Practice: processing data

Statistics and probability, grade 10 (4 units) Lesson 13 of 20Practice

Practice: processing data

1

25 students were asked how many books they read in the holiday:

a

Work out the mean.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

Find the median.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

Find the mode.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

d

Work out the standard deviation.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

2

The final mark in a course is a weighted mean:
the tests during the year – 40% of the mark;
the project – 25%;
the end-of-year exam – 35%.
Dana got 80 in the tests, 92 in the project and 70 in the exam.

What is Dana’s final mark?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

3

The mean hourly pay of six temporary workers is 50 ₪.
The hourly pay of five of them: 50 ₪, 48 ₪, 53 ₪, 52 ₪, 45 ₪.

a

What is the sixth worker’s hourly pay?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

What is the median of the six workers’ hourly pay?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

Two new workers joined, whose hourly pay is 60 ₪ and 56 ₪. What is the new mean of the eight workers?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

4

The histogram shows how many minutes it took 30 students to solve a puzzle:

a

Which is the modal class?

b

Work out the estimated mean.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

Work out the estimated standard deviation, rounded to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

5

The number of hours 20 teenagers slept the night before a test, in order from least to greatest:
3, 4, 5, 5, 6, 6, 7, 7, 7, 8, 8, 8, 8, 9, 9, 10, 10, 11, 12, 15.

a

Find the lower decile.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

Find the upper decile.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

Find the lower quartile $\displaystyle Q_1$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

d

Find the upper quartile $\displaystyle Q_3$ .

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

6

Two runners ran 400 metres five times. Their times, in seconds:
Yoav: 62, 63, 63, 64, 63.
Ido: 61, 65, 62, 64, 63.
Both have a mean time of 63 seconds.

a

Work out the standard deviation of Yoav’s times, rounded to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

Work out the standard deviation of Ido’s times, rounded to two decimal places.

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

c

The coach needs a runner whose time can be relied on. Whom should he choose? Explain.

7

A firm has eight temporary workers.
Their mean hourly pay is 52 ₪, and the standard deviation 4.2 ₪.
In April each of them got a rise of 4 ₪ an hour.
In August each one’s new pay went up by 10%.

a

What is the mean hourly pay after August?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

b

What is the standard deviation after August?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

8

A year group has two classes:
One class has 30 students, and their mean mark in a test is 72.
The other class has 20 students, and their mean mark is 82.

What is the mean mark of all the students in the year group?

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – check and a new line · * – times · ^ – power, ↓ – end of the power

/ – fraction bar · Tab or the arrows – between numerator and denominator, then a note above or below · Enter – a new line · * – times · ^ – power, ↓ – end of the power

9

Write true or false. For “true” explain why; for “false” give a counterexample.

The mean has to be one of the values.
The mean lies between the least value (included) and the greatest value (included).
The median can equal the mean.
Adding a value equal to 0 to the data does not change the mean.
When 10 is added to all the data, the standard deviation grows by 10.

Formulas: statistics and probability, grade 10 (4 units)

Presenting data

Relative frequency – $\displaystyle f$ a value’s frequency, $\displaystyle n$ the number of all the data
$\displaystyle \frac{f}{n}$
Relative frequency in percent
$\displaystyle \frac{f}{n}\cdot100\%$
The angle of a slice in a pie chart
$\displaystyle \frac{f}{n}\cdot360^{\circ}$

Measures of centre and position

The mean of $\displaystyle n$ data
$\displaystyle \bar{x}=\frac{x_1+x_2+\dots+x_n}{n}$
The mean from a frequency table – $\displaystyle x_1, \dots, x_k$ the values, $\displaystyle f_1, \dots, f_k$ their frequencies
$\displaystyle \bar{x}=\frac{x_1f_1+x_2f_2+\dots+x_kf_k}{f_1+f_2+\dots+f_k}$
A weighted mean – $\displaystyle w_1, \dots, w_k$ the weights of the values $\displaystyle x_1, \dots, x_k$
$\displaystyle \bar{x}=\frac{w_1x_1+w_2x_2+\dots+w_kx_k}{w_1+w_2+\dots+w_k}$
The mean of data in classes – $\displaystyle m_1, \dots, m_k$ the classes’ middles, $\displaystyle f_1, \dots, f_k$ their frequencies
$\displaystyle \bar{x}\approx\frac{m_1f_1+m_2f_2+\dots+m_kf_k}{f_1+f_2+\dots+f_k}$
The median’s place in an ordered list of $\displaystyle n$ data
$\displaystyle \frac{n+1}{2}$
The lower decile’s place in an ordered list; the upper decile’s – $\displaystyle 0.9\cdot n$
$\displaystyle 0.1\cdot n$

Measures of spread

The range – the greatest value minus the least
$\displaystyle x_{max}-x_{min}$
The variance – the mean of the squared deviations from the mean
$\displaystyle \sigma^2=\frac{(x_1-\bar{x})^2+(x_2-\bar{x})^2+\dots+(x_n-\bar{x})^2}{n}$
The standard deviation – the square root of the variance
$\displaystyle \sigma=\sqrt{\sigma^2}$
The variance from a frequency table
$\displaystyle \sigma^2=\frac{(x_1-\bar{x})^2f_1+\dots+(x_k-\bar{x})^2f_k}{f_1+\dots+f_k}$

The rules of probability

Equally likely outcomes – $\displaystyle k$ the number of outcomes in the event $\displaystyle A$ , $\displaystyle n$ the number of outcomes in the sample space
$\displaystyle P(A)=\frac{k}{n}$
The probability of any event; $\displaystyle {P(\emptyset)}={0}$ , and $\displaystyle {P(U)}={1}$ for the sample space $\displaystyle U$
$\displaystyle 0\le P(A)\le1$
The complement $\displaystyle \overline{A}$ – $\displaystyle A$ does not happen
$\displaystyle P(\overline{A})$ $\displaystyle =$ $\displaystyle 1-P(A)$
The union – $\displaystyle A$ or $\displaystyle B$ (or both); $\displaystyle A\cap B$ – the intersection, both $\displaystyle A$ and $\displaystyle B$
$\displaystyle P(A\cup B)$ $\displaystyle =$ $\displaystyle P(A)+P(B)-P(A\cap B)$
The union of disjoint events
$\displaystyle P(A\cup B)$ $\displaystyle =$ $\displaystyle P(A)+P(B)$ $\displaystyle A\cap B=\emptyset$

Conditional probability and independence

The probability of $\displaystyle A$ given $\displaystyle B$
$\displaystyle P(A|B)=\frac{P(A\cap B)}{P(B)}$
A path in a tree – the product of the probabilities on its branches
$\displaystyle P(A\cap B)$ $\displaystyle =$ $\displaystyle P(B)\cdot P(A|B)$
The law of total probability – a weighted mean of the conditional probabilities
$\displaystyle P(A)=P(B)\cdot P(A|B)+P(\overline{B})\cdot P(A|\overline{B})$
Bayes’ formula – from $\displaystyle P(B|A)$ to $\displaystyle P(A|B)$
$\displaystyle P(A|B)\cdot P(B)$ $\displaystyle =$ $\displaystyle P(B|A)\cdot P(A)$
Independent events
$\displaystyle P(A\cap B)$ $\displaystyle =$ $\displaystyle P(A)\cdot P(B)$
Independent events – knowing that $\displaystyle B$ happened does not change the probability of $\displaystyle A$
$\displaystyle P(A|B)$ $\displaystyle =$ $\displaystyle P(A)$