# MEP Pupil Text 9. The mean, median and mode are three different ways of describing the average.

Save this PDF as:

Size: px
Start display at page:

Download "MEP Pupil Text 9. The mean, median and mode are three different ways of describing the average."

## Transcription

2 9. MEP Pupil Text 9 Worked Example Five people play golf ad at oe hole their scores are 3, 4, 4, 5, 7. For these scores, fid the mea the media the mode (d) the rage. Solutio The mea is = 3 5 = 46.. (d) The umbers are already i order ad the middle umber is 4. So The score 4 occurs most ofte, so, media = 4. mode = 4. The rage is the differece betwee the smallest ad largest umbers, i this case 3 ad 7, so rage = 7 3 = 4. Exercises. Fid the mea media, mode ad rage of each set of umbers below. 3, 4, 7, 3, 5,, 6, 0 8, 0,, 4, 7, 6, 5, 7, 9, 7, 8, 6, 7, 7, 4,, 5, 6, 7, 4, (d) 08, 99,,, 08 (e) 64, 66, 65, 6, 67, 6, 57 (f), 30,, 6, 4, 8, 6, 7. Twety childre were asked their shoe sizes. The results are give below. 8, 6, 7, 6, 5, 4, 7, 6, 8, 0 7, 5, 5 8, 9, 7, 5, 6, 8 6 For this data, fid the mea the media the mode (d) the rage. 46

4 9. MEP Pupil Text 9 What are the mea ad rage of the data? Richard's fried, Najir, also goes fishig. The mode of the umber of fish he has caught is also 0 ad his rage is 5. What is the largest umber of fish that Najir has caught? 8. A garage ower records the umber of cars which visit his garage o 0 days. The umbers are: 04, 30, 79, 34, 57, 30, 3, 6, 308, 7. Fid the mea umber of cars per day. The ower hopes that the mea will icrease if he icludes the umber of cars o the ext day. If 5 cars use the garage o the ext day, will the mea icrease or decrease? 9. The childre i a class state how may childre there are i their family. The umbers they state are give below.,,, 3,,,, 4,,,, 3,,,,,,, 7, 3,,,,,,,, 3 Fid the mea, media ad mode for this data. Which is the most sesible average to use i this case? 0. The mea umber of people visitig Jae each day over a five-day period is 8. If 0 people visit Jae the ext day, what happes to the mea?. The table shows the maximum ad miimum temperatures recorded i six cities oe day last year. City Maximum Miimum Los Ageles C C Bosto C 3 C Moscow 8 C 9 C Atlata 7 C 8 C Archagel 3 C 5 C Cairo 8 C 3 C Work out the rage of temperature for Atlata. Which city i the table had the lowest temperature? Work out the differece betwee the maximum temperature ad the miimum temperature for Moscow. (LON). The weights, i grams, of seve potatoes are 60, 5, 05, 40, 3, 05, 4. What is the media weight? (SEG) 48

6 9. MEP Pupil Text 9 7. Eight judges each give a mark out of 6 i a ice-skatig competitio. Oksaa is give the followig marks. 5.3, 5.7, 5.9, 5.4, 4.5, 5.7, 5.8, 5.7 The mea of these marks is 5.5, ad the rage is.4. The rules say that the highest mark ad the lowest mark are to be deleted. 5.3, 5.7, 5.9, 5.4, 4.5, 5.7, 5.8, 5.7 (i) Fid the mea of the six remaiig marks. (ii) Fid the rage of the six remaiig marks. Do you thik it is better to cout all eight marks, or to cout oly the six remaiig marks? Use the meas ad the rages to explai your aswer. The eight marks obtaied by Toya i the same competitio have a mea of 5. ad a rage of 0.6. Explai why oe of her marks could be as high as 5.9. (MEG) 9. Fidig the Mea from Tables ad Tally Charts Ofte data are collected ito tables or tally charts. This sectio cosiders how to fid the mea i such cases. Worked Example A football team keep records of the umber of goals it scores per match durig a seaso. No. of Goals Frequecy Fid the mea umber of goals per match. Solutio The table above ca be used, with a third colum added. The mea ca ow be calculated. Mea = =. 85. No. of Goals Frequecy No. of Goals Frequecy = = 0 = = = = 0 TOTALS (Total matches) (Total goals) 50

7 Worked Example The bar chart shows how may cars were sold by a salesma over a period of time. Frequecy Cars sold per day Fid the mea umber of cars sold per day. Solutio The data ca be trasferred to a table ad a third colum icluded as show. Cars sold daily Frequecy Cars sold Frequecy 0 0 = = = = = 5 5 = 0 TOTALS 0 50 (Total days) (Total umber of cars sold) Worked Example 3 Mea = 50 0 = 5. A police statio kept records of the umber of road traffic accidets i their area each day for 00 days. The figures below give the umber of accidets per day Fid the mea umber of accidets per day. 5

8 9. MEP Pupil Text 9 Solutio The first step is to draw out ad complete a tally chart. The fial colum show below ca the be added ad completed. Number of Accidets Tally Frequecy No. of Accidets Frequecy = = 0 = = = = = = = 8 TOTALS Exercises Mea umber of accidets per day = = A survey of 00 households asked how may cars there were i each household The results are give below. No. of Cars Frequecy Calculate the mea umber of cars per household.. The survey of questio also asked how may TV sets there were i each household. The results are give below. No. of TV Sets Frequecy Calculate the mea umber of TV sets per household. 5

9 3. A maager keeps a record of the umber of calls she makes each day o her mobile phoe. Number of calls per day Frequecy Calculate the mea umber of calls per day. 4. A cricket team keeps a record of the umber of rus scored i each over. No. of Rus Frequecy Calculate the mea umber of rus per over. 5. A class coduct a experimet i biology. They place a umber of m by m square grids o the playig field ad cout the umber of worms which appear whe they pour water o the groud. The results obtaied are give below Calculate the mea umber of worms. How may times was the umber of worms see greater tha the mea? 6. As part of a survey, a statio recorded the umber of trais which were late each day. The results are listed below Costruct a table ad calculate the mea umber of trais which were late each day. 53

10 9. MEP Pupil Text 9 7. Haah drew this bar chart to show the umber of repeated cards she got whe she opeed packets of football stickers. q y Frequecy Number of repeats Calculate the mea umber of repeats per packet. 8. I a seaso a football team scored a total of 55 goals. The table below gives a summary of the umber of goals per match. Goals per Match Frequecy I how may matches did they score goals? Calculate the mea umber of goals per match. 9. A traffic warde is tryig to work out the mea umber of parkig tickets he has issued per day. He produced the table below, but has accidetally rubbed out some of the umbers. Tickets per day Frequecy No. of Tickets Frequecy TOTALS 6 7 Fill i the missig umbers ad calculate the mea. 54

