FOR 2015 EXAM AND ONWARDS NEW CONCISE A B FOR LEAVING CERT ORDINARY LEVEL LOUISE BOYLAN, BRENDAN GUILDEA & GEORGE HUMPHREY

Transcription

1 FOR 05 EXAM AND ONWARDS T N E M E s n L o i t a P m r P SU Transfo s h p a r G, s g Statistic Coverin & NEW CONCISE A AND B FOR LEAVING CERT ORDINARY LEVEL LOUISE BOYLAN, BRENDAN GUILDEA & GEORGE HUMPHREY

2 Contents. Transforming Graphs 3. Graphs of Derivatives 0 3. Inferential Statistics 6 Margin of errror 6 Confidence interval 8 Hpothesis testing 4 4. Answers 30 Acknowledgement The authors would like to thank Stace Carter in Our Lad s College Greenhills, Drogheda and Mark Lnch in Monkstown CBC for all their help.

3

4 SECTION TRANSFORMING GRAPHS Changing the equation of a function affects the graph of the function.. = f () ± c + c moves the graph in the positive -direction. c moves the graph in the negative -direction. The function f () is shown. The function f () + 3 represents a translation verticall upwards b 3 units. The function f () 4 represents a translation verticall downwards b 4 units = f ( ± c) f ( + c) moves the graph in the negative -direction. f ( c) moves the graph in the positive -direction. The function f () is shown. The function f ( + ) represents a translation horizontall to the left b unit. The function f ( ) represents a translation horizontall to the right b units. f( + ) 9 f( )

5 NEW CONCISE PROJECT MATHS 3A & 3B 3. = and = f( ) The function f () is shown (in blue color). The graph of = f () is a reflection in the -ais of the graph of = f (). The graph of = f ( ) is a reflection in the -ais of the graph of = f (). f( ) = k, where k 7 0 Multipling a function b a constant will not change the points where the function crosses the -ais, the roots of the function. However, if k 7, the curve will appear to stretch. If k 6, the curve will appear to compress

6 TRANSFORMING GRAPHS The diagram shows the graph of a function, f (). On the same ais and scale, graph the following: (i) f () + 3 (ii) f ( + ) (iii) (iv) f () Solution: (i) f () + 3 Adding 3 units to the f () function moves the graph 3 units upwards. + 3 (ii) f ( + ) Adding units to the part of the function moves the function units to the left. Subtracting unit from the function moves the graph unit downwards f( + ) 5 5

7 NEW CONCISE PROJECT MATHS 3A & 3B (iii) f () A constant in front of the function multiplies all -values b that constant. Therefore, the height of the graph is halved. 4 3 (iv) f () Putting a negative in front of the function reflects the function in the -ais

8 TRANSFORMING GRAPHS Eercise.. The diagram shows the graph of a function k (). Cop the graph into our copbook. On the same ais and scale, graph the following: (i) k() 3 (ii) k( 3) (iii) k( + ) k() The diagram shows the graph of a function g(). 4 Cop the graph into our copbook. On the same ais and scale, graph the following: (i) g() (ii) g( ) (iii) g() (iv) g() g() 7

9 NEW CONCISE PROJECT MATHS 3A & 3B 3. The diagram shows the graph of a function f () = 4. Cop the graph into our copbook. On the same ais and scale, graph the following: (i) g() = f () + 3 (ii) k() = f ( + ) 3 (iii) m() = 4 f () The diagram shows the graph of a cubic function. Cop the graph into our copbook. On the same ais and scale, graph the following: (i) + (ii) f ( + )

10 TRANSFORMING GRAPHS 5. The function A (blue) is defined b the equation =. B observation or otherwise, write down the equation of the functions B, C and D. 7 6 B A C D The graph of = f () is shown in red. Write down the equations of g() and h() in terms of f (). 6 5 g() 4 3 h()

