STA2023 Quiz 1 – 10 Answers

the art and science of learning from data

plan how to obtain the data

summarize the data with graphs and numerical summaries

use data from a random and representative sample to draw conclusions about the population of interest

Confidence and Significance tests

persons, animals, or objects in a study/experiment

the characteristics that we measure on each subject

all subjects of interest

subjects for whom we have data

each member of the population has the same chance of being included in the sample

numerical(proportion) summary of the population

numerical summary of the sample

place observations into groups summarized by percentage (yes or no, eye color)

take on numerical values; key features are center (average) and spread (variability)

Discrete and Continuous

take only a finite list of possible outcomes; a count

has an infinite list of possible values that form an interval; limit to measurability

normal

different values with the same proportions in a population (height)

distinct valley; two centers

direction of the long tail (left)

direction of the long tail (right)

The study of how to collect, organize, analyze, and interpret numerical and categorical information.

A series of steps followed to solve problems including collecting data, formulating a hypothesis, testing the hypothesis, and stating conclusions (choose, perform, develop, design, gather, analyze, formulate)

Numeric values(graphs) calculated from a dataset with the purpose of characterizing the behavior of the variables

involves methods of using information from a sample to draw conclusions regarding the population

set of all the individuals of interest in a particular study

A population in which each individual member can be given a number

collection of objects or individuals that are no boundaries or we can not measure about the total number of individuals in the occupied territories

subset of the population. individuals selected from a population, usually intended to represent the population in a research study

value that describes a population

Value that describes a sample

A characteristic about each individual element of a population or sample

value of the variable associated with one element of a population or sample. This value may be a number, or a symbol

the set of values collected for the variable from each of the elements belonging to the sample

a planned activity whose results yeild a set of data

naturally occuring discrepency between a sample statistic and the cooresponding population parameter

the objects described by a set of data: person (animal), place, and thing. In a medicinal trial, the people in the study referred to as called subjects

one that is relevant or appropriate asa representation of that property.

measurement such that the random error is small

In Population data, the variable is measured for EVERY individual of interest

in sample data, the variable is measured from ONLY SOME of the individuals of interest

variable that quantifies an element of the population. has a numerical measurment for which operations such as addition or averaging make sense.
Ex: Ag, GPA, Tuition, Fees

A variable that categorizes or describes an element of a population. has a nominal measurement that descibes an individual by placing them in a category or group.
Ex: Phone number, college year, addresses.

characterized by gaps or interruptions in the values we assume. gaps implies absence between those values. ALl values are whole numbers
Ex: number of cars in garge, count of males in a group

does not possess the gaps or interruptions characteristic of a discrete variable. decismals are allowed between any two values.
Ex: length of pole, height of basketball player, age of tree

an unordered set of categories only by name. only permit you to determine whether two individuals are the same or different

an ordered set of categories. Ordinal measurements tell you the direction of difference between two individuals

ordered series of equal size categories. Identify direction and magnitude of a difference. The zero point is located arbituarily on an interval scale

an interval scale where a value of zero indicates none of the variable. ratio identifies the direction and magnitude of difference and allows ratio comparisons of measurement.

1. Identify the variable(s) of interest (the focus) and the population of the study.
2. Develop a detailed plan for collecting data. If you use a sample, make sure the sample is representative of the population.
3. Collect the data.
4. Describe the data, using descriptive statistics techniques.
5. Interpret the data and make decisions about the population using inferential statistics.
6. Identify any possible errors.

the degree of dispersion (spread) of data points in a distribution

A variable that has an important effect on the response variable and the relationship amoung the variables in a study but is not one of the explanatory variables studied either because it is unknown or not measured.

