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Grade C RevisionCalculating the Mean from a set of grouped data.Key Fact : ‘10≤ x < 20’ means…..Values that are to be collected are all called ‘x’.Values of x must be between 10 and 20 to go in this group. 10 itself does go into this group but the value 20 itself does not.The group actually means ‘from 10 up to, but not including 20’The next group would be ‘20≤ x < 30’.Estimated Mean: Worked Example.The heights of the 25 students in a class were measured.The table shows there heights, in centimetres.Height (cm)Frequency150 and less than 1602160 and less than 1703170 and less than 18010180 and less than 1909190 and less than 2001Calculate an estimate of the mean height of these students.The data was collected by measuring - it is continuous.We know from the table that 2 values between 150 and 160 were collected, 3 values between 160 and 170 were collected and so on….We do not know, however, what those actual values are !!!To calculate the mean we need to add up all the actual values and divide by how many there are.We don’t know any of the actual values but we do know there were 25 of them.To solve the problem we have to make a guess. We assume that all of the values in each group are the same. We assume that they are all the middle value in each group.So the 2 values that were between 150 and 160 are both assumed to be 155, the 3 values between 160 and 170 are all assumed to be 165 and so on ….We could rewrite the table above as belowHeight (cm)FrequencyAssumed Value150 and less than 1602155160 and less than 1703165170 and less than 18010175180 and less than 1909185190 and less than 2001195To work out the mean we then have to add all the assumed values and divide by 25.Mean=155+155+165+165+165+175+175+175+175+175+175+175+175+175175+185+185+185+185+185+185+185+185+185+195 Mean = 4415÷25 = 176.6cmThis is a correct but long winded method. If there were 200 values to add up it would not be practical to write out all of the values. Instead we use the table.Height (cm)FrequencyMid PointFreq x Mid pointHelpTotal150 and less than 16021552 x 155Two values assumed to be 155310160 and less than 17031653 x 165Three values assumed to be 165495170 and less than 1801017510 x 17510 values assumed to be 1751750180 and less than 19091859 x 1859 values assumed to be 1851665190 and less than 20011951 x 1951 value assumed to be 195195 Total = 4415Mean = 4415÷25 = 176.6cmNote that this value cannot be an exact value of the mean because it uses assumed values. It therefore is an estimated mean. Grade C - Examination QuestionsNow try the examination questions below. If there is any doubt use the help boxes provided.In a competition, a total of 120 fish was caught.The weights of the 120 fish are summarised in the table below.Weight ( w grams)FrequencyMiddle Value0 < w ≤ 2001010010 x 1001000200< w ≤ 40036400< w ≤ 60046600< w ≤ 80025800< w ≤100021000< w ≤ 12001Note that ‘ 800< w ≤1000’means weight values are denoted as w.All values of w between 800 and 1000 go into this group.The value 800 itself does not but the value 1000 does.Total = Calculate an estimate of the mean weight of the fish. Alan keeps a record of the number of minutes his train is late.Number of Minutes late (t)FrequencyYou will have to work out the assumed (middle values) for each group. Beware of the final group !!!Consider your answer. By looking at the table try to decide if your answer is sensible.0≤ t < 102810≤ t < 201320≤ t < 30530≤ t < 40840≤ t < 50450≤ t < 60160≤ t < 1201Total60Calculate an estimate of the mean number of minutes late.A fruit grower keeps a record of the weights ( to the nearest kilogram) of plumbs she picks from each of 60 trees. This table summarises her results.Weight per tree (kg)Frequency (f)Mid Point (m)‘m x f’1 to 5131 x 3 = 36 to1013813 x 8 = 10411 to 152816 to 201521 to 253Modal = most commonWhich is the modal class ?Already plete the mid – point column.c) Calculate an estimate of the mean weight of plums per tree.This frequency polygon shows the distribution of the weights picked from a second set of 60 plum trees which, unlike the first, had been fertilised with GrowLots. Draw a frequency polygon, on the same axes, to show the distribution of the weights picked from the first set.Would you advise the grower to keep using GrowLots? Give a reason for your answer.The