11 SECTION GRAPHS OF DERIVATIVES Following our stud of pages in book 3A, we now proceed to the net section on recognising and sketching graphs of derivatives. When we differentiate a cubic function, we get a quadratic function. When we differentiate a quadratic function, we get a linear function. When we differentiate a linear function, we get a horizontal line. Steps in graphing the derivative of a function. The turning points of graphs are ver important. At a turning point of an graph, the graph of its derivative will cut the -ais. Plot these points first.. Then look for where the given graph is increasing or decreasing: When the graph is increasing, its derivative will be positive and so it will be above the -ais. When the graph is decreasing, its derivative will be negative and so it will be below the -ais. Look at the net eample for a step-b-step solution for graphing the derivative of a function. The diagram shows the graph of a quadratic function, f (). On the same ais and scale, sketch the graph of f (), the derivative of f (). Solution: Since f () is a quadratic function, its derivative, f (), is a linear function (straight line). 0

12 GRAPHS OF DERIVATIVES Step : Start b looking at the turning point of the f () graph. At a turning point, f 9() = 0. Therefore, at the turning point of f (), the graph of its derivative will cut the -ais. Move from the turning point up to the -ais and mark this position on the -ais. Step : To the left of the turning point, the graph of f () is decreasing. This means that it has a negative slope and therefore the value of the derivative will be negative. The slope of f () changes from a steep negative to a shallow negative, so the value for the derivative changes steadil from a large negative, value to a small negative value and eventuall to zero when it reaches the turning point. To graph the derivative we draw a straight line, below the -ais, graduall rising from a large negative value to zero, at the point marked in step. Step 3: To the right of the turning point, the graph of f () is increasing. This means that it has a positive slope and therefore the value of the derivative will be positive. The slope of f () changes from a shallow positive to a steep positive, so the value for the derivative changes steadil from zero, at the turning point, to a small positive value and rising steadil to a large positive value. To graph the derivative we draw a straight line, above the -ais, graduall rising from zero, at the point marked in step, to a large positive value. f9() f9()

13 NEW CONCISE PROJECT MATHS 3A & 3B The diagram shows the graph of a cubic function, g(). On the same ais and scale, sketch the graph of: (i) g () (ii) g () (iii) g (), where g () is the third derivative of g() g() Solution: (i) g () Since g() is a cubic function, its derivative, g (), is a quadratic function. At turning points of g(), g () = 0. Therefore, at the turning point of g(), the graph of its derivative, g (), will cut the -ais. Between the turning points, the graph of the cubic function is decreasing, therefore the graph of g () will be below the -ais between these two points. Outside of the turning points, the graph of the cubic function is increasing, therefore the graph of g () will be above the -ais between these two points. Putting all of this together gives: g () g()

14 GRAPHS OF DERIVATIVES (ii) g () At the turning point of g (), g () = 0. Therefore, at the turning point of g (), the graph of its derivative, g (), will cut the -ais. To the left of the turning points, the graph of the quadratic function, g (), is decreasing, therefore the graph of g () will be below the -ais. To the right of the turning points, the graph of the quadratic function, g (), is increasing, therefore the graph of g () will be above the -ais. Putting all of this together gives: g9() g() g99() (iii) g () The graph of g () is a straight line with a positive slope and therefore it does not have an turning points. It is increasing at a constant rate for all values of, therefore its derivative will be a constant positive value. Hence, an horizontal line above the -ais could represent g (). g9() g() g99() g999() 3

15 NEW CONCISE PROJECT MATHS 3A & 3B Eercise.. The diagrams show the graph of a quadratic function, f (). Cop the sketch of each graph into our copbook. On the same ais and scales, sketch the graph of f (), the first derivative of f (). (i) (ii). The diagrams show the graph of a quadratic function, f (). Cop the sketch of each graph into our copbook. On the same ais and scales, sketch the graph of f (), the first derivative of f (). (i) (ii) 4

