Data and probability

Chapter 12 · 11 lessons · 276 questions

Unit 1: Probability and conditional probability

Key points

Probability = desired outcomes $\div$ total possible outcomes.
Write the fraction in words first, then put in the numbers.
‘Given that’ and ‘of those who’ shrink the total to that group only.
A probability is always between $0$ and $1$ .

See it solved

Example

The table shows the results of a survey of $200$ students. A student in grade $10$ is chosen at random. What is the probability that the student is in a club?

Survey of 200 students

Two-way table of grade and club membership, with totals. Grade 9: 48 in a club, 52 not in a club, 100 in all. Grade 10: 36 in a club, 64 not in a club, 100 in all. Totals: 84 in a club, 116 not in a club, 200 students.
	In a club	Not in a club	Total
Grade 9	48	52	100
Grade 10	36	64	100
Total	84	116	200

Worked by hand

The plan

Determine the probability

Desmos: Desmos does not help you choose the right row or column. It can only turn your fraction into a decimal.

Step 1: Determine the probability
- In words: grade $10$ students in a club $\div$ all grade $10$ students.
- ‘A student in grade $10$ ’ limits the total to the grade $10$ row: $100$ students.
- Of those, $36$ are in a club.
- $\dfrac{36}{100} = \dfrac{9}{25}$ , which is $0.36$ .

The correct answer is

\dfrac{9}{25}

, which is

0.36

The same example in Desmos

new graph
Desmos shows: $0.36$ , at the right of the row. Click the fraction button beside it to see $\dfrac{9}{25}$ .

Desmos expression list: frac{36}{100} — Type

Made with Desmos

Open Desmos

Your turn

The table shows the results of a survey of $200$ students. A student who is in a club is chosen at random. What is the probability that the student is in grade $9$ ?

Survey of 200 students

Two-way table of grade and club membership, with totals. Grade 9: 48 in a club, 52 not in a club, 100 in all. Grade 10: 36 in a club, 64 not in a club, 100 in all. Totals: 84 in a club, 116 not in a club, 200 students.
	In a club	Not in a club	Total
Grade 9	48	52	100
Grade 10	36	64	100
Total	84	116	200

Show the answerHide the answer

Answer

The correct answer is $\dfrac{4}{7}$ .

Step 1: Determine the probability
- In words: club members in grade $9$ $\div$ all club members.
- ‘A student who is in a club’ limits the total to the ‘In a club’ column: $84$ students.
- Of those, $48$ are in grade $9$ .
- $\dfrac{48}{84}$ . Divide the top and the bottom by $12$ : $\dfrac{4}{7}$ .

Traps

Using the grand total. $\dfrac{36}{200} = 0.18$ is the probability that a student chosen from all $200$ is in grade $10$ and in a club. The question chooses only from grade $10$ .
Swapping the condition. ‘A grade $9$ student who is in a club’ and ‘a club member who is in grade $9$ ’ have different totals: $\dfrac{48}{100}$ and $\dfrac{48}{84}$ .

Practice this lesson40 questions

Unit 2: Graphs of relationships

Key points

Each point on the graph pairs an input (how far across) with an output (how far up).
A piece that rises means the amount is going up. A flat piece means no change. A piece that falls means the amount is going down.
The steeper the piece, the faster the change.
Read the axis titles and their units before anything else.

See it solved

Example

The graph shows Leo's distance from home, in miles, during a $60$ -minute bike ride. For how many minutes was Leo not moving?

Worked by hand

The plan

Determine what a flat piece means
Read the flat piece

Desmos: Desmos cannot read a graph for you. It can only check the subtraction.

Step 1: Determine what a flat piece means
- The height of the graph is Leo's distance from home.
- When Leo is not moving, his distance from home does not change, so the graph is flat.
Step 2: Read the flat piece
- The graph is flat from $20$ minutes to $35$ minutes.
- $35 - 20 = 15$ minutes.

