A university uses gas to heat its buildings. Over a period of 10 weekdays during winter, the gas used each day was measured in megawatts (MW) and the average outside temperature each day was recorded in degrees Celsius (°C). Using `x` as the average daily outside temperature and `y` as the total daily gas usage, the equation of the least-squares regression line was found. The equation of the regression line predicts that when the temperature is 0°C, the daily gas usage is 236 MW. The ten temperatures measured were: 0°, 0°, 0°, 2°, 5°, 7°, 8°, 9°, 9°, 10°, The total gas usage for the ten weekdays was 1840 MW. In any bivariate dataset, the least-squares regression line passes through the point `(bar x,bar y)`, where `bar x` is the sample mean of the `x`-values and `bary` is the sample mean of the `y`-values. --- 4 WORK AREA LINES (style=lined) --- --- 2 WORK AREA LINES (style=lined) --- a. b. `y=-10.4x+236` c. `text{Answers could include one of the following:}` `text{→ 23°C is outside the range of the dataset and requires the trend}` `text{to be extrapolated.}` `text{→ At 23°C, the equation predicts negative daily gas usage.}` a. `barx=(0+0+0+2+5+7+8+9+9+10)/10=5^@text{C}` `bary=1840/10=184` `text{Regression line passes through:}\ (0,236) and (5,184)` b. `m=(y_2-y_1)/(x_2-x_1)=(184-236)/(5-0)=-10.4` `text{Equation of line}\ m=-10.4\ text{passing through}\ (0,236):` `text{→ 23°C is outside the range of the dataset and requires the trend}` `text{to be extrapolated.}` `text{→ At 23°C, the equation predicts negative daily gas usage.}`
`(y-y_1)`
`=m(x-x_1)`
`y-236`
`=-10.4(x-0)`
`y`
`=-10.4x+236`
c. `text{Answers could include one of the following:}`
Statistics, 2ADV S2 2022 HSC 24
Jo is researching the relationship between the ages of teenage characters in television series and the ages of actors playing these characters.
After collecting the data, Jo finds that the correlation coefficient is 0.4564.
A scatterplot showing the data is drawn. The line of best fit with equation `y=-7.51+1.85 x`, is also drawn.
Describe and interpret the data and other information provided, with reference to the context given. (4 marks)
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Statistics, STD2 S4 2014* HSC 30b
The scatterplot shows the relationship between expenditure per primary school student, as a percentage of a country’s Gross Domestic Product (GDP), and the life expectancy in years for 15 countries.
- For the given data, the correlation coefficient, `r`, is 0.83. What does this indicate about the relationship between expenditure per primary school student and life expectancy for the 15 countries? (1 mark)
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- For the data representing expenditure per primary school student, `Q_L` is 8.4 and `Q_U` is 22.5.
What is the interquartile range? (1 mark)
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- Another country has an expenditure per primary school student of 47.6% of its GDP.
Would this country be an outlier for this set of data? Justify your answer with calculations. (2 marks)
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- On the scatterplot, draw the least-squares line of best fit `y = 1.29x + 49.9`. (2 marks)
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- Using this line, or otherwise, estimate the life expectancy in a country which has an expenditure per primary school student of 18% of its GDP. (1 mark)
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- Why is this line NOT useful for predicting life expectancy in a country which has expenditure per primary school student of 60% of its GDP? (1 mark)
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Show Answers Only
- `text(It indicates there is a strong positive)`
`text(correlation between the two variables.)`
- `14.1`
- `text(Yes, because it’s > 43.65%)`
-
- `73.1\ text(years)`
- `text(At 60% GDP, the line predicts a life expectancy)`
-
`text(of 127.3. This line of best fit is only accurate)`
-
`text(in a lower range of GDP expediture.)`
Show Worked Solution
| i. | `text(It indicates there is a strong positive)` |
| `text(correlation between the two variables)` |
| ii. | `text(IQR)` | `= Q_U\ – Q_L` |
| `= 22.5\ – 8.4` | ||
| `= 14.1` |
♦ Mean mark 35%
iii. `text(An outlier on the upper side must be more than)`
`Q_u\ +1.5xxIQR`
`=22.5+(1.5xx14.1)`
`=\ text(43.65%)`
`:.\ text(A country with an expenditure of 47.6% is an outlier).`
| iv. |  |
v. `text(Life expectancy) ~~ 73.1\ text{years (see dotted line)}`
♦♦ Mean mark 39%
`text(Alternative Solution)`
`text(When)\ x=18`
`y=1.29(18)+49.9=73.12\ \ text(years)`
♦♦♦ Mean mark 0%. The toughest question on the 2014 paper.
COMMENT: Examiners regularly ask students to identify and comment on outliers where linear relationships break down.
COMMENT: Examiners regularly ask students to identify and comment on outliers where linear relationships break down.
| vi. | `text(At 60% GDP, the line predicts a life)` |
| `text(expectancy of 127.3. This line of best)` | |
| `text(fit is only predictive in a lower range)` | |
| `text(of GDP expenditure.)` |
Statistics, STD2 S4 2013 HSC 28b
Ahmed collected data on the age (`a`) and height (`h`) of males aged 11 to 16 years.
He created a scatterplot of the data and constructed a line of best fit to model the relationship between the age and height of males.
- Determine the gradient of the line of best fit shown on the graph. (1 mark)
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- Explain the meaning of the gradient in the context of the data. (1 mark)
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- Determine the equation of the line of best fit shown on the graph. (2 marks)
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- Use the line of best fit to predict the height of a typical 17-year-old male. (1 mark)
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- Why would this model not be useful for predicting the height of a typical 45-year-old male? (1 mark)
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