two variables such that their effects on the respose variable cannot be distingushed from eachother

any specific experimental condition applied to the subjects

a researcher observees and collects data but does not change existing conditions

a researcher applies a treatment to a part of a population and observes the responses to the treatment. another part of the population is given a placebo called the control group

occurs when a subject recieves no treatment, but believes he or she is in fact recieveing treatment and responds favorably

a random process is used to assign each individual to one of the treatments

individuals are first sorted into blocks, and then a random process is used to assign each individual in the block to one of the treatments

an experiment in which the subjects alone do not know which treatment they are recieving

an experiment in which neither the subjects or the people working know which treatment each is recieving

reduced margin of error

numerical facsimile or representation of a real world phenomenon

used to assign individuals to the treatment groups. this helps prevent bias in the selected members for each group. this includes using an appropriate sampling technique

Experiment, survey, census, sampling frame, sample design

no bias introduces by the sampling technique employed. the process by which the sample data is selected progreses without definite aim, reason, or pattern

measurement mistake caused by the factors that vary from one measurement to another; statistical error due to chance

each element of the pop has an equal probability of being selected. a random number table is utilized to select the individual elements of the pop for the sample

assign each element of the population to a group (stratum). perform simple random sampling from each stratum

assign each element of the population a group or cluster. randomly select the desired number of clusters. every element within the selected cluster is used for the sample

list every member of the target population and uniquely assign a number to each member. randomly select a number. this number will be the starting point of the sample selection. Select member for the sample at equivalent intervals.

a statistical method of drawing representative databy selecting people bc of the ease of their volunteering or selecting units because of their availability or easy access

a sampling method where the pop is divided into a number of primary groups from which samples are drawn. these are then divided into secondary groups from which samples are drawn, and so on

a sampling method that produces data which systematically differs from the sampled population. An unbiased sampling method is not biased.

sample collected from those elements of the pop which chose to contribute the needed information on their own initiative

The science of data.

Objects described by a set of data, they do not necessarily have to be people.

A piece of data about the individual.

The entire group of individuals we want information about.

A group in the population.

A number that describes a characteristic of the population

A number that describes a characteristic of the sample.

A characteristic of an individual.

Separates individuals into categories.

Separates individuals based on numeric values.

Tells us possible values for the variable and how often certain variables occur.

Study where researcher observs but does nothing to influence outcome or responses.

Researcher experiments to influence outcomes and responses.

The way to select samples.

When responses are slanted toward certain outcomes.

Respondeds choose to be included in surveys. Ex: Comment cards at hotels.

Researcher samples those who are willing/available.

Not sampling from the entire population.

Those who don’t respond to surveys, equal to the number surveyed minus the number responded, all divided by the number surveyed.

A form of innacurate responses.

How a question is worded: may have some affect on answers.

A random sample that allows every possible sample of size n the same chance of being selected.

Select a starting point at random and then choose every kth individual, where k = (Population Size)/(Sample Size) rounded DOWN to nearest whole number

Divide the population into groups and take a SRS of each group.

Divide the population into groups, but this time take a SRS of groups

False

True

under coverage

nonresponse bias

response bias

Questions not requiring prior knowledge (should be questions requiring prior knowledge)

parameter

statistic

False (because it does not depend on pop. size)

nonresponse

a number or fact about a population (a population is the entire collection of individuals we’d like to know about)

a number or fact about a sample (a sample is a smaller group of individuals selected from the population) – called a subset of the population

Statistic

Parameter

Parameter

Statistic

Parameter

Statistic

Parameter

Statistic

Parameter

Parameter

95%

0.2358

0.3

2

The IQR is larger for females than males.

It would increase.

The median rain fall for June is higher than for May.

Yes, the range for annual maximum flow pre 1974 is higher than the range for annual maximum flow post 1973.

about 8

It would pretty much stay the same.

0.22

132 / 158

misuse of cause and effect

True

Should not be interpreted.

Each year, the percent that favors capital punishment increases by 0.9753% on average.

As the days go by, the fish tend to be caught less frequently.