16 GRAPHS OF DERIVATIVES 3. The diagrams show the graph of a quadratic function, g(). Cop the sketch of each graph into our copbook. On the same ais and scales, sketch the graph of: (a) g (), the first derivative of g() (b) (i) g (), the second derivative of g() (ii) g() g() 4. Give three reasons wh the given line represents the slope function of the given curve. (i) (ii) 5. The diagram shows the graph of a cubic function, g(). On the same ais and scale, sketch the graph of: (i) g () (ii) g () (iii) g () g() 5

17 3 SECTION INFERENTIAL STATISTICS Inferential statistics is the branch of statistics that uses probabilit and statistics to draw conclusions from data that are affected b random variation. To work on inferential statistics, we should be able to:. Estimate the value of a population proportion. Calculate the margin of error for a sample 3. Construct a confidence interval 4. Test a hpothesis about a population proportion Margin of error and confidence intervals for population proportions Man statistical studies are concerned with obtaining the proportion (percentage) of a population that has a specified attribute. In most cases the population under consideration will be large and hence it would be impractical, if not impossible, to obtain the population proportion b taking a census. Thus, we generall emplo sampling and use the sample data to make inferences about the population. The ke to the validit of an surve is randomness. It is important that the sample be chosen randoml so that the surve results can be generalised to the whole population. The results of a surve are onl an estimate of the quantit of interest. When results of surves are reported in the media, the often include a statement like: 35% of respondents favour Mr Smith in the upcoming election. However, there is a margin of error of 3 percentage points. What this means is that the people who carried out the surve are reasonabl confident that in the real election, the percentage of votes for Mr Smith will be 35% ± 3%. In other words, the are confident that if the election was held now, Mr Smith would receive somewhere between 3% and 38% of the vote. Such intervals are called confidence intervals. An estimate from a surve should be treated with caution. Sampling errors mean that the results in the sample differ from the true results due to the luck of the draw. However, it is important to remember that sampling errors do not make surves useless. Notation We use the letter p to denote the population proportion and it is this value that is to be estimated. We use pn (pronounced p hat ) to denote the sample proportion. pn is the statistic that will be used to estimate the unknown population proportion, p. p = population proportion pn = sample proportion 6

18 INFERENTIAL STATISTICS The population proportion, p, although unknown, is a fied number. On the other hand, the sample proportion, pn, is a random variable and its value depends on chance. Suppose we wanted to know the proportion (percentage) of people in Ireland who are left-handed. We randoml selected 400 people and found that 64 of them are left-handed. pn = Number of people in the sample who are left-handed The number of people sampled If 7 of 400 people sampled were left-handed, then: pn = = 0# 8 (8%) = = 0# 6 (6%) Notice that the value of pn, the sample proportion, changes depending on the sample chosen. If the sample chosen is a good representation of the population, then pn, the sample proportion, will be a good estimate of the true population proportion, p. Margin of error for the estimate of p How should we summarise the strength of the data in a surve? This is where the role of the margin of error comes in. The margin of error is a number that represents the accurac of a surve. The margin of error is denoted b E. On our course, the margin of error, at the 95% level of confidence, is given b: Margin of error = E = where n is the size of the sample n and the confidence interval is alwas 95%. If n = 00: E = If n = 400: E = If n =,000: E = If n = 0,000: E = = 0# = 0% 00 = 0 # 05 = 5% 400 = 0 # = 3 # 6% (correct to two decimal places),000 = 0 # 0 = % 0,000 7

19 NEW CONCISE PROJECT MATHS 3A & 3B Some Notes on Margin of Error On our course, the margin of error is alwas at the 95% level of confidence. As the sample size increases the margin of error decreases At the 95% level of confidence a sample of about (i) 80 has a margin of error approimatel ± % (ii),000 has a margin of error approimatel ± 3 % The size of the (original) population does not matter If the sample size, m, is doubled (sa 500 to,000) we the margin of error, E, is not halved The margin of error estimates how accuratel the resuts of a poll reflect the True feelings of the population Confidence interval The estimated proportion plus or minus its margin of error is called a confidence interval for the true proportion. The 95% confidence for a proportion is given b: sample proportion margin of error true proportion sample proportion + margin of error pn n p p N + n where n is the sample size, p is the population proportion and pn is the sample proportion. We can state with 95% confidence that the true population, p, lies inside this interval. What this means is that if the same population was surveed on numerous occasions and the confidence interval was calculated, then about 95% of these confidence intervals would contain the true proportion and about 5% of these confidence intervals would not contain the true proportion. The 95% confidence interval pˆ Ïn P pˆ + n Ï 8 Note: When working with levels of confidence (or levels of significance), statisticians can use percentages ambiguousl. In particular, the 5% level of significance and the 95% level of confidence mean the same thing, that is to sa, 5% of the time outside the confidence interval or 95% of the time inside the confidence interval. % outside 95% inside % outside