The correct answer is

15

The same example in Desmos

new graph
Desmos shows: $15$ , as a check on the subtraction.

Made with Desmos

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Your turn

The graph shows Leo's distance from home, in miles, during a $60$ -minute bike ride. What was Leo's speed, in miles per minute, during the last $25$ minutes of the ride?

Show the answerHide the answer

Answer

The correct answer is $0.28$ .

Step 1: Read the two ends of the piece
- The last piece runs from $(35, 3)$ to $(60, 10)$ .
Step 2: Determine the slope
- Distance: $10 - 3 = 7$ miles.
- Time: $60 - 35 = 25$ minutes.
- $\dfrac{7}{25} = 0.28$ miles per minute.

Traps

Reading a flat piece as ‘riding straight ahead’. On this graph, flat means the distance from home is not changing: Leo has stopped.
Using the height instead of the change. At $60$ minutes the graph is at $10$ miles, but the last piece covers only $10 - 3 = 7$ miles. $\dfrac{10}{25} = 0.4$ is a wrong answer.

Practice this lesson12 questions

Unit 3: Scatterplots and lines of best fit

Key points

A scatterplot shows pairs of values as dots. The line of best fit shows the trend.
A predicted value is read from the line, not from a dot.
For the slope, use two grid points on the line, not two dots.
The slope is the predicted change in $y$ for each increase of $1$ in $x$ .

See it solved

Example

The scatterplot shows the ages and heights of $10$ trees, with a line of best fit. What height, in feet, does the line of best fit predict for a tree that is $8$ years old?

Worked by hand

The plan

Read the line at the input

Desmos: Reading the line takes a few seconds. Desmos is a check: put two grid points of the line in a table and fit a line through them.

Step 1: Read the line at the input
- Find $8$ on the horizontal axis.
- Go straight up to the line of best fit, not to the dot.
- The line is at $20$ there. The predicted height is $20$ feet.
- The dot above $8$ is at $18$ feet. That is one real tree, not the prediction.

The correct answer is

20

The same example in Desmos

new graph, then Add item + and Table
Table
x₁ y₁
0 4
10 24
Desmos shows: Two grid points on the line of best fit.
Keys: y1~mx1+b
Desmos shows: $m = 2$ and $b = 4$ .
Desmos shows: The line of best fit.
Desmos shows: $20$ , at the right of the row: the predicted height.

Desmos expression list: a table of x1: 0, 10 and y1: 4, 24; y1~ mx1+b; f(x)=mx+b; f(8) — Type

Desmos graph of a table of x1: 0, 10 and y1: 4, 24; y1~ mx1+b; f(x)=mx+b; f(8), with (8, 20) labelled — Look

Made with Desmos

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Your turn

The scatterplot shows the ages and heights of $10$ trees, with a line of best fit. According to the line, by how many feet does the predicted height increase each year?

Show the answerHide the answer

Answer

The correct answer is $2$ .

Step 1: Determine the slope
- Pick two grid points on the line: $(0, 4)$ and $(10, 24)$ .
- Rise: $24 - 4 = 20$ feet.
- Run: $10 - 0 = 10$ years.
- $\dfrac{20}{10} = 2$ feet per year.

Traps

Reading the dot instead of the line. The dot above $8$ years is at $18$ feet. That is one real tree. The prediction comes from the line: $20$ feet.
Taking the slope from two dots. The dots are scattered around the line. Use two points on the line itself, where it crosses grid lines.

Practice this lesson24 questions

Unit 4: Bar graphs, histograms and dot plots

Key points

In a bar graph, the height of a bar is the value for that category.
In a dot plot, each dot is one data value. Count the dots.
In a histogram, each bar counts the values in a range. You cannot see the exact values.
The number of data values is the total of the counts, not the number of bars.

See it solved

Example

The bar graph shows the number of books that each of five students read in March. How many books did the five students read in all?

Books read in March

Worked by hand

The plan

Read the value of each bar
Add the values

Desmos: Desmos cannot read a graph for you. After you have read the values, a list is only a check on the adding.