-0.76

0.54

58 / 62

extrapolation

how we look at data and summarize our findings

makes a statement about a population based on random/representative sample; includes a measure of how confident we are in our statement

researcher assigns subjects to certain experimental treatments

researcher does nothing to subjects but observe x and y

not good; untrustworthy source of data

official government data that can be easily accessible to the general public

fast and cheap way to gather data/evidence of a specific population, but must be documented correctly so as to not imply that data collected may not be an accurate representation of an entire population

BAD; volunteer sample (call in shows, internet polls), convenience sample (in class, libraries, bus stops)

random samples (subjects chosen by chance)

personal interview (good but costly), telephone (cheap but less effective), questionnaires (anonymous but low response rate)

1 / n^2

A poll of 1000 randomly selected voters is conducted a week before the 2016 election. Results show that 51% of the sample is planning on voting for Clinton. Can you be confident Clinton will win?

A poll of 1000 randomly selected voters show that 55% of participants plan on voting for Sanders. Can you be confident Sanders will win?

sampling frame is missing certain parts of population, leading to potentially inaccurate data results from a survey

those unwilling to participate in a survey could have different positions/opinions and complicate the data results

those willing to participate may give untruthful responses due to bad memory retention or simply lying

can influence participants by use of long/confusing questions or leading questions

variable that we measure and can draw conclusions from; basically what we are looking to prove in a given experiment

the actual individuals (subjects) involved in the experiment

experimental conditions given to subjects; parameters of the experiments

compare two or more groups to eliminate confounding and control variability of results

dummy treatment; psychological effects/treatments are important when dealing with experimental units

group that receives placebo; helps determine true effect of treatment. not necessary when comparing more than one form of treatment

subjects unaware of treatment they are receiving

both experimental units and people dealing with subjects don’t know what treatment each unit is being subjected to

use a mechanical method to select subjects and assign them to treatments; can make use of probability to analyze results; avoids selection bias

number of experimental units that get each treatment

sample surveys that just want to take a “snapshot” of the population at current time

retrospective studies in which we match each case (POSITIVE OUTCOME) with a control (NEGATIVE OUTCOME) and then ask questions about the explanatory variable

forward thinking; follow subjects into the future

No, since not all students have local phone numbers.

Survey

influential outlier

The response variable is: the nation’s life expectancy, in years The explanatory variable is: the number of TV sets per person The experimental units are: the nations The lurking variable that acts as a confounding factor in this study is: the nations’ wealth

0.1118

No, since this is an example of voluntary response sample.

Observational Study.

= 1 – P(A)

= P(A) + P(B) – P(A and B)

= P(A and B)/P(A)

P(A) * P(B|A)

disjoint

independent

 

never

sample space

trial

SPACER (use to answer questions below) From the information above, we can identify the following probabilities: Probability that a house has a garage = P(G) = 0.66 Probability that a house has a pool = P(P) = 0.23 Probability that a house has a pool AND a garage = P(G and P) = 0.12

P(pool) = 0.23

P(garage) = 0.66

P(p’ and g’) = 0.23 For this question we will use the general addition rule and the definition of a compliment P(P or G) = P(P) + P(G) – P(P and G) P(P or G) = 0.23 + 0.66 – 0.12 = .77 <- this is the probability that the house has either a garage OR a pool. The opposite of “having a garage or a pool” is “not having a garage and not having a pool”. Thus to solve this we take the compliment of P(P or G). As is stated in the notes, P(P or G)’ (the compliment of having P or G) = 1 – P(P or G) P(P or G)’ = 1 – 0.77 = 0.23

P(G and P’) = 0.54 (get this value directly from the table) The probability that a home has a garage but not a pool is equal to “the probability that a home has a garage” minus “the probability that a home has a garage and a pool” = P(G) – P(G and P) = 0.66-0.12 = 0.54

SPACER (use to answer questions below) *instead of using a contingency table, the above problem can be solved with probability rules* Probability of being accepted by school C = P(C) = 0.6 Probability of being accepted by school D = P(D) = 0.7 Probability of being accepted by both = P(C and D) = 0.5

0.8 This can be solved using the union of C and D P(C or D). P(C or D) = P(C) + P(D) – P(C and D) = 0.6 + 0.7 – 0.5 = 0.8 P(accepted by at least 1) = 1 – P(accepted by neither C or D) = 1 – 0.2 = 0.8

