Question .
The formula $F = 1.8C + 32$ is used to convert a temperature in degrees Celsius, $C$, to degrees Fahrenheit, $F$.
(a) (i) Find a formula for converting a temperature in degrees Fahrenheit to degrees Celsius.
(ii) Find the temperature in degrees Celsius that is recorded as 77 degrees Fahrenheit.
Over one year, the mean daily temperature in Mexico City was calculated to be 17 degrees Celsius with a standard deviation of 9 degrees Celsius.
(b) For the same year, find in degrees Fahrenheit
(i) the mean daily temperature in Mexico City.
(ii) the standard deviation of the daily temperature in Mexico City.
▶️Answer/Explanation
Detailed Solution
Part (a): Temperature Conversion
(i) Formula for converting Fahrenheit to Celsius
The given formula is:
\[ F = 1.8C + 32 \]
where \( F \) is the temperature in degrees Fahrenheit and \( C \) is the temperature in degrees Celsius. We need to solve for \( C \) in terms of \( F \).
Start by isolating \( C \):
\[ F = 1.8C + 32 \]
Subtract 32 from both sides:
\[ F – 32 = 1.8C \]
Divide both sides by 1.8:
\[ C = \frac{F – 32}{1.8} \]
So, the formula to convert Fahrenheit to Celsius is:
\[ C = \frac{F – 32}{1.8} \]
(ii) Temperature in Celsius at 77°F
Using the formula from part (i), substitute \( F = 77 \):
\[ C = \frac{77 – 32}{1.8} \]
Calculate:
\[ C = \frac{45}{1.8} = 25 \]
So, 77°F corresponds to 25°C.
Part (b): Mean and Standard Deviation in Fahrenheit
Over one year, the mean daily temperature in Mexico City is 17°C with a standard deviation of 9°C. We need to convert these to Fahrenheit.
(i) Mean daily temperature in Fahrenheit
Use the original formula \( F = 1.8C + 32 \) with \( C = 17 \):
\[ F = 1.8 \cdot 17 + 32 \]
\[ F = 30.6 + 32 = 62.6 \]
So, the mean daily temperature in Fahrenheit is 62.6°F.
(ii) Standard deviation of the daily temperature in Fahrenheit
Standard deviation measures variability, and when converting temperatures from Celsius to Fahrenheit, only the scaling factor (1.8) applies, not the shift (+32), because the shift is a constant that doesn’t affect the spread.
The standard deviation in Celsius is 9°C. Since the conversion \( F = 1.8C + 32 \) is linear, the standard deviation in Fahrenheit is:
\[ \text{Standard deviation in °F} = \text{Standard deviation in °C} \cdot 1.8 \]
\[ = 9 \cdot 1.8 = 16.2 \]
So, the standard deviation in Fahrenheit is 16.2°F.
……………………………….Markscheme……………………………….
Solution: –
$(a) \ (i) \ \text{attempt to rearrange to isolate C}$
$\text{e.g., subtracting 32 or dividing the equation by 1.8}$
$C = \frac{5}{9}(F-32) \left( C = \frac{F-32}{1.8}, C = 0.556F – 17.8 \right)$
$(ii) \ C = \left(\frac{77-32}{1.8} = \right) 25 (°C)$
$(b) \ (i) \ (1.8 \times 17 + 32 =) 62.6 (°F)$
$(ii) \ \text{recognizing that the “+32” does not affect the SD}$
$(1.8 \times 9 =) 16.2 (°F)$
Question
The prices, in dollars, of 10 different garden chairs are:
79, 139, 255, 99, 50, 209, 229, 193, 69, 49.
(a) Find the range of the prices of the 10 chairs.
(b) Use your graphic display calculator to find:
(i) the mean price of the chairs,
(ii) the standard deviation of the prices of the chairs.
In a sale, the price of each of the 10 garden chairs is reduced by $20.
(c) Write down:
(i) the new mean,
(ii) the new standard deviation.
▶️ Answer/Explanation
Detailed Solution
(a) Finding the Range
- Largest price: \( 255 \)
- Smallest price: \( 49 \)
- Range: \( 255 – 49 = 206 \)
(b) Using the Graphic Display Calculator
- (i) Mean price: \( 137 \) (or \( 137.1 \))
- (ii) Standard deviation: \( 74.5 \) (or \( 74.4693… \))
(c) After reducing each price by $20:
- (i) New mean: \( 137 – 20 = 117 \) (or \( 117.1 \))
- (ii) New standard deviation: \( 74.5 \) (unchanged, as standard deviation is not affected by a constant shift).
…………………………..Markscheme…………………………..
(a)
- Correct calculation of range: \( 206 \).
(b) (i)
- Correct mean: \( 137 \) (or \( 137.1 \)).