Step 1: Read the value of each bar
- Ana: $6$ . Ben: $9$ . Cal: $4$ . Dee: $12$ . Eli: $7$ .
- Ben's bar and Eli's bar end halfway between two numbered lines.
Step 2: Add the values
- $6 + 9 = 15$ .
- $15 + 4 = 19$ .
- $19 + 12 = 31$ .
- $31 + 7 = 38$ books.

The correct answer is

38

The same example in Desmos

new graph
Desmos shows: The five values you read from the graph, as a list.
Desmos shows: $38$ , at the right of the row.

Desmos expression list: L=[6,9,4,12,7]; operatorname{total}(L) — Type

Made with Desmos

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Your turn

The dot plot shows the number of pets owned by each student in a class. How many students own more than $2$ pets?

Number of pets

Show the answerHide the answer

Answer

The correct answer is $6$ .

Step 1: Read the dots
- ‘More than $2$ ’ means $3$ pets or $4$ pets. It does not include $2$ .
- There are $4$ dots above $3$ and $2$ dots above $4$ .
Step 2: Add the counts
- $4 + 2 = 6$ students.

Traps

Counting the bars instead of adding the values. There are $5$ bars, but $5$ is the number of students, not the number of books.
Including the boundary. ‘More than $2$ ’ leaves out the students with exactly $2$ pets. Adding their $5$ dots gives $11$ , a wrong answer.

Practice this lesson32 questions

Unit 5: The mean

Key points

The mean is the sum of the values divided by the number of values.
Turned round: sum = mean $\times$ number of values. This is the key to most mean questions.
For a missing value, find the sum you need, then subtract the values you have.
For two groups together, add the two sums and divide by the total number of values.

See it solved

Example

The mean of five test scores is $82$ . Four of the scores are $75$ , $80$ , $88$ and $90$ . What is the fifth score?

Worked by hand

The plan

Set up the mean
Determine the missing value

Desmos: Use sum = mean $\times$ count by hand. Desmos is a check: put all the values in a list and ask for the mean.

Step 1: Set up the mean
- Sum = mean $\times$ number of values.
- $82 \cdot 5 = 410$ . The five scores add up to $410$ .
Step 2: Determine the missing value
- Add the four known scores: $75 + 80 = 155$ .
- $155 + 88 = 243$ .
- $243 + 90 = 333$ .
- The fifth score is $410 - 333 = 77$ .
- Check: $\dfrac{333 + 77}{5}$ $= \dfrac{410}{5}$ $= 82$ . It matches.

The correct answer is

77

The same example in Desmos

new graph
Desmos shows: All five scores, with your answer as the fifth.
Desmos shows: $82$ , at the right of the row. The mean matches, so $77$ is right.

Desmos expression list: L=[75,80,88,90,77]; operatorname{mean}(L) — Type

Made with Desmos

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Your turn

A class of $20$ students has a mean score of $70$ on a test. A second class of $30$ students has a mean score of $80$ on the same test. What is the mean score of all $50$ students?

Show the answerHide the answer

Answer

The correct answer is $76$ .

Step 1: Determine each sum
- First class: $70 \cdot 20 = 1{,}400$ .
- Second class: $80 \cdot 30 = 2{,}400$ .
Step 2: Set up the mean
- All the scores add up to $1{,}400 + 2{,}400$ $= 3{,}800$ .
- There are $20 + 30 = 50$ students.
- $\dfrac{3{,}800}{50} = 76$ .

Traps

Averaging the four known scores. $\dfrac{333}{4} = 83.25$ is the mean of four scores. It is not the missing fifth score.
Averaging two means without the group sizes. $\dfrac{70 + 80}{2} = 75$ treats the classes as the same size. The bigger class pulls the mean toward $80$ : the answer is $76$ .

