A cumulative frequency table for a data set is shown.
What is the interquartile range of this data set? (2 marks)
--- 3 WORK AREA LINES (style=lined) ---
Aussie Maths & Science Teachers: Save your time with SmarterEd
A cumulative frequency table for a data set is shown.
What is the interquartile range of this data set? (2 marks)
--- 3 WORK AREA LINES (style=lined) ---
`2`
`text(42 data points ⇒ median) = text(21st + 22nd)/2`
`text(Q)_1` | `= 11text(th data point) = 3` |
`text(Q)_3` | `= 32text(nd data point) = 5` |
`:.\ text(IQR)` | `= 5 – 3` |
`= 2` |
The graph displays the mean monthly rainfall in Sydney and Perth.
--- 1 WORK AREA LINES (style=lined) ---
--- 2 WORK AREA LINES (style=lined) ---
i. `text(3 months (Jul, Aug and Sep))`
COMMENT: Extremely volatile result between parts with part (i) producing a 91% mean mark.
ii. `text(The mean monthly rainfall of Sydney is in a)`
`text(much tighter range than Perth.)`
`:.\ text(Sydney has a smaller standard deviation.)`
Ali made monthly deposits of $100 into an annuity for 5 years.
Calculate the total amount Ali deposited into the annuity over this period. (1 mark)
`$6000`
`text(Total deposited)` | `= 5 xx 12 xx 100` |
`= $6000` |
Jeremy rolled a biased 6-sided die a number of times. He recorded the results in a table.
What is the relative frequency of rolling a 3? (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
`8/25`
`text(Rel Freq)` | `= text(number of 3’s rolled)/text(total rolls)` |
`= 48/150` | |
`= 8/25` |
What is the `x`-intercept of the line `x + 3y + 6 = 0`?
`A`
`x text(-intercept occurs when)\ y = 0:`
`x + 0 + 6` | `= 0` |
`x` | `= -6` |
`:. x text{-intercept is}\ (-6, 0)`
`=> A`
A set of data is normally distributed with a mean of 48 and a standard deviation of 3.
Approximately what percentage of the scores lies between 39 and 45?
`A`
During a year, the maximum temperature each day was recorded. The results are shown in the table.
From the days with a maximum temperature less than 25°C, one day is selected at random.
What is the probability, to the nearest percentage, that the selected day occurred during winter?
`text(C)`
`text{P(winter day)}` | `= (text(winter days < 25))/text(total days < 25) xx 100` |
`= 71/223 xx 100` | |
`= 31.8…%` |
`=>\ text(C)`
The table shows the compounded values of $1 at different interest rates over different periods.
Amy hopes to have $21 000 in 2 years to buy a car. She opens an account today which pays interest of 4% pa, compounded quarterly.
Using the table, which expression calculates the minimum single sum that Amy needs to invest today to ensure she reaches her savings goal?
`text(D)`
`text(4% annual)`
`=> (4%)/4 = 1text(% compounded quarterly)`
`=> n = 8`
`=>\ text(Factor) = 1.0829`
`:.\ text(Minimum sum) = 21\ 000 ÷ 1.0829`
`=>\ text(D)`
A set of data is summarised in this frequency distribution table.
Which of the following is true about the data?
`text(B)`
`text{Mode = 7 (highest frequency of 9)}`
`text(Median = average of 15th and 16th data points.)`
`:.\ text(Median = 6)`
`=>\ text(B)`
The diagram shows the positions of towns `A`, `B` and `C`.
Town `A` is due north of town `B` and `angleCAB = 34°`
What is the bearing of town `C` from town `A`?
`C`
`text(Bearing of Town)\ C\ text(from Town)\ A:`
`text(Bearing)` | `= 180 + 34` |
`= 214^@` |
`=>C`
A set of data is displayed in this dot plot.
Which of the following best describes this set of data?
`text(C)`
`text(Data is skewed.)`
`text(S)text(ince the “tail” is on the left had side, the)`
`text(data is negatively skewed.)`
`=>\ text(C)`
A survey asked the following question.
'How many brothers do you have?'
How would the responses be classified?