11 0. Here are the weights, i kg, of 30 studets. 45, 5, 56, 65, 34, 45, 67, 65, 34, 45, 65, 87, 45, 34, 56, 54, 45, 67, 84, 45, 67, 45, 56, 76, 57, 84, 35, 64, 58, 60 Copy ad complete the frequecy table below usig a class iterval of 0 ad startig at 30. Weight Rage (w) Tally Frequecy 30 w < 40 Which class iterval has the highest frequecy? (LON). The umber of childre per family i a recet survey of families is show What is the rage i the umber of childre per family? Calculate the mea umber of childre per family. Show your workig. A similar survey was take i 960. I 960 the rage i the umber of childre per family was 7 ad the mea was.7. Describe two chages that have occurred i the umber of childre per family sice 960. (SEG) 9.3 Calculatios with the Mea This sectio cosiders calculatios cocered with the mea, which is usually take to be the most importat measure of the average of a set of data. Worked Example The mea of a sample of 6 umbers is 3.. A extra value of 3.9 is icluded i the sample. What is the ew mea? 55

12 9.3 MEP Pupil Text 9 Solutio Worked Example Total of origial umbers = 6 3. = 9. New total = = 3. New mea = 3. 7 = 33. The mea umber of a set of 5 umbers is.7. What extra umber must be added to brig the mea up to 3.? Solutio Total of the origial umbers = 5. 7 = Total of the ew umbers = 6 3. = So the extra umber is 5.. Differece = = 5. Exercises. The mea height of a class of 8 studets is 6 cm. A ew girl of height 49 cm jois the class. What is the mea height of the class ow?. After 5 matches the mea umber of goals scored by a football team per match is.8. If they score 3 goals i their 6th match, what is the mea after the 6th match? 3. The mea umber of childre ill at a school is 3.8 per day, for the first 0 school days of a term. O the st day 8 childre are ill. What is the mea after days? 4. The mea weight of 5 childre i a class is 58 kg. The mea weight of a secod class of 9 childre is 6 kg. Fid the mea weight of all the childre. 5. A salesma sells a mea of 4.6 coservatories per day for 5 days. How may must he sell o the sixth day to icrease his mea to 5 sales per day? 6. Adria's mea score for four tests is 64%. He wats to icrease his mea to 68% after the fifth test. What does he eed to score i the fifth test? 7. The mea salary of the 8 people who work for a small compay is Whe a extra worker is take o this mea drops to How much does the ew worker ear? 56

13 8. The mea of 6 umbers is.3. Whe a extra umber is added, the mea chages to.9. What is the extra umber? 9. Whe 5 is added to a set of 3 umbers the mea icreases to 4.6. What was the mea of the origial 3 umbers? 0. Three umbers have a mea of 64. Whe a fourth umber is icluded the mea is doubled. What is the fourth umber? 9.4 Mea, Media ad Mode for Grouped Data The mea ad media ca be estimated from tables of grouped data. The class iterval which cotais the most values is kow as the modal class. Worked Example The table below gives data o the heights, i cm, of 5 childre. Class Iterval 40 h < h < h < h < 80 Frequecy Estimate the mea height. Estimate the media height. Fid the modal class. Solutio To estimate the mea, the mid-poit of each iterval should be used. Class Iterval Mid-poit Frequecy Mid-poit Frequecy 40 h < = h < = h < = h < = 400 Totals 5 85 Mea = 85 5 = 6 (to the earest cm) The media is the 6th value. I this case it lies i the 60 h < 70 class iterval. The 4th value i the iterval is eeded. It is estimated as = 6 (to the earest cm) The modal class is 60 h < 70 as it cotais the most values. 57

14 9.4 MEP Pupil Text 9 Also ote that whe we speak of someoe by age, say 8, the the perso could be ay age from 8 years 0 days up to 8 years 364 days (365 i a leap year!). You will see how this is tackled i the followig example. Worked Example The age of childre i a primary school were recorded i the table below. Age Frequecy Estimate the mea. Estimate the media. Fid the modal age. Solutio To estimate the mea, we must use the mid-poit of each iterval; so, for example for '5 6', which really meas 5 age < 7, the mid-poit is take as 6. Class Iterval Mid-poit Frequecy Mid-poit Frequecy = = = 380 Totals Mea = = 8. (to decimal place) The media is give by the 54th value, which we have to estimate. There are 9 values i the first iterval, so we eed to estimate the 5th value i the secod iterval. As there are 40 values i the secod iterval, the media is estimated as beig 5 40 of the way alog the secod iterval. This has width 9 7 = years, so the media is estimated by 5 40 = 5. from the start of the iterval. Therefore the media is estimated as = 8. 5 years. The modal age is the 7 8 age group. 58

15 Worked Example uses what are called cotiuous data, sice height ca be of ay value. (Other examples of cotiuous data are weight, temperature, area, volume ad time.) The ext example uses discrete data, that is, data which ca take oly a particular value, such as the itegers,, 3, 4,... i this case. The calculatios for mea ad mode are ot affected but estimatio of the media requires replacig the discrete grouped data with a approximate cotiuous iterval. Worked Example 3 The umber of days that childre were missig from school due to sickess i oe year was recorded. Number of days off sick Frequecy Estimate the mea Estimate the media. Fid the modal class. Solutio The estimate is made by assumig that all the values i a class iterval are equal to the midpoit of the class iterval. Class Iterval Mid-poit Frequecy Mid-poit Frequecy = = = = = 69 Totals Mea = = days. As there are 40 pupils, we eed to cosider the mea of the 0th ad st values. These both lie i the 6 0 class iterval, which is really the class iterval, so this iterval cotais the media. As there are values i the first class iterval, the media is foud by cosiderig the 8th ad 9th values of the secod iterval. As there are values i the secod iterval, the media is estimated as beig of the way alog the secod iterval

