elementary probability & statistics midterm

Is the amount of rain measured in cm that falls in mahwah nj a discrete or continuous variable?

this is a continuous variable, since the value can be any real number greater than or equal to zero. It is not limited to integers or even rational numbers

simple random sampling technology examples

-Selecting 3 people from a group of 5 people -Selecting 10 people from a group of 6000 people -Simple random samples require all possible samples to be equally likely to be chosen

Find the range 0-12,7,9,-14,125

r= 125 - (-14) = 139

Find the range -2,0,2,6,8

range= 8 - (-2) =10

Police at a sobriety checkpoint on route 202 pull over every fifth driver to determine whether the driver is sober. What sampling method are the police using?

the sample for this test is systematic since every fifth driver is being sampled

As of January 2020, 48% of governors of the 50 united states are democrats is 48% an example of a statistic or parameter?

this is a parameter since it is a numerical summary of the population of governors of the united states

A salesperson obtained a systematic sample of size 20 from a list of 500 clients. To do so, he randomly selected a number from 1 to 25 obtaining the number 16. He included in the sample the 16th client on the list & every 25h client thereafter. List the numbers that correspond to the 20 clients selected.

Answer is 16, 41, 66, 91, 116, 141, 166, 191, 216, 241, 266, 291, 316, 341, 366, 391, 416, 441, 466, 491 start with the value of 16 & then 25 to get 41, the next number will be 25 after 41, which is 66. The next number will be 25 after 66, which is 91. Continue in this way until you get 20 numbers

Bar graphs

-A bar graph is constructed by labeling each category of data on the horizontal axis -The frequency of relative frequency of the category on the vertical axis -Rectangles of equal width are drawn for each category so that the height of each rectangle represents the categories frequency or relative frequency -Bars don't touch

continuous versus discrete variables

-A discrete variable is a quantitative variable that has either a finite number of possible values or a countable number of possible values. The term countable means that the values result from counting such as 0,1,2,3,..... -A continuous variable is a quantitative variable that has an infinite number of possible values that are not countable -A discrete variable is a quantitative variable that has either a finite number of possible values or a countable number of possible values. The term contable means that the values result from counting, such as 0,1,2,3 -A continuous variable is a quantitative variable that has an infinite number of possible values that are not countable

Organizing Quantitative Data

-A frequency distribution lists each class of data & the number of occurrences for each class -A relative frequency distribution lists each class together with the relative frequency

chapter 1 problem

-A quality control manager randomly selects 50 bottles of coca-cola that are filled on october 15 to assess the calibration of the filling maching -What is the population? = all bottles filled on 10/15 -What is the sample? -Sample = 50 bottle selected

statistics vs parameter

-A statistic is a numerical summary of a sample -Note: we use latin letters (usual) to denote statistics: x, & s are sample mean standard deviation, respectively -A parameter is a numerical summary of a population -Note: we use greek letters to denote parameters: u & o are population mean & population standard deviation, respectively

chapter 3.3 weighted mean objectives

-Be able to discern a weighted mean from an arithmetic mean problem -Compute a weighted mean

Draw a dot plot of the number of miles per gallon achieved on the highway for small cars for the model year 2011

35, 35, 36, 35, 34, 31, 34, 27, 30, 33, 34, 34, 34, 36, 32, 30, 34, 26, 36, 29, 35, 34, 36, 37, 32, 25, 31, 25, 31, 25, 35, 29, 34, 36, 35, 36, 31, 23, 33, 23, 33, 21, 33, 30, 36, 29, 33, 35, 31, 22, 32, 35, 31, 35, 36, 33, 30, 33, 34, 31, 34, 42, 38, 43, 29, 36, 33, 31, 31, 32, 42, 37, 36, 28, 34, 33, 30, 33, 33, 34, 40, 34, 25, 36, 33, 28, 33, 32, 31

stratified sample

A stratified sample is obtained by separating the population into non overlapping groups called strata & then obtaining a simple random sample from each stratum. The individuals in each stratum must be homogenous or similar in some way

law of large numbers

As the number of repetitions of a probability experiment increases, the proportion with which a certain outcome is observed gets closer to the probability of the outcome

descriptive statistics

Consists of organizing & summarizing data. Descriptive statistics describe data through numerical summaries, tables & graphs

sample variance

The sample variance, s2 is the square distance of the data from the mean. It is obtained by taking the square of the sample standard deviation. Conversely, the same standard deviation is obtained by taking the square root of the sample variance.

what is statistics?