list gives the weights, in kilograms, of 30 people attending the keep fit class at the MEGA GYM.444545464748495050525353535656565757597071727475767879797979Find the median of these weights.This table shows the data in a different way.Weight (kg)FrequencyMid Interval Value40 and less than 45145 and less than 50650 and less than 55655 and less than 60660 and less than 65065 and less than 70070 and less than 75475 and less than 807Use the table to calculate an estimate of the mean weight.The manager tells new members that the average weight of those attending is over 75kg.Which measure of average is he using?Which would be a fairer way of giving the average weight? Give a reason for your answer.The weight of each item of baggage accepted by British Medway Airline one Saturday was recorded. The table below gives the distribution of weights.Weight (wkg)Number of items of baggageMid Point0 < w ≤ 52305 < w ≤ 1031610 < w ≤ 1581315 < w ≤ 2062120 < w ≤ 2559725 < w ≤ 30263Total2840Calculate an estimate of the mean weight of an item of baggage.Explain why it is not possible to calculate the exact mean weight of an item of baggage from this information.A grouped frequency distribution table of the heights of 500 adult females is shown.Height (h inches)Frequency54.5 ≤ h < 59.5159.5 ≤ h < 64.511564.5 ≤ h < 69.538069.5 ≤ h < 74.54Calculate an estimate of the mean weight of the 500 females.A frequency polygon showing the distribution of the heights of 500 adult males is shown opposite. On the same axes, draw a frequency polygon for the heights of the females.c) Use the frequency polygons to compare these distributions.Grade D – Frequency DiagramsFrequency diagrams are effectively bar charts. A frequency table is effectively a tally chart.At this level you need to be able to draw and read from three types of graph.Grouped bar chart . Just like a bar chart. Frequency goes on the vertical scale. The horizontal scale is mark continuously. Eg 0102030not 0-1010-2020-30Frequency Polygon – the same as a grouped bar chart but instead you plot points instead of bars. The points are plotted level with the middle of each group. The points are then joined with straight lines. Stem and Leaf diagram. This is a graph made up of a list of numbers – see the example below.The numbers of items bought by 35 customers at a supermarket are shown on the stem and leaf diagram below.17 19Key 1 7 = 171792157832355940375246899602346779701347825a) Find(I) the median[1](the middle number when they are in order)(ii) the range[1](the highest value – lowest value)b) Work out the percentage of customers who bought less than 50 items.[3](write as a fraction /35. Fraction to decimal to percentage)Class 10B has two practice papers for their examination. The results are given in this table.PupilABCDEFGHIJKLMNPQPractice 11524322934221782232213340262112Practice 210293134462020123945343746382415This is the stem and leaf diagram for practice paper 1.081257211224693223440Draw the stem and leaf diagram for practice paper 2[3]Make a comment comparing the results of the two practice papers[1]c) Khaled scored 15 out of 48 marks on a paper. Convert this to a percentage.[2][Fraction to decimal = top bottom. Decimal to percentage = x 100]The times taken by 60 pupils to complete a piece of homework are summarised in the table below.Time( t minutes)5 < t 1010 < t 1515 < t 2020 < t 2525 < t 30Frequency92223425 < t 10 means over 5 and up to (and including) 10Draw a frequency diagram to represent this information.[3][This could be a bar chart or a frequency polygon (line graph) – take care with the horizontal scale]b) Which class interval (group) contains the median time? Explain how you worked it out.[2]Time(seconds)Frequency40 and less than 50550 and less than 60860 and less than 70770 and less than 80380 and less than 902This frequency table shows the times taken by the 25 students in class A to swim 2 lengths of the pool.What is the modal group?[1][mode = most popular]The times taken by the 25 students in class B are shown on the frequency polygon below.On the same axes, draw a frequency polygon to represent the times taken by class A.[2][make sure points are plotted in the middle of a group and the points are joined up with straight lines]Compare the times taken by the two classes.