# Statistics and Data Uncertainty & The Standard Normal Distribution Worksheet

S. Howard, August 2022EPS Foundation Year University of Birmingham LF Statistics and Data

Problems Class Sheet 9

Questions that begin with * are assessed and must be submitted the following Monday at 11:00

UK time. Non-assessed problems should be completed before attending tutorial sessions, to form

the basis of a discussion. Please note that solutions to the assessed problems will not be directly

discussed in tutorials. Instead, related problems will be discussed which will help you to solve the

assessed problems.

Uncertainty

1. The kinetic energy of a projectile with mass ๐ and velocity ๐ฃ is described by the formula:

1

๐ธ = 2 ๐๐ฃ 2

Using a balance and light gate apparatus, a student measured the values of ๐ and ๐ฃ.

Quantity

๐ / kg

๐ฃ/s

Measurement

1.85 ยฑ 0.01

8.3 ยฑ 0.1

(a) Find the percentage uncertainties.

(b) Calculate the kinetic energy of the projectile together with its uncertainty, in Joules, giving the

final answer to an appropriate number of significant figures. You should perform the uncertainty

calculation in the two ways described in the lectures: (i) use the โsimpleโ method for estimating

combined uncertainties (adding percentage uncertainties) and (ii) r.m.s. combination of

percentage uncertainties. Comment on the difference between the two results.

2.* Measurements are made to determine the tension, length and mass per unit length of a string

stretched between two supports. The percentage uncertainties in these measurements are shown

below.

Quantity

Length, L

Tension, T

Mass per unit length, ๐

Uncertainty (%)

0.8

4.0

2.0

A stationary wave is formed on the string. Use this data to find the percentage uncertainty in the

calculated value of the frequency of the fundamental vibration, given by

1

๐

๐ = 2๐ฟ โ๐

You should perform this calculation in the two ways described in the lectures: (i) use the โsimpleโ

method for estimating combined uncertainties (adding percentage uncertainties) and (ii) r.m.s.

combination of percentage uncertainties. Comment on the difference between the two results. [4]

3.* A student carries out an experiment to determine the resistivity of a metal wire. She

๐

determines the resistance ๐
= from measurements of potential difference ๐ between the ends

๐ผ

of the wire and the corresponding current, ๐ผ. The estimated uncertainty in each ๐ and ๐ผ

measurement is 1%. She obtains the following data:

๐/V

๐ผ/A

1.5

0.35

2.0

0.46

2.5

0.54

3.0

0.70

3.5

0.80

4.0

0.91

(a) By either plotting a graph with Excel and observing the equation of the trendline, or using a

calculator to carry out a linear regresson, use this data to find a value for the resistance in ฮฉ. [3]

(b) Use your fitted equation from part (a) to estimate the zero error on the voltmeter in mV.

[1]

(c) Next, the student measures the length of the wire with a ruler and the diameter of the wire

using a micrometer. These two measurement are made with an uncertainty of 1%. The resistivity

is calculated using ๐ = ๐
๐ด/๐ฟ. Which measurement contributes the largest uncertainty to the

calculated value of the resisitivity? Explain your reasoning.

[2]

The Standard Normal Distribution

For the following questions, you may need to use the the Inverse Normal function on your Casio

FX-991EX calculator or an online calculator such as this one by Matt Bognar (Bognar, 2021) from

the University of Iowa, which also calculates inverse normal distributions.

https://homepage.divms.uiowa.edu/~mbognar/applets/normal.html

4. (a) If X ฬด N( 6.5, 25), find the Z-score when X = x = 5.

(b) If X ฬด N( ๐ , 4.32) and P(X < 5) = 0.42, find the mean of X.
(c) If X ฬด N(-14, ๐ 2 ) and P(X > -12) = 0.25, find the standard deviation of X.

(d) If X ฬด N( ๐ , ๐ 2 ), P(X < 21) = 0.29, and P(X > 24) = 0.12, find the mean and standard

deviation of X.

5.* (a) If X ฬด N( ๐ , 2.22) and P(X > 51) = 0.33, find the mean of X.

(b) If X ฬด N( 34, ๐ 2 ) and P(X < 33) = 0.35, find the standard deviation of X.
[2]
[2]
(c) If X ฬด N( ๐ , ๐ 2 ), P(X < 2) = 0.1, and P(X > 4) = 0.2, find the mean and standard deviation

of X.