16 9.4 MEP Pupil Text 9 But the legth of the secod iterval is = 5, so the media is estimated by = 386. from the start of this iterval. Therefore the media is estimated as = The modal class is 5, as this class cotais the most etries. Exercises. A door to door salesma keeps a record of the umber of homes he visits each day. Homes visited Frequecy Estimate the mea umber of homes visited. Estimate the media. What is the modal class?. The weights of a umber of studets were recorded i kg. Mea (kg) 30 w < w < w < w < w < 55 Frequecy Estimate the mea weight. Estimate the media. What is the modal class? 3. A stopwatch was used to fid the time that it took a group of childre to ru 00 m. Time (secods) 0 t < 5 5 t < 0 0 t < 5 5 t < 30 Frequecy Is the media i the modal class? Estimate the mea. Estimate the media. (d) Is the media greater or less tha the mea? 4. The distaces that childre i a year group travelled to school is recorded. Distace (km) 0 d < d < d < d < 0. Frequecy Does the modal class cotai the media? Estimate the media ad the mea. Which is the largest, the media or the mea? 60

17 5. The ages of the childre at a youth camp are summarised i the table below. Age (years) Frequecy Estimate the mea age of the childre. 6. The legths of a umber of leaves collected for a project are recorded. Legth (cm) Frequecy Estimate the mea the media legth of a leaf. 7. The table shows how may ights people sped at a campsite. Number of ights Frequecy Estimate the mea. Estimate the media. What is the modal class? 8. A teacher otes the umber of correct aswers give by a class o a multiple-choice test. Correct aswers Frequecy (i) Estimate the mea. (ii) Estimate the media. (iii) What is the modal class? Aother class took the same test. Their results are give below. Correct aswers Frequecy (i) Estimate the mea. (ii) Estimate the media. (iii) What is the modal class? How do the results for the two classes compare? Iformatio A quartile is oe of 3 values (lower quartile, media ad upper quartile) which divides data ito 4 equal groups. A percetile is oe of 99 values which divides data ito 00 equal groups. The lower quartile correspods to the 5th percetile. The media correspods to the 50th percetile. The upper quartile correspods to the 75th percetile. 6

18 9.4 MEP Pupil Text childre are asked how much pocket moey they were give last week. Their replies are show i this frequecy table. Pocket moey Frequecy f Which is the modal class? Calculate a estimate of the mea amout of pocket moey received per child. (NEAB) 0. The graph shows the umber of hours a sample of people spet viewig televisio oe week durig the summer. 40 Number of people Viewig time (hours) Copy ad complete the frequecy table for this sample. Viewig time Number of (h hours) people 0 h < h < h < h < h < h < 60 Aother survey is carried out durig the witer. State oe differece you would expect to see i the data. Use the mid-poits of the class itervals to calculate the mea viewig time for these people. You may fid it helpful to use the table below. 6

19 Viewig time Mid-poit (h hours) Mid-poit Frequecy Frequecy 0 h < h < h < h < h < h < (SEG). I a experimet, 50 people were asked to estimate the legth of a rod to the earest cetimetre. The results were recorded. Legth (cm) Frequecy Fid the value of the media. Calculate the mea legth. I a secod experimet aother 50 people were asked to estimate the legth of the same rod. The most commo estimate was 3 cm. The rage of the estimates was 3 cm. Make two comparisos betwee the results of the two experimets. (SEG). The followig list shows the maximum daily temperature, i F, throughout the moth of April Copy ad complete the grouped frequecy table below. Temperature, T Frequecy 40 < T < T < T < T 6 6 < T 70 Use the table of values i part to calculate a estimate of the mea of this distributio. You must show your workig clearly. Draw a histogram to represet your distributio i part. (MEG) 63

20 9.5 Cumulative Frequecy Cumulative frequecies are useful if more detailed iformatio is required about a set of data. I particular, they ca be used to fid the media ad iter-quartile rage. The iter-quartile rage cotais the middle 50% of the sample ad describes how spread out the data are. This is illustrated i Example. Worked Example For the data give i the table, draw up a cumulative frequecy table ad the draw a cumulative frequecy graph. Solutio The table below shows how to calculate the cumulative frequecies. Height (cm) Frequecy 90 < h < h 0 0 < h < h < h < h 50 6 Height (cm) Frequecy Cumulative Frequecy 90 < h < h = 7 0 < h = 57 0 < h = < h = < h = A graph ca the be plotted usig poits as show below (50,) (40,06) q y Cumulative Frequecy (0,57) (30,88) 40 0 (0,7) 0 (90,0) (00,5) Height (cm) 64

21 Note A more accurate graph is foud by drawig a smooth curve through the poits, rather tha usig straight lie segmets (50,) (40,06) Cumulative Frequecy (0,57) (30,88) 40 0 (0,7) Worked Example (90,0) (00,5) Height (cm) The cumulative frequecy graph below gives the results of 0 studets o a test Cumulative Frequecy Test Score 65

22 9.5 MEP Pupil Text 9 Use the graph to fid: the media score, the iter-quartile rage, the mark which was attaied by oly 0% of the studets, (d) the umber of studets who scored more tha 75 o the test. Solutio Sice of 0 is 60, the media ca be foud by startig at 60 o the vertical scale, movig horizotally to the graph lie ad the movig vertically dow to meet the horizotal scale. I this case the media is 53. Cumulative Frequecy Start at Score Media = To fid out the iter-quartile rage, we must cosider the middle 50% of the studets. To fid the lower quartile, start at of 0, which is This gives Lower Quartile = 43. To fid the upper quartile, start at 3 of 0, which is This gives Upper Quartile = 67. The iter-quartile rage is the Cumulative Frequecy Upper quartile = Lower quartile = 43 Test Score Iter - quartile Rage = Upper Quartile Lower Quartile = = 4. 66

23 Cumulative Frequecy Here the mark which was attaied by the top 0% is required. 0%of 0 = so start at 08 o the cumulative frequecy scale. This gives a mark of Test Score (d) To fid the umber of studets who scored more tha 75, start at 75 o the horizotal axis This gives a cumulative frequecy of 03. Cumulative Frequecy 60 So the umber of studets with a score greater tha 75 is 0 03 = 7. As i Worked Example, a more accurate estimate for the media ad iter-quartile rage is obtaied if you draw a smooth curve through the data poits. Exercises. Make a cumulative frequecy table for each set of data give below. The draw a cumulative frequecy graph ad use it to fid the media ad iter-quartile rage Test Score 75 Joh weighed each apple i a large box. His results are give i this table. Weight of apple (g) 60 < w < w < w 0 0 < w < w 60 Frequecy Pasi asked the studets i his class how far they travelled to school each day. His results are give below. Distace (km) 0 < d < d < d 3 3 < d 4 4 < d 5 5 < d 6 Frequecy

