Rani and Diksha go shopping for clothes. (i) Rani buys 4 identical vests, 3 identical sweaters and 1 coat. Each vest costs \$5.50 and the coat costs \$90. The mean cost of Rani's 8 items is \$29. Find the cost of a sweater. (ii) Diksha buys 1 hat and 4 identical shirts. The mean cost of Diksha's 5 items is \$26 and the standard deviation is \$0. Explain how you can tell that Diksha spends \$104 on shirts.

(i) 40 dollars · (ii) 104 dollars

Mathematics 9709/62 — May/June 2017

Cambridge AS Level · Probability & Statistics 1 (S1) · worked solutions for every part, with the mark scheme

questions

marks

minutes

Topics Representation of Data · Discrete Random Variables · Probability · The Normal Distribution · Permutations and Combinations

Q1Representation of DataFree sample

Rani and Diksha go shopping for clothes.

(i)

Rani buys 4 identical vests, 3 identical sweaters and 1 coat. Each vest costs $5.50 and the coat costs $90. The mean cost of Rani's 8 items is $29. Find the cost of a sweater.

DifficultyMedium-Easy

Worked solution

Approach

Use the definition of the mean: the mean cost equals the total cost divided by the number of items. Since the mean of 8 items is $29$ dollars, the total cost is $8(29)$ . Let the cost of one sweater be $x$ dollars, write the total cost in terms of $x$ , equate it to $8(29)$ , and solve.

Working

Let the cost of one sweater be $x$ dollars.

The total cost is:

4(5.50) + 3x + 90 = 8(29)

Simplify:

22 + 3x + 90 = 232

112 + 3x = 232

Subtract 112:

3x = 120

Divide by 3:

x = 40

Answer

The cost of a sweater is 40 dollars.

Final answer

40 dollars

Detailed explanation

Walkthrough

The mean of a set of values is found by adding all the values and dividing by how many there are. Here the mean is $29$ and there are $8$ items, so the total cost must be $8(29)=232$ dollars. Rani's items are 4 vests, 3 sweaters and 1 coat. Let one sweater cost $x$ dollars. The total cost is $4(5.50)+3x+90$ . Set this equal to $232$ and solve: $22+3x+90=232$ , so $112+3x=232$ , $3x=120$ , $x=40$ . Thus each sweater costs $40$ dollars.

Key Takeaways

The mean can be used in reverse: if you know the mean and the number of values, you can find the total. Setting up an equation from the total cost is often the key to finding an unknown item price.

Common Mistakes

Forgetting to multiply the mean by the number of items; using $29$ as the total instead of $8(29)$ .
Forgetting the coat in the total cost.
Arithmetic errors when simplifying $4(5.50)$ and adding $90$ .
Not showing the equation; the mark scheme requires a method expression equating total cost to $8(29)$ .

Things to Be Careful About

There are $8$ items, not $4$ or $3$ . The mean applies to all $8$ items.
The vests cost $5.50$ each, so 4 vests contribute $22$ dollars, not $5.50$ .
Units are not required for the final answer, but the value must be $40$ .

Techniques used

form total cost equation using mean = total divided by number of itemssolve linear equation for unknown cost

(ii)

Diksha buys 1 hat and 4 identical shirts. The mean cost of Diksha's 5 items is $26 and the standard deviation is $0. Explain how you can tell that Diksha spends $104 on shirts.

DifficultyMedium-Easy

Worked solution

Approach

A standard deviation of $0$ means there is no spread in the data, so all five items must have the same cost. Since the mean cost is $26$ dollars, each item costs $26$ dollars. Diksha buys $4$ shirts, so the shirts cost $4(26)=104$ dollars.

Working

Because the standard deviation is $0$ , all five items have the same cost.

Mean is $26$ dollars, so each item costs $26$ dollars.

Total cost of all five items: $5(26)=130$ dollars.

The hat costs $26$ dollars, so the four shirts cost $130-26=104$ dollars.

Answer

Since the standard deviation is $0$ , all items cost the same; each costs $26$ dollars, so four shirts cost $104$ dollars.

Final answer

104 dollars

Detailed explanation

Walkthrough

A standard deviation of $0$ means there is no variation between the data values: all values are identical. Since Diksha's five items have mean $26$ , and all five items are the same cost, each item must cost $26$ dollars. Therefore the four shirts cost $4(26)=104$ dollars. Equivalently, the total cost is $5(26)=130$ dollars, and subtracting the hat's $26$ dollars gives $130-26=104$ dollars.

Key Takeaways

Standard deviation measures spread. A standard deviation of $0$ is a special case: every observation is the same. Knowing this lets you deduce individual values from the mean.

Common Mistakes

Thinking standard deviation $0$ means the mean is $0$ .
Thinking standard deviation $0$ means all items are free or that the shirts cost $26$ dollars total.
Not explaining why all items cost the same; the mark scheme requires an interpretation of standard deviation $0$ .

Things to Be Careful About

The standard deviation is $0$ , not the mean. The mean is $26$ .
There are five items total, but the question asks for four shirts, so multiply $26$ by $4$ (or subtract the hat from the total).
The final value $104$ must be stated; the mark scheme allows $4(26)$ or $130-26$ .

Techniques used

interpret zero standard deviation as identical data valuesuse the mean to find each item's cost and multiply by the number of shirts

The rest of this paper

6 more questions

Q2Representation of Data6M
Q3Discrete Random Variables6M
Q4Probability6M
Q5The Normal Distribution9M
Q6Permutations and Combinations9M
Q7Probability · Discrete Random Variables9M

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