The science of collecting, organizing, summarizing & analyzing information to draw conclusions or answer questions. In addition, statistics is about providing a measure of confidence in any conclusions

inferential statistics

Uses methods that take a result from a sample, extend it to the population, & measure the reliability of the result

Create a relative frequency distribution for the favorite color of students enrolled in a past class. Note that some students answered in unconventional colors or gave multiple answers, so you will have to decide how to handle this data. Real data needs to be cleaned up, so this is your chance to see the problems with working with real data.

You must deal with real data that is not always given in a form you like or with non-traditional answers. Some students may keep light blue as a category & others include it with blue. I chose the latter. Additionally, fall colors & red or black needed to be dealt with. There are choices in handling data & this is called cleaning the data. It is important to note what you did, as it could skew your results.

1.3 simple random sampling objectives

Identify if a sampling is random or not Take a simple random sample

chapter 5.1 probability rules objectives

-Identify a probability model -List a sample space -Compute probabilities by the empirical method -Compute probabilities by the classical method

1.4 other effective sampling methods objectives

-Identify a simple random sample -Identify a stratified sample -Identify a systematic sample

procedure for finding the sample 100 times pth percentile

1) Order the data in a non-decreasing order 2) Determine the proportion n times p, where n is the sample size & p is given - Case 1: if n times p is an integer, round up to the nearest integer, say integer k & find the value in the list created in step 1 - Case 2: if n times p is an integer say k, calculate the mean of the kth & (k=1)st values from the lost created in step 1

organizing qualitative data

A frequency distribution lists each category of data & the number of occurrences for each category of data -The relative frequency is the proportion or percent of observations within a category -Relative frequency = frequence divided by sum of all frequencies -A relative frequency distribution lists each category of data together with the relative (frequency) -Ex: frequency divided by 20

definition in blog of statistics

"Statistics is a systematic means of helping us better understand parts of our world we consider important. It also helps us understand why we consider some things important & others not. It is not a substitute for philosophy but fits in neatly with it. It is not a substitute for engineering, but can be an aid to it. It does not replace gut instinct but can inform it" kevin gray, reality science

data versus variables: gender

-Data -Variable -The answer is variable because there are no observed values

Mode

-Defh the mode of a set of data is the element that occurs most often -Ex: 1,3,4,2,5,1,0,3,3 -The mode is 3 -of a variable is the most frequent observation of the variable that occurs in the data set.

Relationship between the mean, median & the distribution shape

-Distribution shape: skewed left, symmetric, skewed right -Mean versus median: mean substantially smaller than median, mean roughly equal to median, mean substantially larger than median

fraction/decimal/percent conversion

-Fraction: 2/13, decimal: 0.153846.... (2 squiggly lines) 0.154, percent: (2 squiggly lines) 15.4% -Fraction: 804/1000= 207/250, decimal: 0.804, percent 80.4% -Fraction: 32/1000= 8/250 = 4/12, decimal: 0.032, percent: 3.2%

z-score

-How can you compare 2 values taken from different populations? -See how they compare relative to their population -The z-score represents the number of standard deviations a data value is from the mean -A z-score has no units, but represents a distribution with mean of 0 & a standard deviation of 1. There is both population z-score & sample z-score;

qualitative versus quantitative variables

-A variable is a characteristic of individuals within the population -Qualitative or categorical, variables allow for classification of individuals based on some attribute or characteristic -Quantitative variables provide numerical measures of individuals -Arithmetic operations such as addition & subtraction can be performed on the values of a quantitative variable & will provide meaningful results

simple random sampling with technology

-Assign each student a number 1 to 6000 -Randomly select 10 numbers from the numbers 1 through 6000 using the random number generator -If you have any repeated numbers, then select the extra numbers until you have 10 different numbers -Record the names associated with those 10 different numbers -Note that technology selects with replacement

The following data represent the ages of the presidents of the united states on their first days in office construct a frequency distribution & histogram