24 9.5 MEP Pupil Text 9 A P.E. teacher recorded the distaces childre could reach i the log jump evet. His records are summarised i the table below. Legth of jump (m) < d < d 3 3 < d 4 4 < d 5 5 < d 6 Frequecy A farmer grows a type of wheat i two differet fields. He takes a sample of 50 heads of cor from each field at radom ad weighs the grais he obtais. Mass of grai (g) 0< m 5 5< m 0 0 < m 5 5< m 0 0 < m 5 5< m 30 Frequecy Field A Frequecy Field B Draw cumulative frequecy graphs for each field. Fid the media ad iter-quartile rage for each field. Commet o your results. 3. A cosumer group tests two types of batteries usig a persoal stereo. Lifetime (hours) < l 3 3 < l 4 4 < l 5 5 < l 6 6 < l 7 7 < l 8 Frequecy Type A Frequecy Type B Use cumulative frequecy graphs to fid the media ad iter-quartile rage for each type of battery. Which type of battery would you recommed ad why? 4. The table below shows how the height of girls of a certai age vary. The data was gathered usig a large-scale survey. Height (cm) 50 < h < h < h < h < h < h < h 85 Frequecy A doctor wishes to be able to classify childre as: Category Percetage of Populatio Very Tall 5% Tall 5% Normal 60% Short 5% Very short 5% Use a cumulative frequecy graph to fid the heights of childre i each category. 68

25 5. The maager of a double glazig compay employs 30 salesme. Each year he awards bouses to his salesme. Bous Awarded to 500 Best 0% of salesme 50 Middle 70% of salesme 50 Bottom 0% of salesme The sales made durig 995 ad 996 are show i the table below. Value of sales ( 000) 0 < V < V < V < V < V 500 Frequecy Frequecy Use cumulative frequecy graphs to fid the values of sales eeded to obtai each bous i the years 995 ad The histogram shows the cost of buyig a particular toy i a umber of differet shops Frequecy Price ( ) Draw a cumulative frequecy graph ad use it to aswer the followig questios. (i) How may shops charged more tha.65? (ii) What is the media price? (iii) How may shops charged less tha.30? (iv) How may shops charged betwee.0 ad.60? (v) How may shops charged betwee.00 ad.50? Commet o which of your aswers are exact ad which are estimates. 69

26 9.5 MEP Pupil Text 9 7. Laura ad Joy played 40 games of golf together. The table below shows Laura's scores. Scores (x) 70 < x < x < x < x 0 0 < x 0 Frequecy O a grid similar to the oe below, draw a cumulative frequecy diagram to show Laura's scores. 40 q y 30 Cumulative Frequecy Score Makig your method clear, use your graph to fid (i) Laura's media score, (ii) the iter-quartile rage of her scores. Joy's media score was 03. The iter-quartile rage of her scores was 6. (i) Who was the more cosistet player? Give a reaso for your choice. (ii) The wier of a game of golf is the oe with the lowest score. Who wo most of these 40 games? Give a reaso for your choice. (NEAB) 8. A sample of 80 electric light bulbs was take. The lifetime of each light bulb was recorded. The results are show below. Lifetime (hours) Frequecy Cumulative Frequecy 4 7 Copy ad complete the table of values for the cumulative frequecy. Draw the cumulative frequecy curve, usig a grid as show below. 70

27 q y Cumulative Frequecy (d) (e) Lifetime (hours) Use your graph to estimate the umber of light bulbs which lasted more tha 030 hours. Use your graph to estimate the iter-quartile rage of the lifetimes of the light bulbs. A secod sample of 80 light bulbs has the same media lifetime as the first sample. Its iter-quartile rage is 90 hours. What does this tell you about the differece betwee the two samples? (SEG) 9. The umbers of joureys made by a group of people usig public trasport i oe moth are summarised i the table. Number of joureys Number of people Copy ad complete the cumulative frequecy table below. Number of joureys Cumulative frequecy (i) Draw the cumulative frequecy graph, usig a grid as below. 40 q y Cumulative Frequecy Number of joureys 7

28 9.5 MEP Pupil Text 9 (ii) (iii) Use your graph to estimate the media umber of joureys. Use your graph to estimate the umber of people who made more tha 44 joureys i the moth. The umbers of joureys made usig public trasport i oe moth, by aother group of people, are show i the graph. 40 Cumulative Frequecy Number of joureys Make oe compariso betwee the umbers of joureys made by these two groups. (SEG) 0. The cumulative frequecy graph below gives iformatio o the house prices i 99. The cumulative frequecy is give as a percetage of all houses i Eglad. 00 q y ( ) Cumulative Frequecy House prices i 99 i ( ) This grouped frequecy table gives the percetage distributio of house prices (p) i Eglad i

### GCSE STATISTICS. 4) How to calculate the range: The difference between the biggest number and the smallest number.

GCSE STATISTICS You should kow: 1) How to draw a frequecy diagram: e.g. NUMBER TALLY FREQUENCY 1 3 5 ) How to draw a bar chart, a pictogram, ad a pie chart. 3) How to use averages: a) Mea - add up all

### 3.1 Measures of Central Tendency. Introduction 5/28/2013. Data Description. Outline. Objectives. Objectives. Traditional Statistics Average

5/8/013 C H 3A P T E R Outlie 3 1 Measures of Cetral Tedecy 3 Measures of Variatio 3 3 3 Measuresof Positio 3 4 Exploratory Data Aalysis Copyright 013 The McGraw Hill Compaies, Ic. C H 3A P T E R Objectives

### Measures of Spread and Boxplots Discrete Math, Section 9.4

Measures of Spread ad Boxplots Discrete Math, Sectio 9.4 We start with a example: Example 1: Comparig Mea ad Media Compute the mea ad media of each data set: S 1 = {4, 6, 8, 10, 1, 14, 16} S = {4, 7, 9,

### Measures of Central Tendency

Measures of Cetral Tedecy A studet s grade will be determied by exam grades ( each exam couts twice ad there are three exams, HW average (couts oce, fial exam ( couts three times. Fid the average if the

### Section 7-3 Estimating a Population. Requirements

Sectio 7-3 Estimatig a Populatio Mea: σ Kow Key Cocept This sectio presets methods for usig sample data to fid a poit estimate ad cofidece iterval estimate of a populatio mea. A key requiremet i this sectio

### Descriptive statistics deals with the description or simple analysis of population or sample data.

Descriptive statistics Some basic cocepts A populatio is a fiite or ifiite collectio of idividuals or objects. Ofte it is impossible or impractical to get data o all the members of the populatio ad a small

### 1 Computing the Standard Deviation of Sample Means

Computig the Stadard Deviatio of Sample Meas Quality cotrol charts are based o sample meas ot o idividual values withi a sample. A sample is a group of items, which are cosidered all together for our aalysis.