0.2 The probability of being rejected by both schools is the compliment (opposite) of being accepted by at least one of these schools. P(rejected by C and D) = 1 – (probability of being accepted by at least 1 school) = 1 – P(C or D) = 1 – 0.8 = 0.2 This can be taken directly from the table; find the intersection of “not accepted by C” and “not accepted by D) = 0.2

0.3 The probability of being accepted by 1 school but not both is equivalent to the probability of C or D but not both. This is written as: P(C or D) – P(C and D) = 0.8 – 0.5 = 0.3 OR We will take the sum of two probabilities here. Find the box where “accepted by c” intersects “not accepted by d”. This is 0.1. Now find the box where “not accepted by c” intersects “accepted by d”. This is 0.2. 0.1 + 0.2 = 0.3

False

False

0.83 This conditional probability can be calculated as: P(D|C) = P(D and C) / P(C) (read “probability of D given C”) P(D|C) = 0.5/0.6 = 0.83

0.0060

5/17

7.93%

0.0039

0.9265

87.08

10/17

0.11%

discrete

continuous

True

miles until empty in your car body temperature money spent at the grocery store

number of pizza slices consumed in one evening number of football games an NFL team plays in a season number of siblings number of class taken in a semester

SPACER

0.15

1.7

0.9

SPACER

$4.50

$9.00

$13.50

$18.00

7.65

4.05

discrete

approximately Normal, with mean 0.6 and standard error 0.04899.

0.896

0.3121

0.9960

True

Normal

0.9713 You need Binomial p(X 60) when n = 100, p(success) = 0.5. Since n is large, use normal approximation for Binomial probability. Need μ = np = 50 ; σ2 = np(1-p) = 25; so σ = 5. Now do 0.5 adjustment; Binomial p(X 60) = normal p(X 59.5) = p(z 1.9) which is the entire area to the left of 1.9. This area is equal to 0.5 + 0.4713 = 0.9713.

0.44 We want p(10 X 15). Find the z-scores for both 10 and 15. These are (10-11)/4 = -0.25 and (15-11)/4 =1. Thus p(10 X 15) = p(-0.25 Z 1). Now draw the bell curve with zero in the middle and -0.25 to the left and 1 to the right of zero. Your answer is the area between -0.25 and 1. Using chart, the area between 0 and -0.25 is 0.0987, and the area between 0 and 1 is 0.3413. The total area is then equal to 0.0987 + 0.3413 = 0.44

0.30 You may look at two ways. Find the probability of winning $0 which is 0.70 and then subtract it from 1 to get the correct answer 0.30. Or find the probability of winning more than zero dollars (in this case $1 with probability = 0.25, $10 with probability 0.04, and $100 with probability 0.01) by adding all three probabilities of winning to get the correct answer 0.25 + 0.04 + 0.01 = 0.30.

0.242 You need to find the probability that the mean FNE score of 49 students is less than 17.5, i.e. you want p( x-bar 17.5). You need z-score for 17.5 using the Z formula involving x-bar since you are dealing with x-bar. The z-score for 17.5 is – 0.7. Now find p( Z – 0.7) using z-table. This is simply the entire area to the left of -0.7 which is equal to (0.5 – area between 0 and 0.7). The answer is then 0.5 – 0.2580 = 0.242.

32/99 you want the probability of one customer wins and the other loses. Here (i) above gives you the probability that only first customer wins and second customer loses, and (ii) above gives the probability that first customer loses and second customer wins. You need to add the two numbers in (i) and (ii) to get the correct answer 16/99 + 16/99 = 32/99 since these two are two disjoint options for the event in the question you are trying to answer.

p(A/B) = p(A) A or B is the same as A union B. Thus the addition law says p(A or B) = p(A) + p(B) – p(A intersection B). Note however that p(A intersection B) = p(A). p(B) since A and B are independent. So if you replace “or” by “intersection” in the left side, then it would be a true statement.