Practice this lesson28 questions

Unit 6: The median

Key points

The median is the middle value when the values are in order.
With an odd number of values, it is the one in the middle.
With an even number of values, it is the mean of the two middle values.
In a frequency table, find the middle position by adding up the counts.

See it solved

Example

The numbers of points a team scored in $8$ games are $12$ , $7$ , $15$ , $9$ , $20$ , $7$ , $11$ and $14$ . What is the median number of points?

Worked by hand

The plan

Sort the numbers
Determine the middle number

Desmos: With a list of values, Desmos is quicker: type the list as given and ask for the median. Desmos sorts it for you.

Step 1: Sort the numbers
- From least to greatest: $7$ , $7$ , $9$ , $11$ , $12$ , $14$ , $15$ , $20$ .
Step 2: Determine the middle number
- There are $8$ values, an even number. The middle two are the $4$ th and the $5$ th.
- They are $11$ and $12$ .
- $\dfrac{11 + 12}{2}$ $= \dfrac{23}{2}$ $= 11.5$ .

The correct answer is

11.5

The same example in Desmos

new graph
Desmos shows: The eight values, in the order given.
Desmos shows: $11.5$ , at the right of the row.

Desmos expression list: L=[12,7,15,9,20,7,11,14]; operatorname{median}(L) — Type

Made with Desmos

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Your turn

The table shows the number of siblings of each of $25$ students. What is the median number of siblings?

Siblings of 25 students

Frequency table. Number of siblings 0: 4 students. 1: 9 students. 2: 7 students. 3: 3 students. 4: 2 students.
Number of siblings	Number of students
0	4
1	9
2	7
3	3
4	2

Show the answerHide the answer

Answer

The correct answer is $1$ .

Step 1: Determine the middle position
- There are $25$ values. The middle one is the $13$ th, because $12$ values are below it and $12$ are above it.
Step 2: Count up to the middle position
- $0$ siblings: students $1$ to $4$ .
- $1$ sibling: the next $9$ students, numbers $5$ to $13$ .
- The $13$ th student has $1$ sibling. The median is $1$ .

Traps

Not sorting first. In the order given, the middle two values are $9$ and $20$ , which gives $14.5$ , a wrong answer.
Taking the middle row of a frequency table. The middle row is $2$ siblings, but the rows hold different numbers of students. Count positions instead: the median is $1$ .

Practice this lesson24 questions

Unit 7: Box plots and quartiles

Key points

A box plot shows five numbers: the minimum, the first quartile $Q_1$ , the median, the third quartile $Q_3$ and the maximum.
The box runs from $Q_1$ to $Q_3$ . It holds the middle half of the data.
Range = maximum $-$ minimum. Interquartile range $= Q_3 - Q_1$ .
A box plot does not show the mean or the number of values.

See it solved

Example

The box plot summarizes the heights, in inches, of the plants in a garden. What is the interquartile range of the heights?

Plant heights (inches)

Worked by hand

The plan

Read the five numbers
Determine the interquartile range

Desmos: Desmos cannot read a box plot for you. Read the five numbers yourself. Desmos can only check the subtraction.

Step 1: Read the five numbers
- Minimum: $10$ . $Q_1$ : $14$ . Median: $17$ . $Q_3$ : $22$ . Maximum: $30$ .
- $Q_1$ and $Q_3$ are the two ends of the box.
Step 2: Determine the interquartile range
- $Q_3 - Q_1$ $= 22 - 14$ $= 8$ inches.

The correct answer is

8

The same example in Desmos

new graph
Desmos shows: $8$ : the interquartile range, as a check on the subtraction.
Desmos shows: $20$ : the range. It is a different measure, from the ends of the whiskers.

Desmos expression list: 22-14; 30-10 — Type

Made with Desmos

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Your turn

The box plots summarize the test scores of two classes. By how many points does the range of the scores in class A exceed the range of the scores in class B?

Test scores

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Answer

The correct answer is $15$ .

