Expert answer:Maths Task-Percentages, mean median mode, standard

Answer & Explanation:Maths Task percentages.docx
maths_task_percentages.docx

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PART 1
All relevant working must be shown.
1.
Are the following data discrete or continuous?
(a)
The number of goals scored.
(b)
The length of life of an electric light.
(c)
The number of road accidents per week.
(d)
The speed of a car.
2.
Daily temperatures at a ski resort
4° –1°
6°
2°
–3°
11°
4°
3°
1°
4°
3°
1°
4°
1°
For the raw data above:
(i)
Find the median, the mode and the mean.
(ii)
Test for outliers using the formula. Indicate the value of any outlier(s).
(iii)
Discard the outlier(s) and recalculate the mean and the median.
(iv)
Draw a table of results, study it and then make a comment?
What is the effect of the outlier(s) on the mean and median?
3.
Weight of pets (kg)
5
7
8
9
11
14
Number of pets
3
2
5
7
2
1
For the data above (shown as a frequency table):
(i)
Find the median weight, the mode and the mean weight.
(ii)
Test for outliers using the formula. Indicate the value of any outlier(s).
(iii)
Discard the outlier(s) and recalculate the mean and the median.
(iv)
Comment on the results?
What is the effect of the outlier(s) on the mean and median?
0°
4. Work out the mean height of the sample below.
Height (cm)
140-
150-
160-
170-
180-
190-
200-<210 Frequency 3 12 20 15 4 0 1 (Hint: For data in intervals, you need to use the midpoints of each interval to calculate the mean. Refer to Lesson 2). The histogram (bar graph) shows how long 51 customers spent waiting in a supermarket queue. Waiting Time in Supermarket Queue 12 10 Customers 5. 8 6 4 2 0 2 4 6 8 10 12 14 16 18 20 22 Minutes (a) From the histogram, how many customers waited less than 4 minutes? Note that the first column of the graph indicates that 9 people waited up to (but not including) 2 minutes. (b) Based on this survey what percentage of customers can expect to wait more than 10 minutes? (c) (i) How is the data skewed? (ii) Which measure of average (mean or median) would give the best indication of the average waiting time? Why? 6. The stem plot on the right displays the average number of decayed teeth in 12-year-old children from 31 countries. (a) Based on the stem plot, describe the shape of the distribution of the average number of decayed teeth for these countries. (b) From the stem plot, read off the median and the quartiles (upper and lower). (c) What is the range? (d) Using the results in (b) and (c), draw a box plot and describe its shape. Before you attempt Q7, it is recommended that you watch the video lesson on Parallel Box plots at this website: http://www.youtube.com/watch?v=k4AVzJpVePg 7. The box plots below show the ratings of three TV stations. Before attempting the questions, refer to Lesson 3 to see what proportion of data values are less than the quartiles and the median. 10 5 15 20 25 30 CBS NBC ABC (a) Which TV station has the: (i) (ii) (iii) (iv) highest rating? smallest interquartile range? highest median rating? largest spread in the top half of the data? (b) What proportion (in percentage) of NBC ratings are over 16? (c) What proportion (in percentage) of CBS ratings are less than 18? (d) Which station has a greater proportion of ratings over 15? 8. The dot plot below displays the difference in ages (in years) between female and male life expectancy for a sample of 20 countries. Calculate the mean and standard deviation of the above data. Exam practice Circle the letter beside the correct answer. 1 People at a party work out that the mean age of the 28 people present is 18.7 years. The host‘s parents then come home, they are aged 44 and 46 respectively. What is the mean age of the people at the party now? A 19.3 years B 20.5 years C 21.9 years D 31.9 years E 36.2 years The following information relates to questions 2 and 3 2 3 The following information relates to questions 4 and 5 Researchers conducted a survey of 403 school leavers who had recently entered the work force. The aim was to determine whether the type of work they undertook was gender related. The data in the table below comes from the survey. Gender Work type trade clerical manual professional Male 104 21 72 8 Female 18 143 31 6 4 In this survey, the variables work type (trade, clerical, manual or professional) and gender (male or female) are A both categorical variables B both numerical variables C categorical and numerical variables respectively D numerical and categorical variables respectively E neither categorical nor numerical variables 5 Of the females surveyed, the percentage who became clerical workers is closest to A 10% B 14% C 35% D 72% E 87% PART 2 1. Complete this question in the spaces and tables provided. (a) Use your CAS calculator to generate a random sample of size 10 from the data set (next page) on Percentage of Sugar in Cereals. Write your data values in the