True

Range

True

(3.72, 5.28) You need to use the large sample confidence interval formula for mean. The z-score for 95% confidence in 1.96. So the lower limit of the confidence interval is 4.5 – (1.96)(4)/10 = 3.716. Similarly the upper limit is 4.5 + (1.96)(4)/10 = 5.284. Thus a 95% confidence interval is (3.716, 5.284).

305 Use the sample size formula for the purpose of estimating true mean. Here z-score for the given 90% confidence is 1.645, sigma is equal to 21.2 and SE = 2. Thus n = (1.645)(1.645)(21.2)(21.2)/(2*2) = 304.05. Since the sample size cannot be fractional number, you need to choose 305 or more. So the minimum n is 305.

t < -2.998 You do a one-tailed t-test with rejection region to the left of the t-distribution with df = n-1 = 7. The t-value from t-table corresponding to row 7 and column t_.01 (since alpha = 0.01) is 2.998 and you insert the negative sign since your test is one-tailed with research hypothesis H_a: mu 0.88.

t > 2.132. You have paired data and you need to use the paired t-test with DF =n-1 = 4. Your test is one-tailed to the right and so the rejection region lies to the right side of the t-distribution. Look at the row 4 and column t_0.05 in the t-table and you get 2.132. The rejection region is t > 2.132.

0.1379. It is a good practice to draw the bell curve with 0 marked in the middle on the horizontal axis. Then mark 1.09 and 4.64 on the horizontal line, color the area between 1.09 and 4.64. This colored area is your answer. To find this area, you need to subtract the area between 0 and 1.09 from the area between 0 and 4.64. Look carefully in your table IV, there is no number more than 3.09; the area between 0 and 3.09 is 0.499. How do we find the area between 0 and 4.64? Since the table does not go over 3.09, we need to approximate it. Since the entire area to the right of zero is o.5 and the area between 0 and 3.09 is approximately 0.499, we can say that the area between 0 and 4.64 is more than 0.499 but less than 0.5. Instead of choosing a number between 0.499 and 0.5, we simply take 0.5 as the approximate area between 0 and 4.64 (NOTE: the area between 0 and any number “k” is always approximated by 0.5 whenever “k” is greater than 3.09). So the area between 1.09 and 4.64 is equal to (0.5 – area between 0 and 1.09) which is 0.5 – 0.3621 = 0.1379.

2.0 You need to add the area between 0 and 1 to 0.1359 to identify the area between 0 and z0 and then find the corresponding z-score. See the explanation below for correct answer: First you need to guess correctly about the position of z0 in relation to zero and 1 in your bell curve (draw the bell curve and mark zero and 1 and then z0 on the horizontal line). Your z0 must be to the right hand side of 1.0 on the horizontal line. Now to find z0, you need to know the area between 0 and z0 which can be obtained by adding 0.3413 (the area between 0 and 1) to 0.1359 (the given area between 1 and z0). This total area is 0.4772 which is the area between 0 and z0. Now look at the chart area for 0.4772 (or the value closest to 0.4772). We have 0.4772 corresponding to z0 = 2 which is the answer.

0.0287

542.40

0.416

0..7580

0.5

0.973

12111.04, 12688.96)

163

H_a: mu < 0.88 (i.e., true mean missed distance for all missiles is less than 0.88)

mu_1 – mu_2 > 0

-1.75

0.1587

0.784

True

29

False

(0.72, 3.85)

Z > 1.645

1.645

3.8

All (exactly 100%) Florida high school students scored between 56 and 98

0.4332

2/5

7/40

312/368

335

t > 1.725

p(x > 8.5)

0.376

0.294

Variance

Median

(0.662, 0.738)

2436

mu_1 – mu_2 > 0

0.9901

1244

0.219

80/99

70 years

335

0.015

0.9239

0.04

At least 75%.

true

0.03

Median, Percentiles and Quartiles are measures of Position

6%

1.02

38/990

80/99

The median of the dataset 1, 4, 6, 5, 8 is equal to 6.

