee of four students be selected from a 15-member club?
15! / (15 - 4)! 4! = 15! / (11! x 4!). Combination because order does not matter.
how many guests were at least 26 but less than 30 years old?
15. Look at the frequency for 26 up to 30
The monthly closing stock prices (rounded to the nearest dollar) for Starbucks Corp. (x) and Panera Bread Co. (y) for the first six months of 2016 are reported in the following table. Panera mean 194 215.11 207 2.78 205 13.44 215 40.11 219 106.78 212 11.11 Total 389.33 calculate the sample mean for Panera
194 + 207 + 205 + 215 + 219 + 212 = 1252 / 6 = 208.67
Consider the following data: 1, 2, 4, 5, 10, 12, 18. The 30th percentile is the closest to ________.
2.8. First arrange the data in ascending order. Then find the approximate position of the percentile using the formula: Lp = (n + 1) P/100 = (7 + 1) 30/ 100= 2.4 we use 7 because there are 7 values in the data set. The location of 2.4 tells us that the percentile is between the 2nd and 3rd number. Calculate by taking the 2nd number in the data set + 0.4 x the 3rd number in the data set - the 2nd number in the data set: 2 + 0.4 x (4 - 2) = 2 + 0.4 x 2 = 2 + 0.8 = 2.8
considering the following frequency distribution: class frequency 12 up to 15 3 15 up to 18 6 18 up to 21 3 21 up to 24. 4 24 up to 27 4 the total number of observations in the frequency distribution is...
20. sum the frequency column to obtain the total number of observations in the frequency distribution
what percent of guests were at least 22 but less than 26 years old?
25%. Look at the relative frequency for 22 up to 26
Students in professor Smith's business statistics course have evaluated the overall effectiveness of the professor's instruction on a five-point scale, where a score of 1 indicates very poor performance and a score of 5 indicates outstanding performance. The raw scores are displayed in the accompanying table: 1 4 4 5 5 3 4 3 4 1 5 5 4 4 2 3 3 2 3 3 4 5 5 5 5 3 2 3 3 2 What is the most common score given in the evaluations?
3. Three occurred nine times and the second-most frequent number was 5 with eight occurrences
How many project teams composed of five students can be created out of a class of 10 students, if each of the five students is assigned a specific position in each group such as president, vice president, secretary, treasurer and social coordinator?
30,240. order matters in this scenario. The formula for permutations is n! / (n-x)! = 10! / (10-5)! = 10! / 5! = 10 x 9 x 8 x 7 x 6 x 5! / 5! 10 x 9 x 8 x 7 x 6 = 30,240
A small company that manufactures juggling equipment makes seven different types of clubs. The company wants to start an ad campaign that emphasizes the myriad combinations the avid juggler can create with the company's clubs. If a juggler wishes to juggle four clubs, each of a different type, how many different combinations of the company's clubs can he or she make?
35. use combinations because order does not matter. 7! / (7-4)! 4!= 7! / 3! 4! = (7 x 6 x 5 x 4!) / 3! x 4! = (7 x 6 x 5) / 3 x 2 x 1 = 210/ 6= 35
Amounts spent by a sample of 200 customers at a retail store are summarized in the following relative frequency distribution. Amount spent (in $) Frequency 0 up to 10 15 10 up to 20 75 20 up to 30 55 30 up to 40 55 The mean amount spent by customers is the closest to _________.
$22.50. first, find midpoint of each interval: (0+10)/2 = 5, (10+20)/2=15, (20+30)/2=25, 30+40)/2=35. Next, multiply each midpoint by frequency given in table and divide by total population (5 x 15) + (15 x 75) + (25 x 55) + (35 x 55) / 200= 4500 / 200= 22.50
a city in California spent $6 million repairing damage to its public buildings in Year 1. The following table shows the categories where the money was directed. Cause: Percent termites 22% water damage 6% mold 12% earthquake 27% other 33% How much more did the city spend to fix damage caused by termites compared to the damage caused by water?
$960,000. The city spent 22% on termite damage and 6% on water damage. The difference is 16%. The total dollar value spent on the difference is 16% of $6 million- 6,000,000 x 0.16= 960,000
The covariance between the returns of A and B is -0.112. The standard deviation of the rates of return is 0.26 for stock A and 0.81 for stock B. The correlation of the rates of return between A and B is the closest to ________.
-0.53. -0.112/(0.26*0.81)=-0.53
The odds against winning $1.00 in the lottery are 19 to 1. What is the probability of winning $1.00 in the lottery?