### NPTEL STRUCTURAL RELIABILITY

NPTEL Course O STRUCTURAL RELIABILITY Module # 0 Lecture 1 Course Format: Web Istructor: Dr. Aruasis Chakraborty Departmet of Civil Egieerig Idia Istitute of Techology Guwahati 1. Lecture 01: Basic Statistics

### Confidence Intervals and Sample Size

8/7/015 C H A P T E R S E V E N Cofidece Itervals ad Copyright 015 The McGraw-Hill Compaies, Ic. Permissio required for reproductio or display. 1 Cofidece Itervals ad Outlie 7-1 Cofidece Itervals for the

### AQA STATISTICS 1 REVISION NOTES

AQA STATISTICS 1 REVISION NOTES AVERAGES AND MEASURES OF SPREAD www.mathsbox.org.uk Mode : the most commo or most popular data value the oly average that ca be used for qualitative data ot suitable if

### Chapter 7: Confidence Interval and Sample Size

Chapter 7: Cofidece Iterval ad Sample Size Learig Objectives Upo successful completio of Chapter 7, you will be able to: Fid the cofidece iterval for the mea, proportio, ad variace. Determie the miimum

### Definition. A variable X that takes on values X 1, X 2, X 3,...X k with respective frequencies f 1, f 2, f 3,...f k has mean

1 Social Studies 201 October 13, 2004 Note: The examples i these otes may be differet tha used i class. However, the examples are similar ad the methods used are idetical to what was preseted i class.

### Lesson 17 Pearson s Correlation Coefficient

Outlie Measures of Relatioships Pearso s Correlatio Coefficiet (r) -types of data -scatter plots -measure of directio -measure of stregth Computatio -covariatio of X ad Y -uique variatio i X ad Y -measurig

### Confidence Intervals. CI for a population mean (σ is known and n > 30 or the variable is normally distributed in the.

Cofidece Itervals A cofidece iterval is a iterval whose purpose is to estimate a parameter (a umber that could, i theory, be calculated from the populatio, if measuremets were available for the whole populatio).

### CHAPTER 7: Central Limit Theorem: CLT for Averages (Means)

CHAPTER 7: Cetral Limit Theorem: CLT for Averages (Meas) X = the umber obtaied whe rollig oe six sided die oce. If we roll a six sided die oce, the mea of the probability distributio is X P(X = x) Simulatio:

### 4.1 Sigma Notation and Riemann Sums

0 the itegral. Sigma Notatio ad Riema Sums Oe strategy for calculatig the area of a regio is to cut the regio ito simple shapes, calculate the area of each simple shape, ad the add these smaller areas

### Example Consider the following set of data, showing the number of times a sample of 5 students check their per day:

Sectio 82: Measures of cetral tedecy Whe thikig about questios such as: how may calories do I eat per day? or how much time do I sped talkig per day?, we quickly realize that the aswer will vary from day

### Definition. Definition. 7-2 Estimating a Population Proportion. Definition. Definition

7- stimatig a Populatio Proportio I this sectio we preset methods for usig a sample proportio to estimate the value of a populatio proportio. The sample proportio is the best poit estimate of the populatio

### NATIONAL SENIOR CERTIFICATE GRADE 11

NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P EXEMPLAR 007 MARKS: 50 TIME: 3 hours This questio paper cosists of pages, 4 diagram sheets ad a -page formula sheet. Please tur over Mathematics/P DoE/Exemplar

### SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES

SECTION 1.5 : SUMMATION NOTATION + WORK WITH SEQUENCES Read Sectio 1.5 (pages 5 9) Overview I Sectio 1.5 we lear to work with summatio otatio ad formulas. We will also itroduce a brief overview of sequeces,

### Determining the sample size

Determiig the sample size Oe of the most commo questios ay statisticia gets asked is How large a sample size do I eed? Researchers are ofte surprised to fid out that the aswer depeds o a umber of factors

### ME 101 Measurement Demonstration (MD 1) DEFINITIONS Precision - A measure of agreement between repeated measurements (repeatability).

INTRODUCTION This laboratory ivestigatio ivolves makig both legth ad mass measuremets of a populatio, ad the assessig statistical parameters to describe that populatio. For example, oe may wat to determie

### I. Chi-squared Distributions

1 M 358K Supplemet to Chapter 23: CHI-SQUARED DISTRIBUTIONS, T-DISTRIBUTIONS, AND DEGREES OF FREEDOM To uderstad t-distributios, we first eed to look at aother family of distributios, the chi-squared distributios.

### Case Study. Normal and t Distributions. Density Plot. Normal Distributions

Case Study Normal ad t Distributios Bret Halo ad Bret Larget Departmet of Statistics Uiversity of Wiscosi Madiso October 11 13, 2011 Case Study Body temperature varies withi idividuals over time (it ca

### Confidence Intervals for One Mean with Tolerance Probability

Chapter 421 Cofidece Itervals for Oe Mea with Tolerace Probability Itroductio This procedure calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) with

### Mathematical goals. Starting points. Materials required. Time needed

Level A1 of challege: C A1 Mathematical goals Startig poits Materials required Time eeded Iterpretig algebraic expressios To help learers to: traslate betwee words, symbols, tables, ad area represetatios

### University of California, Los Angeles Department of Statistics. Distributions related to the normal distribution

Uiversity of Califoria, Los Ageles Departmet of Statistics Statistics 100B Istructor: Nicolas Christou Three importat distributios: Distributios related to the ormal distributio Chi-square (χ ) distributio.

### Confidence Intervals for the Population Mean

Cofidece Itervals Math 283 Cofidece Itervals for the Populatio Mea Recall that from the empirical rule that the iterval of the mea plus/mius 2 times the stadard deviatio will cotai about 95% of the observatios.

### FM4 CREDIT AND BORROWING

FM4 CREDIT AND BORROWING Whe you purchase big ticket items such as cars, boats, televisios ad the like, retailers ad fiacial istitutios have various terms ad coditios that are implemeted for the cosumer

### Confidence Intervals for One Mean

Chapter 420 Cofidece Itervals for Oe Mea Itroductio This routie calculates the sample size ecessary to achieve a specified distace from the mea to the cofidece limit(s) at a stated cofidece level for a

### Statistical Methods. Chapter 1: Overview and Descriptive Statistics

Geeral Itroductio Statistical Methods Chapter 1: Overview ad Descriptive Statistics Statistics studies data, populatio, ad samples. Descriptive Statistics vs Iferetial Statistics. Descriptive Statistics

### CS103X: Discrete Structures Homework 4 Solutions

CS103X: Discrete Structures Homewor 4 Solutios Due February 22, 2008 Exercise 1 10 poits. Silico Valley questios: a How may possible six-figure salaries i whole dollar amouts are there that cotai at least

### Simple Annuities Present Value.

Simple Auities Preset Value. OBJECTIVES (i) To uderstad the uderlyig priciple of a preset value auity. (ii) To use a CASIO CFX-9850GB PLUS to efficietly compute values associated with preset value auities.