20 INFERENTIAL STATISTICS Calculate the margin of error, at the 95% level of confidence when the sample size is (i) 5 (ii) 900 (iii) 5,000 Where relevant, give our answers correct to two decimal places. Solution: E = = margin of error n (i) If n = 5 then E = = 5 5 = 0 # = 0% (ii) If n = 900 then E = = correct to two decimal = 0# Á = 3 # 33% places (iii) If n = 5,000 then E = = 0 # 044 = # 4% correct to two decimal places 5,000 Noah is sitting his Leaving Cert in June. After Christmas he made an estimate of how man CAO points he epected to get. His estimate was 405 CAO points. Noah was not ver confident of his estimate. So he allowed each of his si subject grades go up or down b 0 points. Construct a confidence interval for Noah s CAO points estimate. Solution: 6 subjects b 0 points each = 6 ë 0 = 60 points Noahs lowest estimate would be = 345 How Noahs highest estimate would be = 465 Noah s confidence interval

21 NEW CONCISE PROJECT MATHS 3A & 3B 3 A government department wants to estimate the proportion, p, of its emploees who went on sick leave during the past ear. A random sample of 5 emploees was taken. Twelve of the sample went on sick leave during the past ear. Construct a 95% confidence interval for p. Solution: Step : Calculate the sample proportion pn. pn = 5 = 0# 48 Step : Find the margin of error, E. E = = 0 # (= 0%) 5 Step 3: Construct the 95% confidence interval. or pn 0 # n p p N # p 0 # n # 0 # 8 p 0 # 68 8% p 68% Note: In this eample, the margin of error is large. Margins of error that are this big are of little use. However, to reduce the margin of error we simpl increase the sample size. 4 A random sample of 65 people were given a flu vaccine and 75 of them eperienced headaches. (i) Calculate the sample proportion, pn. (ii) Find the margin of error, E. (iii) Hence, construct a 95% confidence interval for the sample. (iv) Illustrate the 95% confidence interval on a number line. (v) With 95% confidence, what is the highest proportion of people who would eperience headaches? (vi) How could the margin of error have been reduced? Are there an implications of taking this action? 0

22 INFERENTIAL STATISTICS Solution (i) The sample proportion = pn = = 0# 8. (ii) The margin of error = E = = = 0 # 04. n 65 (iii) The 95% confidence interval is given b: (iv) pn n p p N + n 0 # 8 0 # 04 p 0 # # 04 0 # 4 p 0 # 3 95% confidence interval (v) With 95% confidence, the highest proportion who would eperience headaches is 0 3 b observing the confidence interval above. (vi) The margin of error could have been reduced b increasing the sample size. Increasing the sample size incurs etra costs, which might be considered disproportionate to the reduction in the margin of error. It also takes more time to complete the surve with an increase in sample size. These ma not be the onl acceptable reasons. 5 At the 95% confidence level, calculate the sample size, n, to have a margin of error of: (i) # 5% (ii) 3% Solution: (i) # 5% = 0 # 05 = 0#05 n = 0 # 05n (multipl both sides b n) = margin of error n (ii) 3% = 0 # 03 = 0#03 n = 0 # 03n (multipl both sides b n)