Step 1: Read the minimum and maximum of each class
- Class A: minimum $50$ , maximum $95$ .
- Class B: minimum $55$ , maximum $85$ .
Step 2: Determine each range
- Class A: $95 - 50 = 45$ .
- Class B: $85 - 55 = 30$ .
- $45 - 30 = 15$ points.

Traps

Using the whiskers for the interquartile range. $30 - 10 = 20$ is the range, from the minimum to the maximum. The interquartile range uses the ends of the box: $22 - 14 = 8$ .
Reading the line in the box as the mean. That line is the median, $17$ . A box plot never shows the mean.

Practice this lesson12 questions

Unit 8: Standard deviation and range

Key points

Range = greatest value $-$ least value.
Standard deviation measures how spread out the values are around the mean.
Values bunched close to the mean give a small standard deviation. Values far from the mean give a large one.
You never calculate a standard deviation on the test. You compare two data sets by eye.

See it solved

Example

Data set A is $48$ , $49$ , $50$ , $51$ , $52$ . Data set B is $30$ , $40$ , $50$ , $60$ , $70$ . Which statement about the two data sets is true?

AThe means are equal, and set A has the greater standard deviation.
BThe means are equal, and set B has the greater standard deviation.
CSet A has the greater mean and the greater standard deviation.
DSet B has the greater mean, and the standard deviations are equal.

Worked by hand

The plan

Compare the means
Compare the spreads

Desmos: You are not asked to calculate a standard deviation, and comparing by eye is quicker. Desmos can confirm which one is larger, as a check.

Step 1: Compare the means
- Set A: $48 + 49 + 50$ $= 147$ , and $147 + 51 + 52$ $= 250$ .
- The mean of set A is $\dfrac{250}{5} = 50$ .
- Set B: $30 + 40 + 50$ $= 120$ , and $120 + 60 + 70$ $= 250$ .
- The mean of set B is $\dfrac{250}{5} = 50$ .
- The means are equal. That rules out choices C and D.
Step 2: Compare the spreads
- In set A, every value is within $2$ of the mean.
- In set B, the values are up to $20$ away from the mean.
- Set B is more spread out, so it has the greater standard deviation: choice B.

The correct answer choice is B: both means are

50

, and set B is more spread out.

The same example in Desmos

new graph
and
Desmos shows: The two data sets as lists.
and
Desmos shows: $50$ and $50$ : the means are equal.
and
Desmos shows: About $1.58$ and $15.81$ : set B has the greater standard deviation.

Desmos expression list: A=[48,49,50,51,52]; B=[30,40,50,60,70]; operatorname{mean}(A); operatorname{mean}(B); operatorname{stdev}(A); operatorname{stdev}(B) — Type

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Your turn

The daily high temperatures in a town for one week were $61$ , $58$ , $70$ , $66$ , $59$ , $73$ and $64$ degrees. What is the range of these temperatures, in degrees?

Show the answerHide the answer

Answer

The correct answer is $15$ .

Step 1: Determine the greatest and least values
- The greatest value is $73$ . The least value is $58$ .
Step 2: Determine the range
- $73 - 58 = 15$ degrees.

Traps

Thinking that bigger numbers mean a bigger standard deviation. Spread is about distance from the mean, not size. The set $1{,}001$ , $1{,}002$ , $1{,}003$ has a small standard deviation.
Subtracting the first and last values as listed. $64 - 61 = 3$ uses the order in the list. The range uses the greatest and the least: $73 - 58 = 15$ .

Practice this lesson40 questions

Unit 9: What changes when the data change

Key points

Adding the same number to every value moves the mean and the median by that number. The range and the standard deviation stay the same.
Multiplying every value by a number multiplies the mean, the median, the range and the standard deviation by it.
One very large or very small new value pulls the mean toward it. The median barely moves.
Check each measure one at a time.

See it solved

Example

The five workers at a small company earn $40$ , $42$ , $45$ , $48$ and $50$ thousand dollars a year. A manager who earns $150$ thousand dollars a year joins the company. Which statement is true about the six salaries, compared with the first five?