0.05. Given odds against event A occurring of "a to b," the probability of A is b/(a +b). Probability against 19 to 1 odds = 1 / (19 + 1) = 1/20 = 0.05
The following probability table shows probabilities concerning Favorite Subject and Gender. What is the probability of selecting an individual preferring science if she is female? Male: math=0.200 english=0.050 science=0.175 Total= 0.425 Female: math= 0.100 english=0.325 science= 0.150 Total= 0.575 Total: math=0.300 english= 0.375 science= 0.325 total=1.000
0.2609. The contingency table shows frequencies for two qualitative or categorical variables, x and y, where each cell represents a mutually exclusive combination of the pair of x and y values. A more convenient way of calculating relevant probabilities is to convert the contingency table to a joint probability table. P(Science|Female)= 0.150 / 0.575 = 0.2609
The following probability table shows probabilities concerning Favorite Subject and Gender. What is the probability of selecting an individual who is a female or prefers science? Male: math=0.200 english=0.050 science=0.175 Total= 0.425 Female: math= 0.100 english=0.325 science= 0.150 Total= 0.575 Total: math=0.300 english= 0.375 science= 0.325 total=1.000
0.750. P(female U Science)= 0.575 + 0.325 - 0.150 = 0.750
Sales for Adidas grew at a rate of 0.5196 in Year 1, 0.0213 in Year 2, 0.0485 in Year 3, and -0.0387 in Year 4. The average growth rate for Adidas during these four years is the closest to ________.
11.83%
Consider a population with data values of 12 8 28 22 12 30 14 The median is ________.
14. first arrange data in an ascending order. (8, 12, 12, 14, 22, 28, 30). The median is the middle number
The monthly closing stock prices (rounded to the nearest dollar) for Starbucks Corp. (x) and Panera Bread Co. (y) for the first six months of 2016 are reported in the following table. Starbucks mean 61 8.03 58 0.03 60 3.36 55 10.03 57 1.36 58 0.03 Total 22.83 Calculate the sample mean for starbucks
add up the first of values and divide by how many values in that column. 61 + 58 + 60 + 55 + 57 + 58 = 349 / 6 = 58.17
The manager of a nightclub near a local university recorded the ages of the last 100 guests in the following cumulative frequency distribution. Ages: Cumulative Frequency 18 up to 22 45 22 up to 26 70 26 up to 30 85 30 up to 34 96 34 up to 38 100 construct the frequency distribution
ages: frequency 18 up to 22 45 22 up to 26 70-45=25 26 up to 30 85-70=15 30 up to 34 96-85=11 34 up to 38 100-96=4
which of the following is a quantitative variable
all choices (house age, house size, house price). a quantitative variable assumes meaningful numerical values
a population consists of
all items of interest in a statistical problem
The following data concern a sample of employees of the U.S. Marshalls in the state of New York. Identify the qualitative and quantitative variables, the categories associated with each qualitative variable, and the measurement scales for all variables. what type of graphs are most appropriate to describe values of the variable "grade"
bar chart or pie chart since they are for one qualitative variable
how do we find the median if the number of observations in a data set is odd?
by taking the middle value in the sorted data set. If the number of observations is odd, the median is the middle value in the sorted data set
the two branches of the study of statistics are generally referred to as
descriptive and inferential statistics
which of the following variables is qualitative
gender. values corresponding to a qualitative variable are typically expressed in words
the Fahrenheit scale for measuring temperature would be classified as a(n)
interval scale. Zero in Fahrenheit in degrees does not mean "no temperature." We cannot say, for example, that today is twice as warm as six months ago, which characterizes the ratio scale
For k > 1, Chebyshev's theorem is useful in estimating the proportion of observations that fall within ________.
k standard deviations from the mean. For any data set and k > 1, at least (1 - 1/k^2) 100% of observations lie within k standard deviations from the mean
The table below gives the deviations of a portfolio's annual total returns from its benchmark's annual returns, for a six year period ending in Year 6. Year 1 -7.62% Year 2 2.37% Year 3 -9.11% Year 4 0.55% Year 5 5.48% Year 6 -1.67% The arithmetic mean return and median return are closest to...