[1]Pythagoras’ Theorem RevisionPythagoras’ Theorem is used to find the length of a side on a right angled triangle.To do this you need to know two of the three sides which you then use to find the length of the third side.Pythagoras’ Theorem only needs information about sides of the right angled triangle. Pythagoras’ Theoremcaa2 + b2 = c2bNote that c is always the longest side and is always opposite the right angle.It is essential that you know which side is c before you use the theorem.Remember that in order to change from x2 to x you need to square rootWorked Example. Find the length of the side x.12.5m = ca2 + b2 = c2x2 +102 =12.52a = xx2 +100 =156.25x2 =156.25 - 100 x2 = 56.25 x = √56.2510m = b= 7.5mThe diagram shows a wheelchair ramp.The ramp is 3.80m long3.80mThe step is 0.90m high.0.90mSACalculate the distance between the bottom of the ramp, A, and the bottom of the step, S.Round off your answer to an appropriate degree of accuracy. (2dp)The diagram below shows the distance covered by a sponsored walk.The distances are in miles.Calculate the distance BC.BNot to Scale12.6A[3]10.6C4.5mA ladder 4.5 m long leans against a vertical wall. The foot of the ladder is 1.8 m from the wall.1.8mCalculate how far up the wall the ladder reaches.R4. The diagram below shows the position of 3 airports Rotherfield (R), Shelcroft (S) and Thurham (T).Not to Scalea) Calculate the distance from R to T [3]70km45kmST31cm5. xcm14cm12cmCalculate the length marked x.[Hint – find the right angled triangle first]DNot to Scale6.23m6. CBA7.66m5.24m The diagram above shows the plan of a garden.Calculate ABGive your answer to an appropriate degree of accuracy.[5] T (5, 20)S(3,14) and T (5,20) are tow points.Not to ScaleCalculate the length ST.Show you method clearly. [3] S(3, 14)6mBashir decorated his bedroom. The diagram shows the ceiling, which is shaded.2.75m4m3mHe painted the ceiling. Calculate the area he painted[Hint: work out the area of the big rectangle – then subtract the area of the small ‘cut off’ triangle]b) He stuck a border round the top of the walls. Calculate the total length of the border.[Hint: Use Pythagoras’ theorem to find the missing diagonal side – then add up all the sides of the room together]9.10572751905A straight line joins the points A(0,4) and B (5,14).Calculate the length of AB. Show your method clearly[3]Reciprocals1. Copy and complete the table belowNumberReciprocal in fraction form36965102. Copy and complete the table belowFractionReciprocal 1/161/91/101/251/713. Use a calculator to find the reciprocal of the numbers below correct to 2dp.a) 2.5b) 3.2c) 0.5d) 0.2e) 625f) 0.164. Copy and complete the table below and then plot the line y = 1/x on graph paper. This is a reciprocal curve.x-10-5-3-2-1-0.8-0.6-0.4 -0.20.20.40.60.81234510yEXTENSION5. Calculate the reciprocals of the fractions below. Express your answers as fractions or mixed fractions (whole and fractional part)a) 4/5b) 3/8c) 13/5d) 31/3e) 2/25EXAMINATION QUESTIONS6. Work out the reciprocal of 1.6[1]Trial and Improvement and Sequences Exam questions.1aA number pattern begins 1, 1, 2, 3, 5, 8, ….What is the next number in this pattern?The number pattern is continued. Explain how you would find the eighth number in the pattern.bAnother pattern begins 1, 4, 7, 10, 13, …… Write down, in terms of n, the nth term in this pattern.2aLook at this sequence of numbers: 3, 8, 18, 38, ….. The rule that has been used to get each number from the number before isAdd 1 and then multiply by 2Write down the next number in this sequence.Using the same rule but a different starting number, the second number is 16. What is the starting number?bLook at this sequence of numbers: 5, 9, 13, 17, 21, ……Write down, in words, the rule for getting each number from the one before it.Write down a formula, in terms of n, for the nth number of the sequence.Phillipa makes some patterns by linking squares with rods. Here are some of the patterns she makes.a)How many squares are there in the 40th pattern?b)How many squares are there in the nth pattern?How many rods are there in the 40th pattern?How many rods are there in the nth pattern?Find the nth term of the following sequences?3, 6, 9, 12, 15, …..4, 9, 14, 19, 24, ……Solve the equation x? + 2x – 90 = 0 correct to 1 d.p.6.Solve the equation x? -3x + 20 = 0 correct to 2 d.p7.Find the solution to x? = 300 correct to 2 d.p. ................
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