-Class: 40-44, 45-49, 50-54, 55-59, 60-69, 65-69, 70-74 -Frequency: 2, 7, 13, 12, 7, 3, 1

chapter 3.1 measures of central tendency objectives

-Compute the mean for a set of data -Compute the median for a set of data -Understand the relationship between the shape of a distribution & the better measure of center -Determine the mode for a set of data

chapter 3.2 measures of dispersion objectives

-Compute the range of a set of numbers -Compute the population variance & standard deviation -Compute the simple variance & standard deviation -Be able to accurately set up the empirical rule & answer questions about it

chapter 3.4 measure of position objectives

-Compute the z-score to compare 2 values from different populations -Compute percentiles of a list of numbers -Interpret the percentiles of a list of numbers

2.2 organizing quantitative data objectives

-Construct frequency & relative frequency distributions for discrete & continuous data -Construct frequency & relative frequency histogram for discrete & continuous data -Construct a dot plot -Identify the shapes of a distribution

Dot Plot

-Constructed by drawing a circle to represent each piece of data -The observed values are placed on the horizontal axis in increasing order -1 circle will represent 1 data value placed above the value on the axis

continuous versus discrete: graphical depiction age

-Continuous -Discrete -Either answer is possible depending on the justification Continuous because it is a measure of time & time is continuous. We can be 23, 56, 72, 21 years old. -Discrete because we can say we are 17, 20, 23 etc.... This is the measuring convention, so we can argue that the age is discrete

continuous versus discrete: Number of miles my car travels on 1 tank of gas

-Continuous -Discrete -The answer is continuous because distance (miles) is a measurement that archive every value in an interval of values. We do not skip from 10 miles to 11 miles, but rather we hit ever number of miles between 10 & 11

continuous versus discrete: number of points scored in the nba finals

-Continuous -Discrete -The answer is discrete because there are only whole numbers of points in a basketball game. So, some possible values are 204, 245, 231, 253, 249, as in the 5 games of the 2017 nba finals

Chapter 2 (2.1) organizing qualitative data objectives

-Create a frequency distribution -Create a relative frequency distribution -Create a frequency bar graph -Create a relative frequency bar graph create a pie chart

data verus variables: Tommy is 10 years old

-Data -Variable -The answer is data because 10 years old is an observed value of the age of prof b's cat

data versus variables: the teacher for the course is female

-Data -Variable -The answer is data because the variable was gender & this is an observed value

data versus variables: ethnicity

-Data -Variable -The answer is variable because there are no observed values

random sampling example

-If i choose the 3 tallest in the group, is that a simple random sample? no -If i choose the 3 that live closet to me, is that a simple random sample? no -How can we choose a simple random sample of 3 people? -all possible outcomes -ABC -ABD -ABE -ACD -ACE -ADE -BCD -BCE -BDE -CDE -10 groups of 3 -Note ABC = ACB dont double count -5 c 3 = c (5, 3) -= 5 divided by 3 (5-3) = 10

Empirical Rule

-If the distribution is roughly bell shaped, then approximately 68% of the data will lie within 1 standard deviation of the mean -Approximately 95% of the data will lie within 2 standard deviation of the mean -Approximately 99.7% of the data will lie within 3 standard deviation of the mean -Note: we can use the empirical rule based on sample data with the sample mean instead of population mean & sample standard deviation in place of population standard deviation

chapter 1.3 simple random sampling

-In a sample without replacement, an individual who is selected is removed from the population & cannot be chosen again -In a sample with replacement, a selected individual is placed back into the population & could be chosen again

pie chart

-Is a circle divided into sectors each sector represents a category of data. The area of each sector is proportional to the frequency of the category -In other words, the area of each sector is the relative frequency of that category. -is a circle divided into sectors. Each sector represents a category of data. The area of each sector is proportional to the frequency of the category.

Probability

-Is a measure of the likelihood of a random phenomenon or chance behavior. -Probabilities are measured on a scale from 0-1 -Probability of 0 means that the phenomenon will not happen (impossible) -Probability of 1 means that the phenomenon will happen (certain) -As the probability gets closer to 1, the more likely the phenomenon will occur

random sampling

-Is a process of using chance to select individuals from a population to be included in the sample -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 than called a simple random sample

Histogram

-Is constructed by drawing rectangles for each class of data -The height of each rectangle is the frequency or relative frequency of the class -The width of each rectangle is the same & the rectangles touch eachother -When determining the width, we can choose discrete values or intervals. Most histograms have between 5 & 20 bars.

systematic sampling example

-K = 3 -Population: 1 2 3 4 5 6 7 8 9 10 11 12 2 5 8 11 -Sample every 3rd -Random number 1-3 -Select every 3rd person starting with person number 2

populations versus sample example

-Let the 25 registered students in this class be the population -How many people are in our population? -What is an example of a sample? -What is an example of an individual?

graphically Which is larger the mean or the median?