mean = -1.67% & median = -0.56%. mean is calculated as -7.62 + 2.37 - 9.11 + 0.55 + 5.48 - 1.67 / 6 = -1.67 median is calculated by first arranging the data in ascending order (smallest to largest): -9.11, -7.62, -1.67, 0.55, 2.37, 5.48. The median is the middle value, if there is an odd number of values. Here, there is an even number of values, so we take the average of the two values in the middle: -167 & 0.55. -1.67 + 0.55 / 2 = -0.56
the mode is defined as the
most frequent value in a data set. There may be more than one mode. Mode is most useful when describing the most frequently found value in a set of qualitative data
a recent survey of 200 small firms (annual revenue less than $10 million) asked whether an increase in the minimum wage would cause the firm to decrease capital spending. Possible responses to the survey question were: "yes", "no", or "don't know." This data is best classified as
nominal scale. with nominal data all we can do is categorize or group the data
which of the following scales represents the least sophisticated level of measurement
nominal. the nominal scale represents the less sophisticated level of measurement
A respondent of a survey is asked whether the Philadelphia Flyers' performance in the last game was excellent, good, fair, or poor. The person indicates that the performance was "good." This is an example of
ordinal data. The ordinal scale data can be categorized and ranked
positive curve-linear relationship
points curve upward
which of the following is an example of time series data
quarterly housing starts collected over the last 60 years. Time series data refers to data collected by recording a characteristic of a subject over several time periods
The manager of a nightclub near a local university recorded the ages of the last 100 guests in the following cumulative frequency distribution. Ages: Cumulative Frequency 18 up to 22 45 22 up to 26 70 26 up to 30 85 30 up to 34 96 34 up to 38 100 construct the relative frequency distribution
relative frequency = frequency / total frequency. ages: rel freq. 18 up t0 22 45/100=0.45 22 up to 36 25/100=0.25 26 up to 30 15/100=0.15 30 up to 34 11/100=0.11 34 up to 38 4/100=0.04
which of the following represents a population and a sample from that population
residents of Albany, New York, and registered voters in Albany, New York. The registered voters in Albany are clearly a subset of the residents of Albany
The following data concern a sample of employees of the U.S. Marshalls in the state of New York. Identify the qualitative and quantitative variables, the categories associated with each qualitative variable, and the measurement scales for all variables. which variable has the strongest scale of measurement?
salary, because it is ratio scale and can allow all arithmetic operations
The following data concern a sample of employees of the U.S. Marshalls in the state of New York. Identify the qualitative and quantitative variables, the categories associated with each qualitative variable, and the measurement scales for all variables. Identify the quantitative variable from the above data
salary, because it is the only value that assumes meaningful numerical values
Calculate the sample variance and the sample standard deviation for Panera's stock price.
sample var: 389.33 / (6-1) = 389.33/5 = 77.87 standard dev: 8.82
no relationship
scattered points
negative linear relationship
see graph
positive linear relationship
see graph
What is an advantage of the correlation coefficient over the covariance?
that it falls between -1 and 1 and that it is a unit-free measure, assuming the values from the interval [-1, 1]
a company wants to estimate the mean price of oil over the past 10 years. What kind of data does the company need?
time series data. time series data refers to data collected by recording a characteristic of a subject over several time periods
an analyst gathered the following information about the net profit margins of companies in two industries (Industry A & B): Industry A: mean = 15.0% standard deviation = 2.0% Range = 10.0% Industry B: mean = 5.0% standard deviation = 0.8% range = 15% Compared with the other industry, the relative dispersion of net profit margins is smaller for Industry ________.
A, because it has a smaller coefficient of variation. We use the coefficient of variation to measure relative dispersion. Coefficient of Variance is CV = standard deviation / mean. Industry A: CV= 2.0/15.0= 0.1333 Industry B: CV = 0.8/5.0= 0.16 0.13 is smaller than 0.16.
T/F: 0! = 0
F. 0! = 1
T/F: combinations are used when the order in which different objects are arranged matters
F. If the order in which objects are arranged does not matter, we should use combinations
T/F: a discrete variable cannot assume an infinite number of values
F. a continuous variable is characterized by uncountable values within an interval, while discrete variables assume a countable number of values
T/F: a pie chart is a segmented circle that portrays the categories and relative sizes of some quantitative variable
F. a pie chart is a segmented circle whose segments portray the relative (or percent) frequencies of the categories of some qualitative variable
T/F: a qualitative variable assumes meaningful numerical values
F. a quantitative variable assumes meaningful numerical values, while qualitative variables are typically described in labels or names
T/F: cross-sectional data contains values of a characteristic of one subject collected over time
F. cross-sectional data contains values of a characteristic of many subjects at the same point or approximately the same point in time, or without regards to differences in time
T/F: the branch of statistical studies called inferential statistics refers to drawing conclusions about sample data by analyzing the corresponding population
F. inferential statistics refers to drawing conclusions about a large set of data - called a population - based on a smaller set of sample data
T/F: typically, it is possible to examine every member of the population
F. it is too expensive, too time-consuming, or even impossible to examine every member of the population
T/F: a professor's gender (male, female) as well as rank (assistant, associate, full) represent ordinal data
F. professor's gender is nominal, while rank is ordinal. The categories for nominal data do not have any natural natural ordering, while such an ordering exists for ordinal data
T/F: the arithmetic mean is the middle value of a data set
F. the median is the middle value of a data set
T/F: the mathematical operation of addition can be performed on nominal data
F. the only thing we can do with nominal data is to categorize or group the data
T/F: the variance and standard deviation are the most widely used measures of central location
F. the variance and standard deviation are the most widely used measures of dispersion
Which of the following best describes a frequency distribution for qualitative data?