`text(C)`
`text(The number of brothers a person has is)`
`text(an exact whole number.)`
`:.\ text(Classification is numerical, discrete.)`
`=>\ text(C)`
A set of scores has the following five-number summary.
lower extreme = 2
lower quartile = 5
median = 6
upper quartile = 8
upper extreme = 9
What is the range?
`text(D)`
`text(Range)` | `=\ text(upper extreme − lower extreme)` |
`= 9 – 2` | |
`= 7` |
`=>\ text(D)`
What is the gradient of the line `2x + 3y + 4 = 0`?
`A`
`2x + 3y + 4` | `= 0` |
`3y` | `= -2x-4` |
`y` | `= -2/3 x-4/3` |
`:.\ text(Gradient)` | `= -2/3` |
`=> A`
The diagram shows the location of three schools. School `A` is 5 km due north of school `B`, school `C` is 13 km from school `B` and `angleABC` is 135°.
--- 4 WORK AREA LINES (style=lined) ---
--- 6 WORK AREA LINES (style=lined) ---
i. `text(Using cosine rule:)`
`AC^2` | `= AB^2 + BC^2 – 2 xx AB xx BC xx cos135^@` |
`= 5^2 + 13^2 – 2 xx 5 xx 13 xx cos135^@` | |
`= 285.923…` |
`:. AC` | `= 16.909…` |
`= 17\ text{km (nearest km)}` |
ii. |
`text(Using sine rule, find)\ angleBAC:`
`(sin angleBAC)/13` | `= (sin 135^@)/17` |
`sin angleBAC` | `= (13 xx sin 135^@)/17` |
`= 0.5407…` | |
`angleBAC` | `= 32.7^@` |
`:. text(Bearing of)\ C\ text(from)\ A`
`= 180 + 32.7`
`= 212.7^@`
`= 213^@`
A set of data has a lower quartile (`Q_L`) of 10 and an upper quartile (`Q_U`) of 16.
What is the maximum possible range for this set of data if there are no outliers? (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
`24`
`IQR = 16 – 10 = 6`
`text(If no outliers,)`
`text(Upper limit)` | `= Q_U + 1.5 xx IQR` |
`= 16 + 1.5 xx 6` | |
`= 25` |
`text(Lower limit)` | `= Q_L – 1.5 xx IQR` |
`= 10 – 1.5 xx 6` | |
`= 1` |
`:.\ text(Maximum range)` | `= 25 – 1` |
`= 24` |
All the students in a class of 30 did a test.
The marks, out of 10, are shown in the dot plot.
--- 1 WORK AREA LINES (style=lined) ---
Using the dot plot, calculate the percentage of the marks which lie within one standard deviation of the mean. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
--- 2 WORK AREA LINES (style=lined) ---
i. | `text(Median)` | `= text(15th + 16th score)/2` |
`= (4 + 8)/2` | ||
`= 6` |
ii. `text(Lower limit) = 5.4 – 4.22 = 1.18`
`text(Upper limit) = 5.4 + 4.22 = 9.62`
`:.\ text(Percentage in between)`
`= 13/30 xx 100`
`= 43.33…`
`= 43text{% (nearest %)}`
iii. `text(The statement assumes the data is normally)`
`text(distributed which is not the case here.)`
A group of Year 12 students was surveyed. The students were asked whether they live in the city or the country. They were also asked if they have ever waterskied.
The results are recorded in the table.
--- 4 WORK AREA LINES (style=lined) ---
Is this true, based on the survey results? Justify your answer with relevant calculations. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
i. | `P` | `= text(live in city, not skied)/text(total surveyed)` |
`= 2500/3520` | ||
`= 125/176` |
ii. | `P(text(live in country, skied))` | `= 70/((70 + 800))` |
`= 0.0804…` | ||
`= 8text(%)` |
`P(text(live in city, skied))` | `= 150/((150 + 2500))` |
`= 0.0566` | |
`= 6text(%)` |
`text(S)text(ince 8% > 6%, the statement is true.)`
A table of future value interest factors for an annuity of $1 is shown.
An annuity involves contributions of $12 000 per annum for 5 years. The interest rate is 4% per annum, compounded annually.