-Mean -Median -They are about the same -Answer: they are all about the same, the graph appears to be symmetric, so the mean & median are approximately the same. Note that some students say this is skewed left, because of the two bars on the left. If you said this, then that would make the median larger, because the tail is on the left & the mean is pulled towards the tail

-These are the heights in inches of 7 students in this class: 70,62,62,60,65,72,68 -Find the mean height: 70+62+62+60+65+72+68 divided by 7 -Find the median height

-Mean = 65.571 inches -n=7 -mode=62 -median= 65 inches

Find the mean & median of -2,0,2,6,8,100

-Medium = 2+6 divided by 2= 4 -mean= 114 divided by 6 = 19 -We say that the median is resistant while the mean is not -A numerical summary of data is said to be resistant if the extreme value large or small relative to the data do not affect the value substantially

statistic or parameter: We are 98% confident that the average for all math 108 students final exam scores is between 74.5% & 81.5%

-Parameter -Statistic -The answer is parameter because it is giving an interval of values for all students. This indicates it is for the population, which means the summary is a parameter

statistic versus parameter: The average on the final for 12 randomly selected math 108 students last semester was 78%

-Parameter -Statistic -The answer is statistic because there are only 12 students randomly selected. Also, 12 is the sample size, which indicates a sample Sample = 78% is a stat

statistic versus parameter: In a national survey of 1300 high school students (grades 9 to 12), 32% of respondents respondents reported that someone had bullied them at school

-Parameter -Statistic -The answer is statistic because they want to know about all high school bullying & there are more than 300 of them. The 1300 is a sample size, which indicates the summary of 32% is a statistic Sample of high school students

qualitative versus quantiative: grade in previous mathematics course

-Qualitative -Quantitative -Either answer is possible depending on the justification -Qualitative if your examples are a, b, c, d, or f -Quantitative if your examples are 90, 76, 68, etc...

qualitative versus quantiative country of citizenship

-Qualitative -Quantitative -The answer is qualitative because examples are usa, canada, russia, china, france, georgia, norway etc. which are clearly not numbers

qualitative versus quantitative: social security number

-Qualitative -Quantitative -The answer is qualitative because it is an identification number not a measurement of anything. While it is a number, if i took the average of this class's ss number it may not even ve a ss number. If it happened to be the social security number of someone it is unlikely that person would be affiliate with the class in any way. -Operations on the number dont provide anything meaningful

qualitative versus quantiative: age

-Qualitative -Quantitative -The answer is quantitative because examples are 20, 34, 17, 2, etc.... Which are numbers & their mean would make sense as an age of a group

qualitative versus quantiative: time of sunset

-Qualitative -Quantitative -The answer is quantitative because time is a measurement of time for example 11:17am is 11 hours & 17 minutes after midnight. The average time of sunset in september has a meeting of a time of sunset that is related to september

A member of congress wishes to determine her constituencies opinion regarding estate taxes. She divides her constituency into 3 income classes low income households, middle income households, & upper income households. She then takes a simple random sample of households from each income class what type of sample did she obtain?

-Simple random sample -Stratified -Systematic -3 strata: divides her constituency into 3 income -Answer is stratified because the 3 income levels are the 3 strata in which she is selecting randomly from

The presider of a guest lecture series at a university stands outside the auditorium before a lecture begins & hands every fifth person who arrives, beginning with the third, a speaker evaluation survey to be completed & returned a the end of the program. What type of sample did he obtain?

-Simple random sample -Stratified -Systematic -Answer is systematic because he is selecting every 5th starting at 3. Thus, person 3, 8, 13, 18, 23, 28, etc... will be given a survey

24 hour fitness wants to administer a satisfaction survey to its current members. Using its membership roster, the club randomly selects 40 club members & asks them about their level of satisfaction with the club. What type of sample is obtained?