AThe mean and the median increase by the same amount.
BThe mean increases by more than the median.
CThe median increases by more than the mean.
DNeither the mean nor the median changes.

Worked by hand

The plan

Evaluate each measure before the change
Evaluate each measure after the change
Compare the changes

Desmos: Reasoning is quicker: one very large value pulls the mean, not the median. Desmos is a check: put both lists in and compare.

Step 1: Evaluate each measure before the change
- Sum: $40 + 42 + 45$ $= 127$ , and $127 + 48 + 50$ $= 225$ .
- Mean: $\dfrac{225}{5} = 45$ .
- Median: the middle of five sorted values is the $3$ rd, which is $45$ .
Step 2: Evaluate each measure after the change
- Sum: $225 + 150 = 375$ . Mean: $\dfrac{375}{6} = 62.5$ .
- Median: with six values, take the mean of the $3$ rd and the $4$ th. $\dfrac{45 + 48}{2} = 46.5$ .
Step 3: Compare the changes
- Mean: $62.5 - 45 = 17.5$ more.
- Median: $46.5 - 45 = 1.5$ more.
- The mean increases by more than the median: choice B.

The correct answer choice is B: the mean rises by

17.5

and the median by only

1.5

(in thousands of dollars).

The same example in Desmos

new graph
Desmos shows: The first five salaries.
and
Desmos shows: $45$ and $45$ .
Desmos shows: The six salaries.
and
Desmos shows: $62.5$ and $46.5$ . The mean moved by $17.5$ , the median by $1.5$ .

Desmos expression list: L=[40,42,45,48,50]; operatorname{mean}(L); operatorname{median}(L); M=[40,42,45,48,50,150]; operatorname{mean}(M); operatorname{median}(M) — Type

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Your turn

The mean of $30$ quiz scores is $68$ , and the range of the scores is $40$ . Then $5$ points are added to every score. What is the range of the new scores?

Show the answerHide the answer

Answer

The correct answer is $40$ .

Step 1: Evaluate each measure
- Adding $5$ to every score moves the greatest score up by $5$ and the least score up by $5$ .
- The gap between them does not change, so the range is still $40$ .
- The mean does change: it becomes $68 + 5 = 73$ .

Traps

Assuming every measure changes together. One new value moved the mean by $17.5$ but the median by only $1.5$ . Work out each measure by itself.
Adding the $5$ to the range. Both ends move up by $5$ , so the gap stays $40$ . $45$ is a wrong answer.

Practice this lesson12 questions

Unit 10: From a sample to the population: estimates and margin of error

Key points

A random sample stands in for the whole population.
Estimate = sample proportion $\times$ population size.
The margin of error gives a range: from the estimate minus the margin to the estimate plus the margin. The true value is likely to be inside it.
A larger random sample gives a smaller margin of error.

See it solved

Example

A town has $6{,}000$ households. In a random sample of $150$ of them, $45$ households have a dog. Based on the sample, what is the best estimate of the number of households in the town that have a dog?

Worked by hand

The plan

Determine the sample proportion
Determine the estimate

Desmos: The reasoning is done by hand. Desmos only keeps the calculation on one row, as a check.

Step 1: Determine the sample proportion
- $45$ of the $150$ households in the sample have a dog.
- $\dfrac{45}{150} = 0.30$ .
Step 2: Determine the estimate
- Apply the same proportion to the whole town.
- $0.30 \cdot 6{,}000$ $= 1{,}800$ households.

The correct answer is

1{,}800

The same example in Desmos

new graph
Keys: 45/150 → *6000
Desmos shows: $1800$ , at the right of the row.

Desmos expression list: frac{45}{150}·6000 — Type

Made with Desmos

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Your turn

A poll of a random sample of voters estimates that $62\%$ of all voters support a plan, with a margin of error of $4\%$ . Based on the poll, what is the least plausible percent of all voters who support the plan?