--- 1 WORK AREA LINES (style=lined) ---
--- 1 WORK AREA LINES (style=lined) ---
i. `text(FV factor = 5.4163)`
`:.\ text(FV of Annuity)` | `= 12\ 000 xx 5.4163` |
`= $64\ 995.60` |
ii. | `text(Interest earned)` | `=\ text(FV − total repayments)` |
`= 64\ 995.60 – (5 xx 12\ 000)` | ||
`= $4995.60` |
Jamal surveyed eight households in his street. He asked them how many kilolitres (kL) of water they used in the last year. Here are the results.
`220, 105, 101, 450, 37, 338, 151, 205`
--- 1 WORK AREA LINES (style=lined) ---
--- 1 WORK AREA LINES (style=lined) ---
i. | `text(Mean)` | `= (220 + 105 + 101 + 450 + 37 + 338 + 151 + 205) ÷ 8` |
`= 200.875` |
ii. | `text(Std Dev)` | `= 127.357…\ \ text{(by calc)}` |
`= 127.4\ \ text{(1 d.p.)}` |
The heights of Year 12 girls are normally distributed with a mean of 165 cm and a standard deviation of 5.5 cm.
What is the `z`-score for a height of 154 cm?
A. `−2`
B. `−0.5`
C. `0.5`
D. `2`
`text(A)`
`ztext(-score)` | `= (x – mu)/sigma` |
`= (154 – 165)/5.5` | |
`= −2` |
`=>A`
Which of the data sets graphed below has the largest positive correlation coefficient value?
A. | B. | ||
C. | D. |
\(C\)
\(\text{Largest positive correlation occurs when both variables move}\)
\(\text{in tandem. The tighter the linear relationship, the higher the}\)
\(\text{correlation.}\)
\(\Rightarrow C\)
\(\text{(Note that B is negatively correlated)}\)
A single amount of $10 000 is invested for 4 years, earning interest at the rate of 3% per annum, compounded monthly.
Which expression will give the future value of the investment?
A. `10\ 000 xx (1 + 0.03)^4`
B. `10\ 000 xx (1 + 0.03)^48`
C. `10\ 000 xx (1 + 0.03/12)^4`
D. `10\ 000 xx (1 + 0.03/12)^48`
`D`
`text(Compounding rate)` | `= 3/100 ÷ 12` |
`= 0.03/12` |
`text(Compounding periods)` | `= 4 xx 12=48` |
`:.\ text(FV) = 10\ 000 xx (1 + 0.03/12)^48`
\(\Rightarrow D\)
What is the value of `x` in the equation `(5-x)/3 = 6`?
`A`
`(5-x)/3` | `= 6` |
`5-x` | `= 18` |
`x` | `= 5-18` |
`= -13` |
`=>A`
A factory’s quality control department has tested every 50th item produced for possible defects.
What type of sampling has been used?
A. Random
B. Stratified
C. Systematic
D. Numerical
`C`
`text(A systematic sample divides a population)`
`text(into equal sample sizes and then selects)`
`text(equally among them.)`
`=> C`
In a survey of 200 randomly selected Year 12 students it was found that 180 use social media.
Based on this survey, approximately how many of 75 000 Year 12 students would be expected to use social media?
A. 60 000
B. 67 500
C. 74 980
D. 75 000
`B`
`text(Expected number)` | `= 180/200 xx 75\ 000` |
`= 67\ 500` |
`=> B`
The graph shows the relationship between infant mortality rate (deaths per 1000 live births) and life expectancy at birth (in years) for different countries.
What is the life expectancy at birth in a country which has an infant mortality rate of 60?
\(A\)
The graph shows the life expectancy of people born between 1900 and 2000.
--- 1 WORK AREA LINES (style=lined) ---
--- 4 WORK AREA LINES (style=lined) ---
i. \(\text{68 years}\)
ii. \(\text{Using (1900,60), (1980,80):}\)
\(\text{Gradient}\) | \(= \dfrac{y_2-y_1}{x_2-x_1}\) |
\(= \dfrac{80-60}{1980-1900}\) | |
\(= 0.25\) |
\(\text{After 1900, life expectancy increases by 0.25 years for}\)
\(\text{each year later that someone is born.}\)
The ages of members of a dance class are shown in the back-to-back stem-and-leaf plot.
Pat claims that the women who attend the dance class are generally older than the men.