box below. (b) Explain clearly how you generated your random numbers using your calculator. Which CAS calculator are you using: TI-nSpire or Casio Classpad? (c) Using your calculator, find the summary statistics for the percentage of sugar in the cereals of your sample. Write the values in the second table below. Random Number Cereal % sugar Summary Statistics Mean Sx Mode Q1 Median Q3 IQR Minimum Maximum Range (d) Work out the values for the lower and upper fences. Hence, conclude whether there is any outlier in your sample. Calculation for Lower Fence Calculation for Upper Fence Conclusion: Is there any outlier? Where? (e) Construct a boxplot for your sample. Label the five summary numbers. (f) Describe the distribution represented by the box plot in terms of shape, centre and spread. DATA SET: PERCENTAGE OF SUGAR IN CEREALS The cereals are numbered from 1 to 62. Product % Sugar Product % Sugar 1 Sugar Smacks 56.0 32 Kellogg Raisin Bran 29.0 2 Apple Jacks 54.6 33 C. W. Post, Raisin, 29.0 3 Froot Loops 48.0 34 C. W. Post 28.7 4 General foods Raisin Bran 48.0 35 Frosted Mini Wheats 26.0 5 Sugar Corn Pops 46.0 36 Country Crisp 22.0 6 Super Sugar Crisp 46.0 37 Life, cinnamon 21.0 7 Crazy cow, chocolate 45.6 38 100% Bran 21.0 8 Corny snaps 45.5 39 All Bran 19.0 9 Frosted Rice Krinkles 44.0 40 Fortified Oat Flakes 18.5 10 Frankenberry 43.7 41 Life 16.0 11 Cookie Crisp, Vanilla 43.5 42 Team 14.1 12 Cap’n Crunch, Crunch Berries 43.3 43 40% Bran 13.0 13 Cocoa Krispies 43.0 44 Grape Nuts Flakes 13.3 14 Cocoa Pebbles 42.6 45 Buckwheat 12.2 15 Fruity Pebbles 42.5 46 Product 19 9.9 16 Lucky Charms 42.2 47 Concentrate 9.3 17 Cookie Crisp, Chocolate 41.0 48 Total 8.3 18 Sugar Frosted Flakes of Corn 41.0 49 Wheaties 8.2 19 Quisp 40.7 50 Rice Krispies 7.8 20 Crazy Cow, Strawberry 40.1 51 Grape Nuts 7.0 21 Cookie Crisp, Oatmeal 40.1 52 Special K 5.4 22 Cap’n Crunch 40.0 53 Corn Flakes 5.3 23 Count Chocula 39.5 54 Post toasties 5.0 24 Alpha Bits 38.0 55 Kix 4.8 25 Honey Comp 37.2 56 Rice Chex 4.4 26 Frosted Rice 37.0 57 Corn Chex 4.0 27 Trix 35.9 58 Wheat Chex 3.5 28 Coca Puffs 33.3 59 Cheerios 3.0 29 Cap’n Crunch, Peanut Butter 32.2 60 Shredded Wheat 0.6 30 Golden Grahams 30.0 61 Puffed Wheat 0.5 31 Cracklin’ Bran 29.0 62 Puffed Rice 0.1 Source : US Department of Agriculture 2. Use your calculator to help you work through this question. (refer back to week 1 also) A company survey recorded the petrol cost of 16 employees $21, $32, $26, $18, $31, $23, $20, $35, $33, $85, $16, $28, $19, $37, $25, $26. (a) Find the median petrol cost. (b) Find the mean and standard deviation. (c) Draw the boxplot (using the calculator) and identify the outlier(s). State the value of the outlier(s). (look back to week 1 pg 42) (d) Disregard and recalculate the median, mean and standard deviation without the outlier(s). (e) Compare the values of the mean, median and standard deviation with and without the outlier(s). What effect do outliers have on them in this case? Which measure of average – the mean or median is more significantly affected by the outliers. 3. A conservation park in Thailand is home to 49 elephants, of which 26 are females and 23 are males. The parallel boxplots below show the distribution of their ages by sex. Based on the information contained in the parallel boxplots, which one of the following statements is incorrect? A The youngest elephant is male B There are fewer female elephants under the age of 15 years than male elephants under the age of 15 C There are no female elephants over the age of 40 years D The median age of the female elephants is approximately the same as the median age of the male elephants. E Approximately 25% of the male elephants are 30 years of age or older. 4. Using the 68–95–99.7% rule, work out the various percentages of the distribution which lie between the mean and 1 standard deviation from the mean and between the mean and 2 standard deviations from the mean and so on. Write the percentage in each section A- G of the distribution below, the middle sections have been done for you. 99.7% 95% 68% 5. Some IQ tests are set so that on average, people taking the test score 100 points with a standard deviation of 15 points. IQ scores from this test are known to be approximately normally distributed. From this information we can conclude that: (a) almost all people taking the test will score between (b) if you scored 90 points your score would be score. points and _points (above/below) average test (c) if you scored between 85 and 115 you would be in the middle taking the test. (d) 50% of people taking the test will score more than % of people points (e) 99.85% of people taking the test will score more than (f) 84% of people will score less than 6. points . The distribution of SAC scores in a VCE subject studied by more than 10 000 students is bell-shaped with a mean of 34 and a standard deviation of 9. From this information, it can be concluded that the percentage of students between 25 and 43 is closest to: A 5% 7. points. B 16% C 50% D 68% E 95% The mean length of a large batch of broom handles is 120 cm. The data have a standard deviation of 3 cm. The percentage of broom handles in this batch which are shorter than 114 cm is A 0.15% B 2.5% C 13.5% D 16% E 34% 8. (a) Find the standard score for a the data value 9 from a normal distribution which has a mean of 16.8 and a standard deviation of 2.3 (b) Interpret the standard score. 