23 NEW CONCISE PROJECT MATHS 3A & 3B 0#05 = n (divide both sides b 0#05) a 0 # 05 b = n (square both sides) 6,400 = n 0#03 = n (divide both sides b 0#03) a 0 # 03 b = n (square both sides), # = n, = n Note: Alwas use the net whole number value of n, not to the nearest whole number. Eercise 3. In this eercise, where necessar, give all decimal answers correct to four decimal places and all percentages correct to two decimal places.. Write down the sample proportion, pn, in each of the following. (i) A random sample of 75 people are given a flu vaccine and 75 of them eperienced headaches. (ii) A random sample of 546 people found that 84 of them were left handed. (iii) Out of 55 cars parked in a car park. 84 were fitted with an anti-theft device. (iv) In a surve carried out in a large cit, 78 households out of a random sample of 80 owned at least one pet. (v) Tedd tosses his luck coin 87 times and a head occurs 58 times. (vi) An insurance compan conducted a surve of,800 car crashes. It found that,708 of the crashes occurred within 8-kilometers of the drivers home.. For each given random sample of size n, calculate the margin of error at the 95% confidence level. (i) n = 0,000 (ii) n =,890 (iii) n = 976 (iv) n = 86 (v) n = For each of the following, calculate the sample size, n, at the 95% confidence level to have a margin of error of: (i) 5% (ii) 4% (iii) % (iv) 5% (v) % Comment on the relationship between the margin of error and the sample size. 4. Moll is sitting her Leaving Cert in June. After Christmas she made an estimate of how man CAO points she epected to get. Her estimate was 35 CAO points. Moll was not totall confident of her estimate so she allowed each of her si subject grades to go up or down b 5 points. Construct a confidence interval for Molls CAO points estimate.

24 INFERENTIAL STATISTICS 5. Show on separate diagrams the following 95% confidence intervals. (i) In a surve, 5% of voters supported a certain candidate with a margin of error of 5%. (ii) 0 # 8 p 0 # 6 (iii) 6% p 67% 5 (iv) 8 p (v) In a clinical stud, 86% of patients reported relief after taking a new drug. The margin of error was calculated as 4 5%. 6. In a surve carried out in a large cit, 450 households out of a random sample of 65 owned at least one pet. Calculate the 95% confidence interval for the proportion of households that own at least one pet. 7. In a market research surve, 34 people out of a random sample of 00 from a certain area said that the use a particular brand of toothpaste. Find the 95% confidence interval for the proportion of people in this area who use this brand of toothpaste. 8. Out of 76 cars parked in a car park, 07 were fitted with an anti-theft device. (i) Assuming that the cars form a random sample of parked cars, construct the 95% confidence interval for the population of parked cars fitted with an anti-theft device. (ii) If the margin of error is required to be 5%, calculate the associated sample size. 9. In an opinion poll carried out before a local election, 53 people out of a random sample of 950 declare that the will vote for a particular one of the two candidates contesting the election. Construct the 95% limits for the true proportion of all voters that will vote for this candidate. In our opinion, is there significant evidence that this candidate will win the election? 0. During the making of a movie, a surve found that out of 400 etras, onl 0 were suitable for parts in a major movie. (i) Find the 95% confidence interval for the proportion of all film etras that ma be suitable for parts in major movies. (ii) With 95% confidence, what is the highest proportion of etras who would be suitable for parts in a major movie? (iii) If onl 50 etras were surveed: (a) What effect would this have on the margin of error? (b) What are the implications of taking this action?. When designing a large auditorium for a universit, an architect made the assumption that % of the students in the universit are left-handed. As a result, he wanted to design the auditorium so that % of the built-in desks in the auditorium were suitable for left-handed people. Before the auditorium was built, the head of the universit commissioned a surve to see if the architect s assumption was reasonable. 400 students were randoml selected from the student register and 7 identified as left-handed. 3