(b) (ii)
- Correct standard deviation: \( 74.5 \) (or \( 74.4693… \)).
(c) (i)
- Correct new mean: \( 117 \) (or \( 117.1 \)).
(c) (ii)
- Correctly stating that the standard deviation remains unchanged.
Question
The formula for converting a temperature in degrees Celsius, \(C\), to degrees Fahrenheit, \(F\), is:
\[ F = 1.8C + 32 \]
(a)
(i) Find a formula for converting a temperature in degrees Fahrenheit to degrees Celsius.
(ii) Find the temperature in degrees Celsius that is recorded as 77 degrees Fahrenheit.
Over one year, the mean daily temperature in Mexico City was calculated to be 17 degrees Celsius with a standard deviation of 9 degrees Celsius.
(b) For the same year, find in degrees Fahrenheit:
(i) The mean daily temperature in Mexico City.
(ii) The standard deviation of the daily temperature in Mexico City.
▶️Answer/Explanation
Detailed Solution
(a) (i) Finding the formula for \(C\)
We start with the given equation:
\[ F = 1.8C + 32 \]
To isolate \(C\):
1. Subtract 32 from both sides:
\[ F – 32 = 1.8C \]
2. Divide by 1.8:
\[ C = \frac{F – 32}{1.8} \]
Thus, the formula to convert Fahrenheit to Celsius is:
\[ C = \frac{5}{9}(F – 32) \]
(a) (ii) Converting 77°F to Celsius
Using the derived formula:
\[ C = \frac{77 – 32}{1.8} \]
\[ C = \frac{45}{1.8} = 25 \]
Thus, 77°F is equivalent to 25°C.
(b) (i) Finding the Mean Temperature in Fahrenheit
Using the given mean temperature \( C = 17 \), we convert to Fahrenheit:
\[ F = 1.8(17) + 32 \]
\[ F = 30.6 + 32 = 62.6 \]
Thus, the mean daily temperature in Mexico City is 62.6°F.
(b) (ii) Finding the Standard Deviation in Fahrenheit
The standard deviation in Fahrenheit follows the rule:
\[ \text{SD}_F = 1.8 \times \text{SD}_C \]
Given \( \text{SD}_C = 9 \), we calculate:
\[ \text{SD}_F = 1.8 \times 9 = 16.2 \]
Thus, the standard deviation of the daily temperature in Mexico City is 16.2°F.
……………………………Markscheme……………………………….
(a) (i)
\( C = \frac{5}{9} (F – 32) \)
(ii)
\( C = 25°C \)
(b) (i)
\( \text{Mean} = 62.6°F \)
(ii)
\( \text{Standard deviation} = 16.2°F \)
Question
The following data show the heights, in metres, of six players in a basketball team.
1.67, 1.60, 1.68, 2.31, 2.31, 2.19
(a) For these six players, find
(i) the mean height.
(ii) the median height.
(iii) the modal height.
(iv) the range of the heights.
A new player, Gheorghe, joins the team. Their height is measured as 1.98 metres to the nearest centimetre.
(b) Write down the shortest possible height of Gheorghe.
▶️Answer/Explanation
Detailed solution
(a) Finding the Measures of Central Tendency
(i) Mean Height:
The mean height is calculated by summing all the heights and dividing by the total number of players.
\[ \text{Mean} = \frac{1.67 + 1.60 + 1.68 + 2.31 + 2.31 + 2.19}{6} \]
\[ = \frac{11.76}{6} = 1.96 \text{ m} \]
(ii) Median Height:
To find the median, arrange the heights in ascending order:
\[ 1.60, 1.67, 1.68, 2.19, 2.31, 2.31 \]
Since there are 6 numbers (even count), the median is the average of the middle two values:
\[ \text{Median} = \frac{1.68 + 2.19}{2} = \frac{3.87}{2} = 1.935 \text{ m} \approx 1.94 \text{ m} \]
(iii) Modal Height:
The mode is the most frequently occurring value. Here, 2.31 appears twice, making it the modal height.
\[ \text{Mode} = 2.31 \text{ m} \]
(iv) Range:
The range is found by subtracting the smallest height from the largest height:
\[ \text{Range} = 2.31 – 1.60 = 0.71 \text{ m} \]
(b) Shortest Possible Height of Gheorghe
Since Gheorghe’s height is given as 1.98 m to the nearest centimetre, the actual height could be any value within ±0.005 m of 1.98 m.
The shortest possible height is:
\[ 1.975 \text{ m} = 197.5 \text{ cm} \]
……………………………Markscheme……………………………….
(a)
(i) \(1.96\) (m)
(ii) \(1.94\) (m) (\(1.935\))
(iii) \(2.31\) (m)
(iv) \(2.31 – 1.60 = 0.71\) (m)
(b)
\(1.975\) (m) OR \(197.5\) (cm)