It groups data into categories and records the number of observations in each category
The following data concern a sample of employees of the U.S. Marshalls in the state of New York. Identify the qualitative and quantitative variables, the categories associated with each qualitative variable, and the measurement scales for all variables. List all categories for the variable "station"
New York, NY, New York-Kings, Buffalo, Syracuse
Which firm's stock price had greater variability as measured by the standard deviation?
Panera's stock price had greater variability as indicated by a higher standard deviation
Calculate the sample variance and the sample standard deviation for Starbucks stock price.
Sample Var: 22.83/(6-1) = 22.83/5 = 4.57 Standard Dev: take the square root of sample var answer =2.14
T/F: for quantitative data, a cumulative relative frequency distribution records the proportion (fraction) of values that fall below the upper limits of each class
T. a cumulative relative frequency distribution represents the proportion of values that fall below the upper limit of each class
T/F: a stem and leaf diagram is useful in that it gives an overall picture of where quantitative data are centered and how the data are dispersed from the center
T. a stem and leaf diagram is a visual method for displaying quantitative data and gives an idea how data are centered and dispersed from the center. It also maintains the original data values in the chart
T/F: approximately 60% of the observations in a data set fall below the 60th percentile
T. percentile is defined as the approximate percentage of the observations have values below the percentile value
T/F: the geometric mean is a multiplicative average of a data set used to measure values over a period of time
T. the geometric mean is a multiplicative average, as opposed to an additive average
T/F: the relative frequency of a category is calculated by dividing the category's frequency by the total number of observations
T. the relative frequency of each category equals the proportion of observations in each category and is calculated by dividing the frequency by the total number of observations
T/F: the variance is an average squared deviation from the mean
T. the variance is computed as:
which of the following variables is not continuous
The number of obtained heads when a fair coin is tossed 20 times. although in practice the exact values of such variables as height, time, and temperature are approximated, they are continuous in nature. If a fair coin is tossed 20 times, the possible numbers of obtained heads are 0, 1, 2, ..., 20
which firms stock price has the greater relative dispersion
To find relative dispersion, take the standard dev / sample mean. Starbucks: 2.14/58.17=0.037 Panera: 8.82 / 208.67= 0.042. Panera has the greater relative dispersion, with a higher coefficient of variation
the following is a list of five of the world's busiest airports by passenger traffic for Year 1. 1. Name: Hartsfield-Jackson, Location: Atlanta, Georgia, U.S., # of passengers (in millions): 89 2. Name: Capital International, Location: Beijing, China, # of passengers (in millions): 74 3. Name: London Heathrow, Location: London, United Kingdom, # of passengers (in millions): 67 4. Name: O'Hare, Location: Chicago, Illinois, U.S., # of passengers (in millions): 66 5. Name: Tokyo, Location: Tokyo, Japan, # of passengers (in millions): 64 The percentage of passenger traffic in the five busiest airports that occurred in Asia is the closest to ?
38%. 74 million passengers flew out of Beijing, 64 million passengers flew out of Tokyo, and there is a total of 360 million passengers: (74 + 64)/360 = 38.33%.
in the accompanying stem and leaf diagram, the values in the stem and leaf portions represent 10's and 1's digits respectively. Stem Leaf 1 3 5 6 8 8 9 2 0 1 2 2 3 5 6 6 8 8 8 9 3 0 1 2 2 8 4 2 2 Which of the following numbers appears in the stem and leaf diagram?
38. add the left most digits (1, 2, 3, 4) + the last digits (9 + 9 + 8 + 2) = 38
The following histogram represents the number of pages in each book within a collection. What is the frequency of books containing at least 250 but fewer than 300 pages? 100-150: 1 150-200: 6 200-250: 5 250-300: 7 300-350: 3 350-400: 0 400-450: 0 450-500: 1 500-550: 1 550-600: 0 600-650: 1
7
the following table shows the number of payroll jobs the government added during the years it added jobs (since 1973). The jobs are in thousands. Jobs added: frequency: 100 up to 200 5 200 up to 300 8 300 up to 400 7 400 up to 500 5 500 up to 600 1 approximately what percent of the time did the government add 200,000 or more jobs?
81%. sum the frequency of the intervals 200 up to 300, 300 up to 400, and so on, and divide by the total of 26: (8 + 7 + 5 + 1) =21 / 26= 0.81= 81%
what percent of the guests were younger than 34 years old?
96/100=0.96=96%
A group of students has 12 girls and 10 boys. A project team, including three girls and two boys, must be created. Find the number of possible project teams.
9900. order does not matter. combination (12,3) x combination 10,2) = 9900