-Simple random sample -Stratified -Systematic -Current members is population -Answer is Simple random sample because they are just selecting 40 from the entire membership list

Populations versus samples

-The entire group to be studied is called the population -An individual is a person or object that is a member of the population of being studied -A sample is a subset of the population that is being studied -Pop = individual sample

important percentiles

-The first quartile denoted q1 is the 25th percentile denoted p25 -The second quartile denoted q2 is the 50th percentile denoted p50 -The third quartile denoted q3 is the 75th percentile denoted p75 -Inter quartile range (IQR)= interval from q__1 to q__3

example frequency distribution

-The following data represent the diagnoses of a random sample of 20 patients admitted to a hospital -Cancer -Gunshot wound (1) -Gunshot wound (2) -Assault -Gunshot wound (3) -Motor vehicle accident -Fall -Motor vehicle accident -Motor vehicle accident -Motor vehicle accident -Congestive heart failure -Gunshot wound -Gunshot wound -Gunshot wound -Gunshot wound -Motor vehicle accident -Motor vehicle accident -Fall -Gunshot wound -Fall frequency distribution solutions -Diagnosis: cancer, frequency: 1 -Diagnosis: gunshot, frequency: 8 -Diagnosis: assault, frequency: 1 -Diagnosis: MVA, frequency: 6 -Diagnosis: fall, frequency: 3 -Diagnosis: heart, frequency: 1 Sum (foeq) = 20

example relative frequency distribution

-The following data represent the diagnosis of a random sample of 20 patients admitted to a hospital -Cancer -Gunshot wound -Gunshot wound -Assault -Gunshot wound -Motor vehicle accident -Fall -Motor vehicle accident -Motor vehicle accident -Motor vehicle accident -Congestive heart failure -Gunshot wound -Gunshot wound -Gunshot wound -Gunshot wound -Motor vehicle accident -Motor vehicle accident -Fall -Gunshot wound -Fall solution: -Diagnosis: cancer, frequency: 1, relative frequency 1/20 -Diagnosis: gunshot wound, frequency: 8, relative frequency: 8/20 -Diagnosis: assault, frequency: 1, relative frequency: 1/20 -Diagnosis: motor vehicle accident, frequency: 6, relative frequency: 6/20 -Diagnosis: fall, frequency: 3, relative frequency: 3/20 -Diagnosis: congestive heart failure, frequency: 1, relative frequency: 1/20 -Sum, frequency: 20, relative frequency: 20/20=1

data versus variables

-The list observed values for a variable is data -Note: the variables are the characteristics of the individuals, while data are the specific values of the variables

Median

-The median of a variable is the value that lies in the middle of the data when arranged in ascending order. -If the number of observations is odd, then there is an exact center value & that is the median -If the number of observations is even, then we average the 2 numbers that are the next to the middle to determine the median -is the value that lies in the middle of the data when arranged in ascending order. We use M to represent the median

how do you determine k in systematic sampling?

-The owner of a private food store is concerned about employee morale. She decides to survey all 29 employees & wants to conduct a systematic sample. 4 greater than 29/6 greater than 5 -Determine k if the sample size is 6. k=4 -Determine the individuals who will be administered the survey. More than 1 answer is possible -Start with a random number between 1 & k -2, 6, 10, 14, 18, 22 systematic sample size is 6

stratified sample example

-The owner of a private food store is concerned about employee morale. She decides to survey the managers & hourly employees to see if she can learn about work environment & job satisfaction. There are 8 managers & 21 hourly employees. How should she go about selecting 6 people to survey? -solution: Managers 8 Hourly 21 Total 29 Choose 3 randomly from each group

mean

-The population arithmetic mean of a variable is what we often call the average -X1,x2,x3..... -If x1,x2,x3 are the N values of a variable from a population, then the parameter known as arithmetic mean is found by -The sample arithmetic mean of a variable is what we often call the average. -X1,x2,x3,.....are -If x1,x2,x3 are the n values of a variable from a sample, then the statistic known as arithmetic mean is found by -Population -Parameter Iu -Sample -Statistic - x

population standard deviation

-The population standard deviation, o is a measure of distance the vales are from the mean -Recall: m = mean of the population -of a variable is the square root of the sum of squared deviations about the population mean divided by the number of observations in the population, N. that is, it is the square root of the mean of the squared deviations about the population mean

percentiles

-The sample 100-pth percentile is a value such that after the data values are arranged in a non-decreasing order, at least p of the data lie at or below it, & at least 1-p of the data lie at or above it. -0 < p < 1

sample space

-The sample space, s of a probability experiment is the collection of all possible outcomes. -An event is any collection of outcomes from a probability experiment. An event may consist of 1 outcome or more than 1 outcome -Notation: s = sample space -a= 1 outcome -a= event -S of a probability experiment is the collection of all possible outcomes