9. The scores for a large number of golf rounds played on two courses were recorded. Both sets of data were normally distributed. Course A Course B Mean 78 81 Standard Deviation 7 14 Janet scored 70 on both courses. Using the standardising method, show on which course she performed better? (Note: In golf, the lower the score, the better is the performance.) Exam Practice Circle the letter beside the correct answer. 1 The percentage investment returns of seven superannuation funds for the year 2006 are −4.6%, −4.7%, 2.9%, 0.3%, −5.5%, −4.4%, −1.1% The range of investment returns is A 2.6% B 3.5% C 4.0% D 5.5% E 8.4% 2 The distribution of study scores in a particular VCE subject is known to be approximately normal, with a mean of 30 and a standard deviation of 7. This means that the percentage of students who gain a study score of more than 37 is approximately: A 95% B 68% C 32% D 16% E 5% 3 The amount of money (in dollars) spent by each of 15 students on leisure activities in the past week is as follows: 10 30 30 30 35 38 45 45 47 49 50 51 52 58 100 The mean and the standard deviation of the weekly expenditure on leisure are respectively: A Mean = $44.67; Standard deviation = $19.59 B Mean = $45.35; Standard deviation = $18.45 C Mean = $38.68; Standard deviation = $14.56 D Mean = $44.68; Standard deviation = $18.92 E Mean = $43.24; Standard deviation = $19.86 4 When starting university, a student’s age is 18.8 years. The mean age of starting students in her course is 18.4 years with a standard deviation of 0.3 years. With respect to her peers, the student’s standardized age (z-score) is closest to A –1.3 B –0.4 C 1.3 D 2.6 E 3.9 5 Test scores obtained when 2500 students sit for an examination follow a normal distribution with a mean of 64 and a standard deviation of 12. From this information we can conclude that the number of these students who obtained marks between 52 and 76 is closest to A 68 B 95 C 850 D 1700 E 2375 PART 3 1. Data were collected to investigate whether a person’s pulse rate can be predicted from their sex. The two parallel box plots below show the distribution of pulse rates for 21 adult females and 22 adult males. (a) There are two variables here: Pulse rate and Sex (male, female). (i) Which is the numerical and which is the categorical variable? Numerical: Categorical: (ii) Which is the explanatory and which is the response variable? (EV): (RV): (b) Do the parallel box plots support the contention that there is an association between a person’s sex and their pulse rate? Explain your answer by comparing appropriate summary statistics from the above boxplots (make sure to use percentages and statistical terminology to give hard evidence for your claims). 2. Complete the following: (a) If the variables x and y are positively associated then as x increases, y (b) If there is negative association between the variables m and n, then as m increases, n (c) If there is no association between two variables then the points on the scatterplot appear to be . 3. Fill in the table below by performing the following analysis of the scatterplot in each row. (i) First check if there is any sign of an association between the variables and, if there is, state whether the form of the association is linear or non-linear. (Hint: try drawing a line through the data and look to see if there is any general pattern which moves away from this line.); (ii) If there are signs of an association between the variables, assess the strength of the association (strong, moderate or weak). (iii) Also, if the trend is linear, give the direction of the trend. Is it positive, negative? (iv) Check for outliers. Are any points significantly distant from the main trend? Scatterplot Form Strength Direction Outliers (linear/nonlinear/no association (a) (b) (c) (d) (strong-weak) (positive/negative) (Yes/No) 4. For each of the following lines (a) and (b): (i) Is the gradient positive or negative? (ii) Calculate the gradient of the line. (iii) Find the equation of the line. (a) (b) y (–1,4) y (3, 2) x (0, –2) (2, 0) x 2. Find the x and y intercepts of the following straight lines, then sketch their graphs. (a) y = 1 – 3x (b) 2y = 5x + 8 Exam Practice Circle the letter beside the correct answer. Question 1 Question 2 Question 3 Question 4 Question 5 Question 5 ... Purchase answer to see full attachment

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