25 NEW CONCISE PROJECT MATHS 3A & 3B (i) At the 95% confidence level: (a) Calculate the margin of error (b) Construct the confidence interval. (ii) Using the results of the surve, would ou accept the architect s assumption? Justif our answer.. High-visibilit vests come in five different sizes:,, 3, 4 and 5. A road maintenance compan issued vests to 40 randoml selected emploees. The sizes issued to these 40 emploees were as follows (i) Find pn, the proportion in the sample of emploees that were issued with size 3. (ii) Calculate the 95% confidence interval for the proportion of all emploees in the compan that require size 3. (iii) How could the compan find a more precise 95% confidence interval? (iv) The estimate of p is pn. How large a sample would be needed in order to obtain an approimate 95% confidence interval of the form pn ± 0 #? 3. The results of two polls on the government s budget plan appeared in a national newspaper. Poll A stated that 79% supported the budget plan. The margin of error was plus or minus 5%. Poll B stated that 74% supported the budget plan. The margin of error was plus or minus 4%. Is it possible that both of these polls were correct in their conclusion? Justif our answer. 4. (i) A catalogue sales compan promises to deliver orders placed on the internet within five das. Follow-up calls to n randoml selected customers show that a 95% confidence interval for the proposition that all orders arrive on time (within five das) is 88% ± 6%. Find the value of n. (ii) Which, if an, of the following conclusions is correct? (In each case, justif our answer.) (a) Between 8% and 94% of all orders arrive on time. (b) 95% of all random samples of customers will show that 88% of all orders will arrive on time. (c) On 95% of the das, between 8% and 94% of the orders will arrive on time. 4 Hpothesis testing A hpothesis is a statement or conjecture whose truth has et to be proven or disproven. Eamples of hpotheses: More than half the population is satisfied with EU membership. Drinking fizz drinks causes tooth deca. The age at marriage has increased over the past 0 ears.

26 INFERENTIAL STATISTICS Null hpothesis The statement being tested in a test of significance is called the null hpothesis. The test of significance is designed to assess the strength of the evidence against the null hpothesis. Usuall the null hpothesis is a statement of no effect or no difference. We abbreviate null hpothesis as H 0. Statistics help to make decisions We can use statistics to reject claims.. Is global temperature increasing? The null hpothesis, H 0, is that global temperature is not increasing, i.e. there is no difference in temperature. The alternative hpothesis, H A, is that global temperature is increasing.. Is a new drug effective for treating HIV/AIDS? The null hpothesis, H 0, is that the new drug is not effective. The alternative hpothesis, H A, is that the new drug is effective. 3. Is a surve on left-handed people biased if it indicates that 4% of people are left-handed? The null hpothesis, H 0, is that 4% of people are left-handed, i.e. the surve is not biased. The alternative hpothesis, H A, is that the surve is biased. Often the people investigating the data hope to reject H 0. The hope: Their new drug is better than the old one or The new ad campaign is better than the old one, etc. However, in statistics, it is essential that our attitude is one of skepticism. Until we are convinced otherwise, we accept H 0. In other words, we cling to the idea that there is no change, no improvement, no deterioration, no effect. In a courtroom, the null hpothesis is that the defendant did not commit a crime. A verdict of guilt means we reject the null hpothesis, that is to sa, the defendant committed a crime. However, a verdict of not guilt does not mean the defendant did not commit a crime, but simpl that the case has not been proven. Appling this logic to hpothesis testing, we either reject H 0 or fail to reject H 0. The reasoning behind hpothesis testing is that we usuall prefer to think about getting things right rather than getting them wrong. In testing a hpothesis, data ma be given or collected. 5