1.3

True

0.9312

0.421

0.74

0.4332

0.18

0.6

101

0.0188

Data must be from a simple random sample. Data is categorical. Counts of successes and failures at least 15 each.

not approximately normal

Normal

averages based on small n, if the population is Normal individuals, if the population is Normal sample proportion of successes out of n independent trials, when np and n(1-p) is large enough averages based on large n, if the population is Normal averages based on large n, if the population is NOT Normal

sigma/sqrt(n)

(0.2871, 0.3337)

response

explanatory

False

True

explanatory

response

True

–> perfect negative linear association –> perfect positive linear association –> no linear association

missing your Thursday morning class –> response going out on Wednesday night explanatory –> explanatory

cost of airfare –> explanatory number of flights you take in a year –> response

Amount of data used by your phone –> response amount of time spent watching YouTube –> explanatory

SPACER

Graph 1 – 0.28 Graph 2 – 0.89 Graph 3 – -0.92

0.28 –> very weak positive correlation 0.89 –> strong positive correlation -0.92 –> very strong negative correlation

False

number value assigned to a population parameter based on the value of a SAMPLE statistic EX: The average rent in Islamorada, Fl for a 1 bed/1 bath place is $1,200 based on a sample of 180 available apartments, townhomes and houses.

the actual sample statistic used to estimate a population parameter EX: for a population mean the sample statistic is x-bar for a sample proportion the sample statistic is p-hat If the average rent in Islamorada, Fl is based on a sample of 180 places and is said to be $1200 then we would label this x-bar.

1. Select a sample 2. Gather data 3. Calculate the statistic 4. Assign the value of the parameter * Procedure above assumes that the sample is a simple random sample

1. point estimate = value of a sample statistic computed from the sample (either noted as x-bar or p-hat) NOTE: point estimates are single value each sample selected from the population can yield a different value of the sample statistic values assigned to the population mean are based on the point estimate from the sample which is dependent on the sample that is drawn point estimate almost always differs from the true value of the entire population mean 2. interval estimate = interval is constructed around the point estimate by adding and subtracting the same value to create an interval where we believe the population mean will fall within NOTE: this includes a lower and upper limit instead of giving only one value we have a range (i.e. cost of rent for a 1BD/1BA place in Islamorada, Fl is $800-$1400)

the number that is added and subtracted from the point estimate 2 considerations to determine this number value: 1. The standard deviation of the sample mean 2. Confidence level selected to be attached to the interval (i.e. 90%, 95%, etc.)

point estimate plus or minus the margin or error NOTE: picture includes x-bar formula but it is also possible with p-hat as well

level of confidence that is selected for the interval or sample statistic denoted by (1-alpha) * 100% NOTE: alpha is also called the significance level EX: 90% confidence level would have an alpha of 0.10

0.0903

p-value is between 0.002 and 0.01

t table: small n and no outliers in the sample (so population could be normal)
neither the Z nor the t tables are appropriate: small n, non-normal population (there is at least one extreme outlier in the sample)
either the t or the Z tables would work: large n, any shape population

-0.148

Ho: mu = 350
Ha: mu does not equal 350

p-value greater than 0.20

-0.28

-0.22

1 standard deviation of the mean 2 standard deviations of the mean 3 standard deviations of the mean

unbounded

3

invNorm normalcdf

The total area under the pdf must equal one The probability that the random variables falls between any two values is given by the area under the density curve between those two values. The probability that a continuous random variable is equal to a single specific value is considered to be zero.

SPACER

0.764

0.432

3,121.6

??

2,881.5

3,650.5

IQR = Q3 – Q1 for female: IQR = 3,650.5 – 2,881.5 = 769 for male: IQR = ??

the values of statistics vary from sample to sample there is a recognizable long-term pattern to the variation between samples

decreases

30

σ/√n

N = 45 and the sampling distribution of the sample mean is unknown N = 45 and the sampling distribution of the sample mean is normal N = 30 and the sampling distribution of the sample mean is unknown N = 15 and the sampling distribution of the sample mean is normal

x ¯ ~ N ( μ , σ/√n )

SPACER

20

0.924

0.948

??

yes

0.052

increase

decrease