weighted mean

-The weighted mean is a mean where each outcome is not equally likely -It can be expressed as where w, is the weight of the ith observation & xi is the value of the ith observation -found by multiplying each value of the variable by its corresponding weight, adding these products, & dividing this sum by the sum of the weights. It can be expressed using the formula

chapter 1.1 introduction to the practice of statistics objectives

-Understand the vocabulary & identify examples of each -Statistics-inferential & descriptive -Population -Sample -Parameter -Statistic -Qualitative variable -Quantitative variable -Continuous -Discrete

identifying the same of distribution

-Uniform, bell-shaped (symmetric) -Skewed right -Skewed left

relative frequencies

-We represent relative frequencies in 1 of 3 equivalent ways -Fraction -Decimal -Percentage -Lets recall how to change between each of these

does this mean the average of all students taking math 108 final is 78%?

-Yes -No -The answer is no because it does not account for all students, just those 12. There is a lot of variability that could occur based on which 12 were selected

Find the median of the sample: -2,0,2,6,8

-n=5, median=2 -Find the median of the population: -2,0,2,6,8,10 -n=6, median = 2+6 divided by 2 = 4

Compute the mean of population. Do not use any statistical features of your calculator -1,4,5,7,-2,0

-n=6 -m=-1+4+5+7-2+0 divided by 6= 13/6

Compute the median of the sample. Do not use any statistical features of your calculator: -1,4,5,7,-2,0

-n=6, medium = 0+4 divided by 2 = 2 -Answer: 4.2

-We asked 8 members of class to give their heights the table represents their heights in inches -60, 62, 62, 65, 67, 68, 71, 72 -71, 62, 62, 60(4th), 65, 72(6th), 68(7th), 67 -Determine the 3rd quartile. Interpret your result

-q3=p75 p=0.75 n=8 np=6 q3= 71+68 divided by 2 = 69.5 -Determine the 40th percentile. Interpret your result: p= 0.4 n=8 np=3.2 p40=65in 40% others is shorter than 65in

A dividend is a payment from a publicly traded company to its shareholders. The dividend yield of a stock is determined by dividing the annual dividend yields in percent of a random sample of 28 publicly traded stocks of companies with a value of at least 5 billion. With the first class having a lower class limit of 0 & a class width of 0.40 construct a frequency histogram.

1.7, 2.83, 2.59, 2.04 0, 2.16, 0, 0, 1.15, 1.05, 1.7, 0, 0.62, 1.22, 0.64, 1.35, q.06, q.68, 0.67, 0, 2.45, 0.89, 2.07, 0, 2.38, 0, 0.94, 0.41

The organizers of a board game convention want to survey the 8,763 attendees about their experience at the event. The organizers have a list of the attendees organized by their registration number. They want the survey to be conducted using a systematic sample of size 100. The first person is selected randomly to be number 42. List the registration numbers of the first 3 people selected & the last person selected to survey.

First = 42, second = 42+87 = 129 + 87 = 216. Last = 42 + 99 (87) = 8,635. 42, 129, 216,.....8,655

systematic sample

Is obtained by selecting every kth individual from the population the first individual selected corresponds to a random number between 1 & k

The organizers of a board game convention want to survey the 8,763 attendees about their experience at the event. The organizers have a list of the attendees organized by their registration number. They want the survey to be conducted using a systematic sample of size 100. What is k?

K is less than or equal to 8,763 divided by 100= 87.63 so k=87

notation

Note: a shortcut notation for a 100 times pth percentile is P100p for instance, a shortcut notation for 25th percentile is P25 -P (A) is the probability of event A occurring. It is a number between 0 & 1 -The event E = even numbers less than 5 is a set of outcomes. It can be represented by a list of values, it is not a number -Note that P (0.73) makes no sense.

extension to simple random sampling

Suppose we want to find a sample random of 10 students from a large population, like all students at ramapo college. What would we do?

Police at a sobriety checkpoint on route 202 pull over every fifth driver to determine whether the driver is sober What is the population of this test?

The population of this test is the drivers on route 202

population variance

The population variance, o2 is the square distance the data is from the mean. It is obtained by taking the square of the population standard deviation. Conversely, the population standard deviation is obtained by taking the square root of the population variance.

range

The range, R of a variable is the difference between the largest data vale & the smallest data value. That is, range=R=(largest data value) -(smallest data value)