### 5.4 Amortization. Question 1: How do you find the present value of an annuity? Question 2: How is a loan amortized?

5.4 Amortizatio Questio 1: How do you fid the preset value of a auity? Questio 2: How is a loa amortized? Questio 3: How do you make a amortizatio table? Oe of the most commo fiacial istrumets a perso

### .04. This means \$1000 is multiplied by 1.02 five times, once for each of the remaining sixmonth

Questio 1: What is a ordiary auity? Let s look at a ordiary auity that is certai ad simple. By this, we mea a auity over a fixed term whose paymet period matches the iterest coversio period. Additioally,

### Week 3 Conditional probabilities, Bayes formula, WEEK 3 page 1 Expected value of a random variable

Week 3 Coditioal probabilities, Bayes formula, WEEK 3 page 1 Expected value of a radom variable We recall our discussio of 5 card poker hads. Example 13 : a) What is the probability of evet A that a 5

### Approximating Area under a curve with rectangles. To find the area under a curve we approximate the area using rectangles and then use limits to find

1.8 Approximatig Area uder a curve with rectagles 1.6 To fid the area uder a curve we approximate the area usig rectagles ad the use limits to fid 1.4 the area. Example 1 Suppose we wat to estimate 1.

### In nite Sequences. Dr. Philippe B. Laval Kennesaw State University. October 9, 2008

I ite Sequeces Dr. Philippe B. Laval Keesaw State Uiversity October 9, 2008 Abstract This had out is a itroductio to i ite sequeces. mai de itios ad presets some elemetary results. It gives the I ite Sequeces

### sum of all values n x = the number of values = i=1 x = n n. When finding the mean of a frequency distribution the mean is given by

Statistics Module Revisio Sheet The S exam is hour 30 miutes log ad is i two sectios Sectio A 3 marks 5 questios worth o more tha 8 marks each Sectio B 3 marks questios worth about 8 marks each You are

### ARITHMETIC AND GEOMETRIC PROGRESSIONS

Arithmetic Ad Geometric Progressios Sequeces Ad ARITHMETIC AND GEOMETRIC PROGRESSIONS Successio of umbers of which oe umber is desigated as the first, other as the secod, aother as the third ad so o gives

### Listing terms of a finite sequence List all of the terms of each finite sequence. a) a n n 2 for 1 n 5 1 b) a n for 1 n 4 n 2

74 (4 ) Chapter 4 Sequeces ad Series 4. SEQUENCES I this sectio Defiitio Fidig a Formula for the th Term The word sequece is a familiar word. We may speak of a sequece of evets or say that somethig is

### Hypothesis Tests Applied to Means

The Samplig Distributio of the Mea Hypothesis Tests Applied to Meas Recall that the samplig distributio of the mea is the distributio of sample meas that would be obtaied from a particular populatio (with

### Statistics Lecture 14. Introduction to Inference. Administrative Notes. Hypothesis Tests. Last Class: Confidence Intervals

Statistics 111 - Lecture 14 Itroductio to Iferece Hypothesis Tests Admiistrative Notes Sprig Break! No lectures o Tuesday, March 8 th ad Thursday March 10 th Exteded Sprig Break! There is o Stat 111 recitatio

Ca you aswer these questios? A savigs accout gives % iterest per aum.. If 000 is ivested i this accout, how much will be i the accout at the ed of years? A ew car costs 16 000 ad its value falls by 1%

### Geometric Sequences and Series. Geometric Sequences. Definition of Geometric Sequence. such that. a2 4

3330_0903qxd /5/05 :3 AM Page 663 Sectio 93 93 Geometric Sequeces ad Series 663 Geometric Sequeces ad Series What you should lear Recogize, write, ad fid the th terms of geometric sequeces Fid th partial

### Key Ideas Section 8-1: Overview hypothesis testing Hypothesis Hypothesis Test Section 8-2: Basics of Hypothesis Testing Null Hypothesis

Chapter 8 Key Ideas Hypothesis (Null ad Alterative), Hypothesis Test, Test Statistic, P-value Type I Error, Type II Error, Sigificace Level, Power Sectio 8-1: Overview Cofidece Itervals (Chapter 7) are

### Stat 104 Lecture 2. Variables and their distributions. DJIA: monthly % change, 2000 to Finding the center of a distribution. Median.

Stat 04 Lecture Statistics 04 Lecture (IPS. &.) Outlie for today Variables ad their distributios Fidig the ceter Measurig the spread Effects of a liear trasformatio Variables ad their distributios Variable:

### The following example will help us understand The Sampling Distribution of the Mean. C1 C2 C3 C4 C5 50 miles 84 miles 38 miles 120 miles 48 miles

The followig eample will help us uderstad The Samplig Distributio of the Mea Review: The populatio is the etire collectio of all idividuals or objects of iterest The sample is the portio of the populatio

### when n = 1, 2, 3, 4, 5, 6, This list represents the amount of dollars you have after n days. Note: The use of is read as and so on.

Geometric eries Before we defie what is meat by a series, we eed to itroduce a related topic, that of sequeces. Formally, a sequece is a fuctio that computes a ordered list. uppose that o day 1, you have

### Biology 171L Environment and Ecology Lab Lab 2: Descriptive Statistics, Presenting Data and Graphing Relationships

Biology 171L Eviromet ad Ecology Lab Lab : Descriptive Statistics, Presetig Data ad Graphig Relatioships Itroductio Log lists of data are ofte ot very useful for idetifyig geeral treds i the data or the

### Searching Algorithm Efficiencies

Efficiecy of Liear Search Searchig Algorithm Efficiecies Havig implemeted the liear search algorithm, how would you measure its efficiecy? A useful measure (or metric) should be geeral, applicable to ay

### Arithmetic Sequences and Partial Sums. Arithmetic Sequences. Definition of Arithmetic Sequence. Example 1. 7, 11, 15, 19,..., 4n 3,...

3330_090.qxd 1/5/05 11:9 AM Page 653 Sectio 9. Arithmetic Sequeces ad Partial Sums 653 9. Arithmetic Sequeces ad Partial Sums What you should lear Recogize,write, ad fid the th terms of arithmetic sequeces.