27 NEW CONCISE PROJECT MATHS 3A & 3B With given data in this course, we accept the information at face value and proceed to analse the data and answer the question. To collect data, a questionnaire could be used to carr out a surve. The ke to the validit of an surve is randomness. Procedure for carring out a hpothesis test The procedure for carring out a hpothesis test will involve the following steps:. Clearl state H 0, the null hpothesis, and H A, the alternative hpothesis.. Write down or calculate the sample proportion, pn. 3. Find the margin of error. 4. Write down the confidence interval for p, using pn n p p N + n. In addition, we ma illustrate the confidence interval with a diagram. 5. (i) If the value of the population proportion stated is within the confidence interval, we fail to reject H 0. (ii) If the value of the population proportion is outside the confidence interval, reject the null hpothesis, H 0. A poll carried out b a newspaper indicated that 48% of the voting population would support a candidate in a presidential election. Three weeks later, a rival newspaper surveed,800 voters and 98 said the would support the candidate. Investigate at the 5% level of significance whether support for the candidate changed. Solution. State H 0 and H A. H 0 : The support for the candidate has remained at 48%. H A : The support for the candidate is not at 48%, i.e. the support has changed.. Sample proportion pn = 98,800 = 0# 5 3. Margin of error = E = = = 0#04 n,800 6

28 INFERENTIAL STATISTICS 4. Confidence interval 5. pn n p p N + n 0 # 5 0 # 04 p 0 # # 04 0 # 486 p 0 # # 6% p 53 # 4% 95% confidence interval 48% 48 6% 5% 53 4% The claimed voter support of 48% is not within the confidence interval, so we reject the null hpothesis, H 0. We conclude that voter support has changed. In fact, it appears to have increased. Hpothesis testing a summar In the final analsis, testing the null hpothesis, H 0 simpl involves a confidence interval and a red dot Either OR Confidence interval Confidence interval If the red dot is inside the confidence interval we fail to reject H 0 If the red dot is outside the confidence interval we reject H 0 Eercise 3.. A drugs compan produced a new pain-relieving drug for migraine sufferers and claimed that the drug had an 80% success rate. A group of doctors doubted the compan s claim. The prescribed the drug for a group of,600 patients. After one ear,,3 of these patients said that their migraine smptoms had been relieved b the drug. Calculate: (i) The sample proportion (ii) The margin of error at the 95% level of confidence. 7

29 NEW CONCISE PROJECT MATHS 3A & 3B (iii) The 95% confidence interval for the proportion of patients in the sample who had their migraine smptoms relieved (iv) State the null hpothesis, H 0. (v) Is this result consistent with the compan s claim at the 95% level of confidence? Justif our answer. A national newspaper is investigating a claim made b the CEO of a large multinational compan. The CEO claims that 90% of the compan s one million customers are satisfied with the service the receive. Using simple random sampling, the newspaper surveed 300 customers. Among the sampled customers, 55 said the were satisfied with the compan s service. (i) Construct a 95% confidence interval for the proportion of satisfied customers. (ii) Eplain what the 95% confidence interval means in the contet of the question. (iii) State the null hpothesis, H 0, and the alternative hpothesis, H A. (iv) Based on these findings, can we reject the CEO s claim? (v) How could the investigation be made more accurate? Eplain our reasoning. 3. An insurance compan conducted a surve of 4,000 car crashes. It found that 8,330 of the crashes occurred within 8 km of the driver s home. The compan claims that 60% of car crashes occur within 8 km of home. (i) State the null hpothesis, H 0. (ii) Use a hpothesis test at the 5% level of significance to decide whether there is sufficient evidence to justif the compan s claim. State our conclusion clearl. 4. Tedd tosses his luck coin,000 times and a head occurs 450 times. Tedd claims his coin is biased. Use a hpothesis test at the 5% level of significance to decide whether there is sufficient evidence to justif Tedd s claim. State the null hpothesis and state our conclusion clearl. 5. A pharmaceutical compan has developed and tested a new pain-killing drug. The compan s records show that the old drug provided relief for 7% of all patients who were administered it. A random sample of,5 were administered the new drug and 900 of these claimed that the new drug provided relief. (i) State the null hpothesis, H 0, and the alternative hpothesis, H A. (ii) Use a hpothesis test at the 5% level of significance to decide if the new drug is different from its old counterpart. State our conclusion clearl. 6. A soccer manager has a hpothesis that oung European soccer plaers born in the first three months of the ear have an advantage in being selected to represent their countr at the under-7 level over plaers born later in the same ear. (i) Estimate the epected percentage of plaers born in the first three months of the ear. (ii) Find the margin of error, at the 95% confidence level, for a sample size of