Show the answerHide the answer

Answer

The correct answer is $58\%$ .

Step 1: Determine the plausible range
- Estimate minus margin: $62 - 4 = 58$ .
- Estimate plus margin: $62 + 4 = 66$ .
- The plausible range is from $58\%$ to $66\%$ . The least is $58\%$ .

Traps

Treating the estimate as exact. $1{,}800$ is the best estimate, not a certain count. A different random sample would give a slightly different number.
Using the margin on one side only. The range is $4$ either side of $62$ , so it runs from $58\%$ to $66\%$ . ‘From $62\%$ to $66\%$ ’ is a wrong answer.

Practice this lesson28 questions

Unit 11: What a study can conclude

Key points

Ask two questions about any study.
Was the sample chosen at random from a population? Then the results apply to that population, and only to it.
Were the treatments assigned at random? Then the study can show that the treatment caused the result.
Without random assignment, a link between two things is only an association, not a cause.
Be wary of the words ‘all’, ‘always’ and ‘proves’.

See it solved

Example

A researcher chose $100$ students at random from the students at one high school. She assigned $50$ of them at random to use a study app for a month. The other $50$ did not use it. On a test at the end of the month, the app group had a mean score of $84$ and the other group had a mean score of $76$ . Which conclusion is best supported?

AThe app causes higher test scores for all high school students.
BThe app caused higher test scores for students at this high school.
CUsing the app is linked to higher scores at this high school, but the study cannot show cause.
DNo conclusion is possible, because only $100$ students took part.

Worked by hand

The plan

Check how the sample was chosen
Check how the treatment was assigned
Check the size of the effect

Desmos: Desmos does not help here: the question is about how the study was run, not about a calculation. It can only check the subtraction.

Step 1: Check how the sample was chosen
- The $100$ students were chosen at random from one high school.
- So the results apply to the students at that school, and only to them.
- That rules out choice A, which talks about all high school students.
Step 2: Check how the treatment was assigned
- The students were assigned to the app at random.
- Random assignment lets a study show cause. That rules out choice C.
Step 3: Check the size of the effect
- $84 - 76 = 8$ points.
- A random sample of $100$ students is large enough to support a conclusion. That rules out choice D.

The correct answer choice is B: the app raised the mean score by about

8

points, for students at this high school.

The same example in Desmos

new graph
Desmos shows: $8$ : the difference between the two mean scores, as a check on the subtraction.

Made with Desmos

Open Desmos

Your turn

A survey of a random sample of $500$ adults in a city asked about walking and sleep. Of the $200$ adults who walk every day, $140$ said they sleep well. Of the $300$ who do not, $150$ said they sleep well. Which conclusion is best supported?

AWalking every day causes better sleep for adults in the city.
BWalking every day causes better sleep for all adults.
CAdults in the city who walk every day are more likely to say they sleep well, but the survey does not show cause.
DNo conclusion is possible, because the adults chose for themselves whether to walk.

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Answer

The correct answer choice is C: $70\%$ of the walkers and $50\%$ of the others sleep well. That is a link, not a cause.

Step 1: Check how the sample was chosen
- The adults were chosen at random from one city.
- So the results apply to adults in that city. That rules out choice B.
Step 2: Check how the treatment was assigned
- Nobody assigned the walking. The adults chose for themselves.
- Without random assignment, the survey cannot show cause. That rules out choice A.
Step 3: Check the size of the effect
- Walkers: $\dfrac{140}{200} = 0.70$ , which is $70\%$ .
- The others: $\dfrac{150}{300} = 0.50$ , which is $50\%$ .
- The walkers are more likely to say they sleep well. That link is a real finding, so choice D is wrong too.

Traps

Stretching the population. A sample from one school says nothing certain about other schools. ‘All high school students’ is choice A, a wrong answer.
Claiming cause without random assignment. In Your turn, the adults chose for themselves whether to walk. Something else, such as their health or their free time, could explain both the walking and the sleep.