### Overview. Learning Objectives. Point Estimate. Estimation. Estimating the Value of a Parameter Using Confidence Intervals

Overview Estimatig the Value of a Parameter Usig Cofidece Itervals We apply the results about the sample mea the problem of estimatio Estimatio is the process of usig sample data estimate the value of

### TIEE Teaching Issues and Experiments in Ecology - Volume 1, January 2004

TIEE Teachig Issues ad Experimets i Ecology - Volume 1, Jauary 2004 EXPERIMENTS Evirometal Correlates of Leaf Stomata Desity Bruce W. Grat ad Itzick Vatick Biology, Wideer Uiversity, Chester PA, 19013

### 2-3 The Remainder and Factor Theorems

- The Remaider ad Factor Theorems Factor each polyomial completely usig the give factor ad log divisio 1 x + x x 60; x + So, x + x x 60 = (x + )(x x 15) Factorig the quadratic expressio yields x + x x

### Hypothesis testing. Null and alternative hypotheses

Hypothesis testig Aother importat use of samplig distributios is to test hypotheses about populatio parameters, e.g. mea, proportio, regressio coefficiets, etc. For example, it is possible to stipulate

### Riemann Sums y = f (x)

Riema Sums Recall that we have previously discussed the area problem I its simplest form we ca state it this way: The Area Problem Let f be a cotiuous, o-egative fuctio o the closed iterval [a, b] Fid

### Basic Elements of Arithmetic Sequences and Series

MA40S PRE-CALCULUS UNIT G GEOMETRIC SEQUENCES CLASS NOTES (COMPLETED NO NEED TO COPY NOTES FROM OVERHEAD) Basic Elemets of Arithmetic Sequeces ad Series Objective: To establish basic elemets of arithmetic

### Predictive Modeling Data. in the ACT Electronic Student Record

Predictive Modelig Data i the ACT Electroic Studet Record overview Predictive Modelig Data Added to the ACT Electroic Studet Record With the release of studet records i September 2012, predictive modelig

### CHAPTER 11 Financial mathematics

CHAPTER 11 Fiacial mathematics I this chapter you will: Calculate iterest usig the simple iterest formula ( ) Use the simple iterest formula to calculate the pricipal (P) Use the simple iterest formula

### ORDERS OF GROWTH KEITH CONRAD

ORDERS OF GROWTH KEITH CONRAD Itroductio Gaiig a ituitive feel for the relative growth of fuctios is importat if you really wat to uderstad their behavior It also helps you better grasp topics i calculus

### UNIT 3 SUMMARY STATIONS THROUGHOUT THE NEXT 2 DAYS, WE WILL BE SUMMARIZING THE CONCEPT OF EXPONENTIAL FUNCTIONS AND THEIR VARIOUS APPLICATIONS.

Name: Group Members: UNIT 3 SUMMARY STATIONS THROUGHOUT THE NEXT DAYS, WE WILL BE SUMMARIZING THE CONCEPT OF EXPONENTIAL FUNCTIONS AND THEIR VARIOUS APPLICATIONS. EACH ACTIVITY HAS A COLOR THAT CORRESPONDS

### Hypergeometric Distributions

7.4 Hypergeometric Distributios Whe choosig the startig lie-up for a game, a coach obviously has to choose a differet player for each positio. Similarly, whe a uio elects delegates for a covetio or you

### 8.1 Arithmetic Sequences

MCR3U Uit 8: Sequeces & Series Page 1 of 1 8.1 Arithmetic Sequeces Defiitio: A sequece is a comma separated list of ordered terms that follow a patter. Examples: 1, 2, 3, 4, 5 : a sequece of the first

### Chapter 6: Variance, the law of large numbers and the Monte-Carlo method

Chapter 6: Variace, the law of large umbers ad the Mote-Carlo method Expected value, variace, ad Chebyshev iequality. If X is a radom variable recall that the expected value of X, E[X] is the average value

### 1. C. The formula for the confidence interval for a population mean is: x t, which was

s 1. C. The formula for the cofidece iterval for a populatio mea is: x t, which was based o the sample Mea. So, x is guarateed to be i the iterval you form.. D. Use the rule : p-value

### Math C067 Sampling Distributions

Math C067 Samplig Distributios Sample Mea ad Sample Proportio Richard Beigel Some time betwee April 16, 2007 ad April 16, 2007 Examples of Samplig A pollster may try to estimate the proportio of voters

### 1 The Binomial Theorem: Another Approach

The Biomial Theorem: Aother Approach Pascal s Triagle I class (ad i our text we saw that, for iteger, the biomial theorem ca be stated (a + b = c a + c a b + c a b + + c ab + c b, where the coefficiets

### Ch 7.1 pg. 364 #11, 13, 15, 17, 19, 21, 23, 25

Math 7 Elemetary Statistics: A Brief Versio, 5/e Bluma Ch 7.1 pg. 364 #11, 13, 15, 17, 19, 1, 3, 5 11. Readig Scores: A sample of the readig scores of 35 fifth-graders has a mea of 8. The stadard deviatio

### Review for 1 sample CI Name. MULTIPLE CHOICE. Choose the one alternative that best completes the statement or answers the question.

Review for 1 sample CI Name MULTIPLE CHOICE. Choose the oe alterative that best completes the statemet or aswers the questio. Fid the margi of error for the give cofidece iterval. 1) A survey foud that

### 3. If x and y are real numbers, what is the simplified radical form

lgebra II Practice Test Objective:.a. Which is equivalet to 98 94 4 49?. Which epressio is aother way to write 5 4? 5 5 4 4 4 5 4 5. If ad y are real umbers, what is the simplified radical form of 5 y

### Trigonometric Form of a Complex Number. The Complex Plane. axis. ( 2, 1) or 2 i FIGURE 6.44. The absolute value of the complex number z a bi is

0_0605.qxd /5/05 0:45 AM Page 470 470 Chapter 6 Additioal Topics i Trigoometry 6.5 Trigoometric Form of a Complex Number What you should lear Plot complex umbers i the complex plae ad fid absolute values

### 1 Correlation and Regression Analysis

1 Correlatio ad Regressio Aalysis I this sectio we will be ivestigatig the relatioship betwee two cotiuous variable, such as height ad weight, the cocetratio of a ijected drug ad heart rate, or the cosumptio

### 3 Data Analysis and Interpretation

Chapter 3: Data and Interpretation 3 Data Analysis and Interpretation 3.1 Mean, Median, Mode and Range You have already looked at ways of collecting and representing data. In this section, you will go

### NATIONAL SENIOR CERTIFICATE GRADE 12

NATIONAL SENIOR CERTIFICATE GRADE MATHEMATICS P FEBRUARY/MARCH 009 MARKS: 50 TIME: 3 hours This questio paper cosists of 0 pages, a iformatio sheet ad 3 diagram sheets. Please tur over Mathematics/P DoE/Feb.