Is Pat correct? Justify your answer by referring to the median and skewness of the two sets of data. (3 marks)
--- 6 WORK AREA LINES (style=lined) ---
`text(Women:)`
`text(The median is 55 in a data)`
`text(set that is negatively skewed.)`
`text(Men:)`
`text(The median is 45 in a data)`
`text(set that is positively skewed.)`
`:.\ text(Pat is correct.)`
`text(Women:)`
`text(The median is 55 in a data)`
`text(set that is negatively skewed.)`
`text(Men:)`
`text(The median is 45 in a data)`
`text(set that is positively skewed.)`
`:.\ text(Pat is correct.)`
The mass `M` kg of a baby pig at age `x` days is given by `M = A(1.1)^x` where `A` is a constant. The graph of this equation is shown.
--- 1 WORK AREA LINES (style=lined) ---
--- 2 WORK AREA LINES (style=lined) ---
i. `text(When)\ x = 0,`
`1.5` | `= A(1.1)^0` |
`:. A` | `= 1.5\ text(kg)` |
ii. `text(Daily growth rate)`
`= 0.1`
`= 10text(%)`
The table gives the contribution per period for an annuity with a future value of $1 at different interest rates and different periods of time.
Margaret needs to save $75 000 over 6 years for a deposit on a new apartment. She makes regular quarterly contributions into an investment account which pays interest at 3% pa.
How much will Margaret need to contribute each quarter to reach her savings goal? (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
`$2865`
`text(Periods) = 6 xx 4 = 24`
`text(Interest rate) = 1/4 xx 3 = 0.75text(%)`
`=>\ text(Table factor = 0.0382)`
`(text(i.e. 3.82 cents contributed per)`
`text(quarter = $1 after 6 years))`
`:.\ text(Quarterly contribution)`
`= 75\ 000 xx 0.0382`
`= $2865`
The heights of 400 students were measured. The results are displayed in this cumulative frequency polygon.
Use the polygon to estimate the interquartile range. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
`36\ text(cm)`
A small population consists of three students of heights 153 cm, 168 cm and 174 cm. Samples of varying sizes can be taken from this population.
What is the mean of the mean heights of all the possible samples? Justify your answer. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
`165\ text(cm)`
`text(If sample is 1 person,)`
`text{Possible mean(s): 153, 168 or 174.}`
`text(If sample is 2 people,)`
`text{Possible mean(s):}quad(153 + 168)/2` | `= 160.5` |
`(153 + 174)/2` | `= 163.5` |
`(168 + 174)/2` | `= 171` |
`text(If sample is 3 people,)`
`text(Mean:)quad(153 + 168 + 174)/3 = 165`
`:.\ text(Mean of all mean heights)`
`= (153 + 168 + 174 + 160.5 + 163.5 + 171 + 165)/7`
`= 165\ text(cm)`
The diagram shows towns `A`, `B` and `C`. Town `B` is 40 km due north of town `A`. The distance from `B` to `C` is 18 km and the bearing of `C` from `A` is 025°. It is known that `∠BCA` is obtuse.
What is the bearing of `C` from `B`?
(A) `070°`
(B) `095°`
(C) `110°`
(D) `135°`
`=> D`
`text(Using the sine rule,)`
`(sin∠BCA)/40` | `= (sin25^@)/18` |
`sin angle BCA` | `= (40 xx sin25^@)/18` |
`= 0.939…` | |
`angle BCA` | `= 180 – 69.9quad(angleBCA > 90^@)` |
`= 110.1°` |
`:. text(Bearing of)\ C\ text(from)\ B`
`= 110.1 + 25qquad(text(external angle of triangle))`
`= 135.1`
`=> D`
A group of 485 people was surveyed. The people were asked whether or not they smoke. The results are recorded in the table.
A person is selected at random from the group.
What is the approximate probability that the person selected is a smoker OR is male?
`=> C`
`P(text(Smoker or a male))`
`= (text(Total males + female smokers))/(text(Total surveyed))`
`= (264 + 68)/485`
`= 0.684…`
`=> C`
The box-and-whisker plots show the results of a History test and a Geography test.
In History, 112 students completed the test. The number of students who scored above 30 marks was the same for the History test and the Geography test.