Practice this lesson24 questions

Typing tips for Desmos

In the keys lines, → means the right-arrow key.
Typing / starts a fraction, and everything after it goes into the denominator until you press →. So 42/120*100 typed straight through becomes 42 ÷ (120 × 100). Type 42/120 → *100 instead.
Type y1 and Desmos shows y₁: a digit typed straight after a letter becomes a subscript. If you type the underscore yourself (y_1), press → after the digit: a ~ typed next lands inside the subscript (+, − and = leave it by themselves).
The ~ key is Shift + ` on most keyboards (the key left of 1). Use ~, not =: ~ asks Desmos to find the unknown letters; = only draws or defines.
A letter that already has a value, such as a slider m = 1 left over from earlier, is held fixed by a ~ fit, with no warning. Start a new graph, or delete old sliders, before each question.
Typing ^ or sqrt works the same way: press → to come back down from an exponent or out of a square root before typing the rest.

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Lesson 1, Probability and conditional probability, 40 questions
Lesson 2, Graphs of relationships, 12 questions
Lesson 3, Scatterplots and lines of best fit, 24 questions
Lesson 4, Bar graphs, histograms and dot plots, 32 questions
Lesson 5, The mean, 28 questions
Lesson 6, The median, 24 questions
Lesson 7, Box plots and quartiles, 12 questions
Lesson 8, Standard deviation and range, 40 questions
Lesson 9, What changes when the data change, 12 questions
Lesson 10, From a sample to the population: estimates and margin of error, 28 questions
Lesson 11, What a study can conclude, 24 questions

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x₁	y₁
0	4
10	24

1Unit 1: Probability and conditional probability

Key points

See it solved

Worked by hand

Step 1: Determine the probability

The same example in Desmos

Your turn

Step 1: Determine the probability

Traps

2Unit 2: Graphs of relationships

Key points

See it solved

Worked by hand

Step 1: Determine what a flat piece means

Step 2: Read the flat piece

The same example in Desmos

Your turn

Step 1: Read the two ends of the piece

Step 2: Determine the slope

Traps

3Unit 3: Scatterplots and lines of best fit

Key points

See it solved

Worked by hand

Step 1: Read the line at the input

The same example in Desmos

Your turn

Step 1: Determine the slope

Traps

4Unit 4: Bar graphs, histograms and dot plots

Key points

See it solved

Worked by hand

Step 1: Read the value of each bar

Step 2: Add the values

The same example in Desmos

Your turn

Step 1: Read the dots

Step 2: Add the counts

Traps

5Unit 5: The mean

Key points

See it solved

Worked by hand

Step 1: Set up the mean

Step 2: Determine the missing value

The same example in Desmos

Your turn

Step 1: Determine each sum

Step 2: Set up the mean

Traps

6Unit 6: The median

Key points

See it solved

Worked by hand

Step 1: Sort the numbers

Step 2: Determine the middle number

The same example in Desmos

Your turn

Step 1: Determine the middle position

Step 2: Count up to the middle position

Traps

7Unit 7: Box plots and quartiles

Key points

See it solved

Worked by hand

Step 1: Read the five numbers

Step 2: Determine the interquartile range

The same example in Desmos

Your turn

Step 1: Read the minimum and maximum of each class

Step 2: Determine each range

Traps

8Unit 8: Standard deviation and range

Key points

See it solved

Worked by hand

Step 1: Compare the means

Step 2: Compare the spreads

The same example in Desmos

Unit 1: Probability and conditional probability

Unit 2: Graphs of relationships

Unit 3: Scatterplots and lines of best fit

Unit 4: Bar graphs, histograms and dot plots

Unit 5: The mean

Unit 6: The median

Unit 7: Box plots and quartiles

Unit 8: Standard deviation and range

Unit 9: What changes when the data change

Unit 10: From a sample to the population: estimates and margin of error

Unit 11: What a study can conclude