30 INFERENTIAL STATISTICS (iii) A surve of 400 plaers, selected at random, who plaed soccer at the under-7 level for their countr showed that 35% of these plaers were born in the first three months of the ear. Use a hpothesis test at the 5% level of significance to decide whether there is sufficient evidence to justif the manager s claim. State the null hpothesis and state our conclusion clearl. 9

31 ANSWERS Section Transforming Graphs k() k() (iii) 6 5 k( + ) 4 k( 3) (iv) (i) (ii) 3. 5 (ii) (i) 4 3 (iii) f( + ) Graph B: = Graph C: = ( ) Graph D: = ( + 3) g() = 3 h() = f( + ) +

32 ANSWERS Section Graphs of Derivatives. (i) (ii) f 9() f 9(). (i) (ii) f'() f'() 3. (i) (ii) g'() g () g() g''() g () g() 4. Curve is decreasing for 6, therefore the slope is negative. Turning point at =, therefore the slope is zero when =. Curve is increasing for 7, therefore the slope is positive. 3

33 NEW CONCISE PROJECT MATHS 3A & 3B 5. g() g () g () g () Section 3 Inferential Statistics Eercise 3.. (i) % (ii) 3% (iii) 3 % (iv) 3 5% (v) 4 8%. (i) 400 (ii) 65 (iii),500 (iv) 4,444 (v) 6, % to 76% 5. 4% to 44% 6. (i) 69% to 8% (ii), % to 57 %; es 8. (i) 5% to 35% (ii) 35% (iii) (a) Made wider (b) The results are less useful 9. (i) (a) 3 5% (b) (ii) Architect s assumption is too low. 0. (i) 40% (ii) 4 % to 55 8% (iii) Increase the number of emploees selected (iv) 00. Yes, because the two confidence intervals partl overlap. (i) 78 (ii) (c) True 3 Eercise 3.. (i) 0 77 (ii) 0 05 (iii) 74 5% to 79 5% (iv) H 0 : The success rate of the drug is 80% (v) No. Reject H 0.. (i) 79 % to 90 8% (iv) Do not reject CEO s claim (v) Increase the sample size 3. (i) 60% of car crashes occur within 8 km from the driver s home (ii) Within the confidence interval means do not reject H 0 4. H 0 : The coin is not biased. Reject H (i) H 0 : The success rate for the new drug is 7%. H A : The success rate for the new drug is not 7%. (ii) 70 64% to 76 36% means the new drug is 7%, i.e. not different to old drug. 6. (i) 5% (ii) ± 5% (iii) Do not reject the manager s hpothesis

34 New Concise Project Maths 3A and 3B Supplement The content covered in this supplement addresses recent changes to the Leaving Certificate Ordinar Level Project Maths sllabus. This includes additional Statistics material, which was deferred for eamination until the June 05 eam. This supplement is laid out in the same stle as the New Concise Project Maths tetbooks. It contains eplanator material, worked eamples and eercise questions, with answers given at the end of the supplement. In order to complete the Leaving Certificate Ordinar Level sllabus, the entiret of this supplement must be covered, as an addition to both of the New Concise Project Maths 3A and 3B tetbooks. The Authors Louise Bolan Louise Bolan is a maths teacher in Mount Sackville Secondar School, Dublin 0. She is an eperienced eaminer at both Junior and Leaving Certificate level and holds a Masters in ICT in Education. She regularl leads revision workshops for state eaminations candidates. Brendan Guildea is one of Irelands leading maths teachers. He presents revision seminars for students and inservice courses for teachers throughout Ireland. He features regularl in the media leading discussions on maths teaching, eams and sllabus. He also teaches DEIS Junior and Leaving Certificate students under the Trinit College Access Program. George Humphre is the author of several bestselling tetbooks and revision books for maths. He is a livel and enthusiastic teacher who conducts workshops and seminars throughout Ireland for teachers and students. GILL & MACMILLAN Cover design: Aisli Madden