How many students completed the Geography test?
`=> C`
`text{In History} \ => \ text{Q}_3 = 30\ \text{marks}`
`:.\ text{Scoring over 30}\ = 25text(%) xx 112 = 28\ \text{students}`
`text{In Geography} \ => \ text{Median}\ = 30\ \text{marks}`
`:.\ text{Students completing Geography}\ =2 xx 28 = 56\ \text{students}`
`=> C`
A grouped data frequency table is shown.
What is the mean for this set of data?
(A) 6.5
(B) 10.5
(C) 11.9
(D) 12.4
`=> D`
`text(Using the centre of each class interval:)`
`text(Mean)` | `= (3 xx 3 + 8 xx 6 + 13 xx 8 + 18 xx 9)/(3 + 6 + 8 + 9)` |
`= 12.42…` |
`=> D`
A soccer referee wrote down the number of goals scored in 9 different games during the season.
`2, \ 3, \ 3, \ 3, \ 5, \ 5, \ 8, \ 9, \ ...`
The last number has been omitted. The range of the data is 10.
What is the five-number summary for this data set?
`=> A`
`text{Since range is 10} \ => \ text{Last data point = 12}`
`text{Q}_1 = 3`
`text{Q}_3 = (8 + 9)/2 = 8.5`
`text(Median = 5)`
`=> A`
The speed limit outside a school is 40 km/h. Year 11 students measured the speed of passing vehicles over a period of time. They found the set of data to be normally distributed with a mean speed of 36 km/h and a standard deviation of 2 km/h.
What percentage of the vehicles passed the school at a speed greater than 40 km/h?
(A) `text(2.5%)`
(B) `text(5%)`
(C) `text(47.5%)`
(D) `text(95%)`
`=> A`
The table shows the future value of an investment of $1000, compounding yearly, at varying interest rates for different periods of time.
Based on the information provided, what is the future value of an investment of $2500 over 3 years at 4% pa?
(A) $1124.86
(B) $2812.15
(C) $3624.86
(D) $5312.15
`=> B`
`text(Table factor) = 1124.86`
`:. FV` | `= 2.5 xx 1124.86` |
`= $2812.15` |
`=> B`
Which set of data is classified as categorical and nominal?
`A`
`text(Categorical and nominal data is)`
`text(qualitative and not ordered.)`
`=> A`
The pulse rates of a large group of 18-year-old students are approximately normally distributed with a mean of 75 beats/minute and a standard deviation of 11 beats/minute.
The percentage of 18-year-old students with pulse rates less than 53 beats/minute or greater than 86 beats/minute is closest to
(A) `2.5text(%)`
(B) `5text(%)`
(C) `16text(%)`
(D) `18.5text(%)`
`D`
`mu=75,\ \ \ sigma=11`
`z text{-score (53)}` | `=(x-mu) /sigma` |
`=(53-75)/11` | |
`= – 2` |
`z text{-score (86)}` | `= (86-75)/11` |
`=1` |
`text(From the diagram, the % of students that have a)`
`z text(-score below – 2 or above 1)`
`=2.5+16`
`=18.5 text(%)`
`=>D`
The head circumference (in cm) of a population of infant boys is normally distributed with a mean of 49.5 cm and a standard deviation of 1.5 cm.
Four hundred of these boys are selected at random and each boy’s head circumference is measured.
The number of these boys with a head circumference of less than 48.0 cm is closest to
(A) `3`
(B) `10`
(C) `64`
(D) `272`
`C`
`mu=49.5,\ \ sigma=1.5`
`z text{-score (49.5)` | `=(x-mu)/sigma` |
`=(48.0-49.5)/1.5` | |
`=– 1` |
`:.\ text(Number of boys with a head under 48.0 cm)`
`=16text(%) xx 400`
`=64`
`=> C`
If `(y-3)/3 =5`, find `y`. (1 mark)
`18`
`(y3)/3` | `= 5` |
`y3` | `= 15` |
`y` | `= 18` |
Find the value of `r` given `r/7-4 = 3`. (1 mark)
`49`
`r/7-4` | `= 3` |
`r/7` | `= 7` |
`:.r` | `= 49` |
The table gives the present value interest factors for an annuity of $1 per period, for various interest rates `(r)` and numbers of periods `(N)`.
