Practice: the normal distribution

Science and society, grade 12 (3 units) Lesson 13 of 14Practice

Practice: the normal distribution

1

The weight of loaves of bread at a bakery is normally distributed, with mean 750 grams and standard deviation 10 grams.

a

What percent of the loaves weigh between 740 and 760 grams?

/ – 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 percent of the loaves weigh less than 730 grams?

/ – 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 bakery bakes 400 loaves a day. How many of them are expected to weigh more than 760 grams?

/ – 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 score of a loaf weighing 765 grams.

/ – 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 journey time to work in a city, in minutes, is normally distributed, with mean 40 and standard deviation 10.

a

What percent of the workers have a journey time in the shaded range?

/ – 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

The city has 50,000 workers. How many of them take more than an hour to get to work?

/ – 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

Omer got 81 in a physics exam and 85 in a chemistry exam.
In physics the mean is 72 and the standard deviation 6.
In chemistry the mean is 65 and the standard deviation 10.

a

Work out his standard score in physics.

/ – 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 his standard score in chemistry.

/ – 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

In which subject did Omer do better compared with the others? Explain.

4

The (upper) blood pressure of healthy adults is normally distributed, with mean 120 and standard deviation 10.

a

A doctor sends the 2.5% with the highest blood pressure for a further test. From what value?

/ – 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

The doctor advises the 16% with the highest blood pressure to eat less salt. From what value?

/ – 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 standard score of a patient’s blood pressure is $\displaystyle -0.5$ . What is her blood pressure?

/ – 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 weight of oranges from two groves is normally distributed:
in grove A – mean 200 grams, standard deviation 15 grams.
In grove B – mean 200 grams, standard deviation 30 grams.

a

What percent of the oranges in each grove weigh more than 230 grams?

Grove A

/ – 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

Grove B

/ – 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 percent of the oranges in each grove weigh less than 170 grams?

Grove A

/ – 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

Grove B

/ – 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

A supermarket chain wants oranges of an even size. Which grove should it buy from? Explain.

Formulas: science and society, grade 12 (3 units)

A quadratic model

A quadratic model – $\displaystyle a<0$ : it has a highest point, $\displaystyle a>0$ : a lowest point
$\displaystyle y=ax^{2}+bx+c$ $\displaystyle a\neq0$
The vertex’s $\displaystyle x$ – the axis of symmetry
$\displaystyle x=-\frac{b}{2a}$
The vertex’s $\displaystyle x$ – midway between two values of $\displaystyle x$ with the same $\displaystyle y$
$\displaystyle x=\frac{x_1+x_2}{2}$
The quadratic formula for $\displaystyle {ax^{2}+bx+c}={0}$ – the zeros
$\displaystyle x_{1,2}=\frac{-b\pm\sqrt{b^{2}-4ac}}{2a}$

Other models

A linear relation – the differences in the table are equal
$\displaystyle y=mx+b$
An exponential relation – the ratio of two neighbouring values in the table is fixed, $\displaystyle b$
$\displaystyle y=a\cdot b^{x}$

The normal distribution

The standard deviation of $\displaystyle n$ data – $\displaystyle \bar{x}$ their mean
$\displaystyle \sigma=\sqrt{\frac{(x_1-\bar{x})^{2}+(x_2-\bar{x})^{2}+\dots+(x_n-\bar{x})^{2}}{n}}$
In a normal distribution – $\displaystyle \mu$ the mean, $\displaystyle \sigma$ the standard deviation: about 68% of the data within one standard deviation of the mean (34% on each side)
$\displaystyle P(\mu-\sigma<x<\mu+\sigma)\approx68\%$
About 95% within two standard deviations of the mean (another 13.5% on each side)
$\displaystyle P(\mu-2\sigma<x<\mu+2\sigma)\approx95\%$
About 2.5% above $\displaystyle \mu+2\sigma$ , and about 2.5% below $\displaystyle \mu-2\sigma$
$\displaystyle P(x>\mu+2\sigma)\approx2.5\%$
How many of $\displaystyle n$ data are $\displaystyle p$ percent of them
$\displaystyle n\cdot p\%$

The standard score

The standard score of the value $\displaystyle x$ – how many standard deviations it is above the mean (or below it, when $\displaystyle z<0$ )
$\displaystyle z=\frac{x-\mu}{\sigma}$
The value whose standard score is $\displaystyle z$
$\displaystyle x=\mu+z\cdot\sigma$