### Incremental calculation of weighted mean and variance

Icremetal calculatio of weighted mea ad variace Toy Fich faf@cam.ac.uk dot@dotat.at Uiversity of Cambridge Computig Service February 009 Abstract I these otes I eplai how to derive formulae for umerically

### CHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Means and Proportions

CHAPTER 8: CONFIDENCE INTERVAL ESTIMATES for Meas ad Proportios Itroductio: We wat to kow the value of a parameter for a populatio. We do t kow the value of this parameter for the etire populatio because

### 5: Introduction to Estimation

5: Itroductio to Estimatio Cotets Acroyms ad symbols... 1 Statistical iferece... Estimatig µ with cofidece... 3 Samplig distributio of the mea... 3 Cofidece Iterval for μ whe σ is kow before had... 4 Sample

### SENIOR CERTIFICATE EXAMINATIONS

SENIOR CERTIFICATE EXAMINATIONS MATHEMATICS P1 016 MARKS: 150 TIME: 3 hours This questio paper cosists of 9 pages ad 1 iformatio sheet. Please tur over Mathematics/P1 DBE/016 INSTRUCTIONS AND INFORMATION

### x : X bar Mean (i.e. Average) of a sample

A quick referece for symbols ad formulas covered i COGS14: MEAN OF SAMPLE: x = x i x : X bar Mea (i.e. Average) of a sample x i : X sub i This stads for each idividual value you have i your sample. For

### Institute for the Advancement of University Learning & Department of Statistics

Istitute for the Advacemet of Uiversity Learig & Departmet of Statistics Descriptive Statistics for Research (Hilary Term, 00) Lecture 5: Cofidece Itervals (I.) Itroductio Cofidece itervals (or regios)

### hp calculators HP 12C Platinum Statistics - correlation coefficient The correlation coefficient HP12C Platinum correlation coefficient

HP 1C Platium Statistics - correlatio coefficiet The correlatio coefficiet HP1C Platium correlatio coefficiet Practice fidig correlatio coefficiets ad forecastig HP 1C Platium Statistics - correlatio coefficiet

### Non-life insurance mathematics. Nils F. Haavardsson, University of Oslo and DNB Skadeforsikring

No-life isurace mathematics Nils F. Haavardsso, Uiversity of Oslo ad DNB Skadeforsikrig Mai issues so far Why does isurace work? How is risk premium defied ad why is it importat? How ca claim frequecy

### Chapter 9: Correlation and Regression: Solutions

Chapter 9: Correlatio ad Regressio: Solutios 9.1 Correlatio I this sectio, we aim to aswer the questio: Is there a relatioship betwee A ad B? Is there a relatioship betwee the umber of emploee traiig hours

### Lesson 15 ANOVA (analysis of variance)

Outlie Variability -betwee group variability -withi group variability -total variability -F-ratio Computatio -sums of squares (betwee/withi/total -degrees of freedom (betwee/withi/total -mea square (betwee/withi

### 7 b) 0. Guided Notes for lesson P.2 Properties of Exponents. If a, b, x, y and a, b, 0, and m, n Z then the following properties hold: 1 n b

Guided Notes for lesso P. Properties of Expoets If a, b, x, y ad a, b, 0, ad m, Z the the followig properties hold:. Negative Expoet Rule: b ad b b b Aswers must ever cotai egative expoets. Examples: 5

### PSYCHOLOGICAL STATISTICS

UNIVERSITY OF CALICUT SCHOOL OF DISTANCE EDUCATION B Sc. Cousellig Psychology (0 Adm.) IV SEMESTER COMPLEMENTARY COURSE PSYCHOLOGICAL STATISTICS QUESTION BANK. Iferetial statistics is the brach of statistics

### Review for Test 3. b. Construct the 90% and 95% confidence intervals for the population mean. Interpret the CIs.

Review for Test 3 1 From a radom sample of 36 days i a recet year, the closig stock prices of Hasbro had a mea of \$1931 From past studies we kow that the populatio stadard deviatio is \$237 a Should you

### Math 105: Review for Final Exam, Part II - SOLUTIONS

Math 5: Review for Fial Exam, Part II - SOLUTIONS. Cosider the fuctio fx) =x 3 l x o the iterval [/e, e ]. a) Fid the x- ad y-coordiates of ay ad all local extrema ad classify each as a local maximum or

### Bridging Units: Resource Pocket 4

Bridgig Uits: Resource Pocket 4 Iterative methods for solvig equatios umerically This pocket itroduces the cocepts of usig iterative methods to solve equatios umerically i cases where a algebraic approach

### Present Value Factor To bring one dollar in the future back to present, one uses the Present Value Factor (PVF): Concept 9: Present Value

Cocept 9: Preset Value Is the value of a dollar received today the same as received a year from today? A dollar today is worth more tha a dollar tomorrow because of iflatio, opportuity cost, ad risk Brigig

### PENSION ANNUITY. Policy Conditions Document reference: PPAS1(7) This is an important document. Please keep it in a safe place.

PENSION ANNUITY Policy Coditios Documet referece: PPAS1(7) This is a importat documet. Please keep it i a safe place. Pesio Auity Policy Coditios Welcome to LV=, ad thak you for choosig our Pesio Auity.

### 23.3 Sampling Distributions

COMMON CORE Locker LESSON Commo Core Math Stadards The studet is expected to: COMMON CORE S-IC.B.4 Use data from a sample survey to estimate a populatio mea or proportio; develop a margi of error through

### Time Value of Money. First some technical stuff. HP10B II users

Time Value of Moey Basis for the course Power of compoud iterest \$3,600 each year ito a 401(k) pla yields \$2,390,000 i 40 years First some techical stuff You will use your fiacial calculator i every sigle

### 1.3 Binomial Coefficients

18 CHAPTER 1. COUNTING 1. Biomial Coefficiets I this sectio, we will explore various properties of biomial coefficiets. Pascal s Triagle Table 1 cotais the values of the biomial coefficiets ( ) for 0to

### Explore Identifying Likely Population Proportions

COMMON CORE Locker LESSON Cofidece Itervals ad Margis of Error Commo Core Math Stadards The studet is expected to: COMMON CORE S-IC.B.4 Use data from a sample survey to estimate a populatio mea or proportio;