Use the information in the table to calculate the present value of this annuity. (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
Calculate the amount of each monthly repayment, correct to the nearest dollar. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
i. `N = 74, r = 0.0080`
`PVtext{(annuity) table factor}\ = 55.68446`
`:.PV\ text(of annuity)`
`= $200 xx 55.68446`
`= $11\ 136.892`
`= $11\ 136.89\ \ text{(nearest cent)}`
ii. `text(Over 6 years)`
`N = 6 xx 12 = 72\ text(months)`
`r = 10.8/12 = text(0.9%) = 0.009`
`PVtext{(annuity) table factor}\ =52.82118`
`text(Let)\ $M =\ text(monthly repayment)`
`text(Loan)\ = PV\ text(of annuity)`
`$21\ 500` | `= M xx 52.82118` |
`:.\ M` | `= $407.033…` |
`= $407\ \ text{(nearest dollar)}` |
Data from 200 recent house sales are grouped into class intervals and a cumulative frequency histogram is drawn.
--- 1 WORK AREA LINES (style=lined) ---
--- 2 WORK AREA LINES (style=lined) ---
The shoe size and height of ten students were recorded.
\begin{array} {|l|c|c|}
\hline \rule{0pt}{2.5ex} \text{Shoe size} \rule[-1ex]{0pt}{0pt} & \text{6} & \text{7} & \text{7} & \text{8} & \text{8.5} & \text{9.5} & \text{10} & \text{11} & \text{12} & \text{12} \\
\hline \rule{0pt}{2.5ex} \text{Height} \rule[-1ex]{0pt}{0pt} & \text{155} & \text{150} & \text{165} & \text{175} & \text{170} & \text{170} & \text{190} & \text{185} & \text{200} & \text{195} \\
\hline
\end{array}
--- 1 WORK AREA LINES (style=lined) ---
--- 2 WORK AREA LINES (style=lined) ---
ii. `text{Shoe size 7½ gives a height estimate of 162 cm (see graph)}`
`text{Shoe size 9 gives a height estimate of 175 cm (see graph)}`
`:.\ text(Height difference)` | `= 175-162` |
`= 13\ text{cm (or close given LOBF drawn)}` |
iii. `text(A correlation co-efficient of 1 would mean)`
`text(that all data points occur on the line of best)`
`text(fit which clearly isn’t the case.)`
The results of two tests are normally distributed. The mean and standard deviation for each test are displayed in the table.
Kristoff scored 74 in Mathematics and 80 in English. He claims that he has performed better in English.
Is Kristoff correct? Justify your answer using appropriate calculations. (2 marks)
--- 4 WORK AREA LINES (style=lined) ---
`text(He is correct.)`
`text(In Maths)`
`ztext{-score(74)}` | `= (x − mu)/sigma` |
`= (74 − 70)/6.5` | |
`= 0.6153…` |
`text(In English)`
`ztext{-score(80)}` | `= (80 − 75)/8` |
`= 0.625` |
`=>\ text(Kristoff’s)\ ztext(-score in English is higher than)`
`text(his)\ z text(-score in Maths.)`
`:.\ text(He is correct. He performed better in English.)`
In a small business, the seven employees earn the following wages per week:
\(\$300, \ \$490, \ \$520, \ \$590, \ \$660, \ \$680, \ \$970\)
--- 6 WORK AREA LINES (style=lined) ---
What effect will this have on the standard deviation? (1 mark)
--- 2 WORK AREA LINES (style=lined) ---
i. \(\text{See Worked Solutions.} \)
ii. \(\text{The standard deviation will remain the same.}\)
i. \(300, 490, 520, 590, 660, 680, 970\)
\(\text{Median}\) | \(= 590\) |
\(Q_1\) | \(= 490\) |
\(Q_3\) | \(= 680\) |
\(IQR\) | \(= 680-490 = 190\) |
\(\text{Outlier if \$970 is greater than:} \)
\(Q_3 + 1.5 x\times IQR = 680 + 1.5 \times 190 = \$965 \)
\(\therefore\ \text{The wage \$970 per week is an outlier.}\)
ii. \(\text{All values increase by \$20, but so too does the mean.} \)
\(\text{Therefore the spread about the new mean will not change} \)
\(\text{and therefore the standard deviation will remain the same.} \)
Ariana’s parents have given her an interest‑free loan of $4800 to buy a car. She will pay them back by paying `$x` immediately and `$y` every month until she has repaid the loan in full.
After 18 months Ariana has paid back $1510, and after 36 months she has paid back $2770.
This information can be represented by the following equations.
`x + 18y = 1510`
`x + 36y = 2770`
i.
`:.\ text(Solution is)\ \ x = 250, \ y = 70`
ii. `text(Let)\ \ A = text(the amount paid back after)\ n\ text(months)`
`A = 250 + 70n`
`text(Find)\ n\ text(when)\ A = 4800`
`250 + 70n` | `= 4800` |
`70n` | `= 4550` |
`n` | `= 65` |
`:.\ text(It will take Ariana 65 months to repay)`
`text(the loan in full.)`
The table shows the relative frequency of selecting each of the different coloured jelly beans from packets containing green, yellow, black, red and white jelly beans.
--- 1 WORK AREA LINES (style=lined) ---
--- 1 WORK AREA LINES (style=lined) ---
i. `text(Relative frequency of red)`
`= 1 − (0.32 + 0.13 + 0.14 + 0.24)`
`= 1 − 0.83`
`= 0.17`
ii. `Ptext{(not selecting black)}`
`= 1 −Ptext{(selecting black)}`
`= 1 − 0.14`
`= 0.86`
A family currently pays $320 for some groceries.
Assuming a constant annual inflation rate of 2.9%, calculate how much would be paid for the same groceries in 5 years’ time. (2 marks)
`$369.17\ \ text{(nearest cent)}`
`FV` | `= PV(1 + r)^n` |
`= 320(1.029)^5` | |
`= $369.1703…` | |
`= $369.17\ \ text{(nearest cent)}` |
Consider the equation `(2x)/3-4 = (5x)/2 + 1`.
Which of the following would be a correct step in solving this equation?
`B`
`(2x)/3-4` | `= (5x)/2 + 1` |
`(2x)/3` | `= (5x)/2 + 5` |
`=>B`
A machine produces cylindrical pipes. The mean of the diameters of the pipes is 8 cm and the standard deviation is 0.04 cm.
Assuming a normal distribution, what percentage of cylindrical pipes produced will have a diameter less than 7.96 cm?
(A) `text(16%)`
(B) `text(32%)`
(C) `text(34%)`
(D) `text(68%)`
`A`
The table shows the life expectancy (expected remaining years of life) for females at selected ages in the given periods of time.
In 1975, a 45‑year‑old female used the information in the table to calculate the age to which she was expected to live. Twenty years later she recalculated the age to which she was expected to live.
What is the difference between the two ages she calculated?
(A) 2.7 years
(B) 3.1 years
(C) 3.7 years
(D) 5.8 years
`D`
`text(In 1975, her life expectancy)`
`=\ text(age + remaining years)`
`= 45 + 34`
`= 79`
`text(In 1995, her life expectancy)`
`= 65 + 19.8`
`= 84.8`
`:.\ text(Difference)` | `= 84.8 − 79` |
`= 5.8\ text(years)` |
`⇒ D`
What amount must be invested now at 4% per annum, compounded quarterly, so that in five years it will have grown to $60 000?
`C`
`text(Using)\ \ FV = PV(1 + r)^n`
`r` | `= text(4%)/4` | `= text(1%) = 0.01\ text(per quarter)` |
`n` | `= 5 xx 4` | `= 20\ text(quarters)` |
`60\ 000` | `= PV(1 + 0.01)^(20)` |
`:.PV` | `= (60\ 000)/1.01^(20)` |
`= $49\ 172.66…` |
`⇒ C`
The times, in minutes, that a large group of students spend on exercise per day are presented in the box‑and‑whisker plot.
What percentage of these students spend between 40 minutes and 60 minutes per day on exercise?
`C`
`text{Q}_1 = 40, \ text(Median) = 60`
`:.\ text(% Students between 40 and 60)`
`= 50text{%}-25text{%}`
`=25 text{%}`
`=>C`