[6]

Week 8

Foundation Year

Stats & Data

Lecture 8 Uncertainty Calculations

N. Drury (Edgbaston), S. Howard (Edgbaston), R. Irfan (Dubai)

Delivered by: Rimsha Irfan โ Dubai campus

(r.irfan.1@bham.ac.uk)

Learning Aims

โข Understand the nature of uncertainty

โข State results with uncertainty

โข Understand the relationship between uncertainty and standard deviation

โข Standard deviation of the mean

โข Propagation of uncertainty (uncertainty in derived quantities)

Random errors in measurement

โข Suppose we ask a large class of students to measure the length of the

same metal bar which is, in fact, exactly 100 mm long, using a ruler.

โข They would make small random errors.

โข This is due to the intrinsic limitation of the measuring equipment. It isnโt

possible to measure the length very precisely with a ruler.

โข However, the average of all their measurements should be close to the

true value.

Random errors in measurement

โข If we group their measurements and

plot a histogram, this is what we get.

โข The blue line shows the normal

distribution for the errors in their

measurements. The normal

distribution is a well-known

mathematical shape.

โข The normal distribution is

characterized by its mean value ยต

and a measure of width called the

standard deviation (s).

Types of errors

โข Systematic โ these are due to faults in the measuring instrument or in the

techniques used in the experiment

โข These cause readings to differ from the true value by a consistent amount each time

a measurement is made.

โข Random โ these are associated with unpredictable variations in the

experimental conditions, or a deficiency in defining the quantity being

measured

โข These cause readings to be spread about the true value, due to results varying in an

unpredictable way from one measurement to the next. Random errors are present

when any measurement is made, and cannot be corrected. The effect of random

errors can be reduced by making more measurements and calculating a new mean.

Random and systematic errors

Random errors only

True value

Random + systematic errors

โข Systematic errors are due to e.g. miscalibration of equipment.

โข A result is said to be accurate if it is largely free from systematic error.

โข A result is said to be precise if the random errors are small.

Accuracy and precision

โข Random uncertainty decreases

the precision of an experiment

โข Systematic uncertainty decreases

the accuracy of an experiment

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Errors and Uncertainty

โข โErrorโ is the difference between one

particular measured value and the true

value. It may be positive or negative.

โข Measured values tend to be spread

around the true value (unless there is

systematic error).

โข โUncertaintyโ is a measure of the spread

of these measured values. The measure

of uncertainly is the standard deviation.

It is never negative.

Stating Uncertainties

โข The calculated population standard deviation (๐๐ฅ ) for a set of monovariate

data {๐ฅ} represents the uncertainty (likely error) in a single measurement.

โข Absolute uncertainties are usually reported with the result as ( ๐ฅ ยฑ ๐๐ฅ ) unit.

โข For example: ๐ = (9.4 ยฑ 0.3) g

โข When using scientific notion, the factor of ten multiplier should come after

the significant digits and uncertainty, e.g.

๐ = 9.4 ยฑ 0.3 ร 10โ3 kg

Stating Uncertainties

โข The uncertainty in a result may also be given as a percentage e.g.

๐๐ฅ

ร 100%

๐ฅ

โข In this case, it is stated as e.g. ๐ = 9.4 ยฑ 3%

โข But it is less common to state uncertainties in this way.

Comparing values with uncertainties

โข An experiment to calculate g at 52o North gives ๐ = (9.72 ยฑ 0.05) m๐ โ2 .

โข Does it agree with the accepted (true) value value of 9.81 m๐ โ2 ?

โข No, it doesnโt.

โข The measured value is more than one standard deviation from

the true one.

โข In general, two measurements of the same quantity agree if their

distributions of values (represented by their uncertainties) overlap.

Comparing values with uncertainties

โข Two experimental groups measured the speed of sound under the same

conditions (sea level, 10000 kPa and 20 oC.)

โข One group got ๐ฃ = 341 ยฑ 2 m๐ โ1 ; the other got ๐ฃ = (344 ยฑ 2) m๐ โ1

โข Do these two results agree with each other?

โข Yes, they do.

โข The two distributions of values overlap.

Standard deviation of the mean

โข The standard deviation represents the uncertainty in one measurement.

โข If a stated result is a mean ๐ฅาง , then a different (smaller) standard deviation

should be stated with it. This is the standard deviation of the mean and it is

๐

given by

.

๐

Example:

The masses of six Digestive biscuits were measured and gave (in g):

{ 14.7, 13.8, 14.3, 14.5, 13.9, 14.8 }

0.377

mean = 14.33 g ; ๐ = 0.377 g; s.d. of mean =

= 0.15 ๐

6

mean mass of a Digestive biscuit = 14.33 ยฑ 0.15 g

Example 1

โข Two groups both make five measurements of the speed of sound under the

same conditions. Their results are as follows:

Group 1 (๐ฃ / ๐๐ โ1 ):

342.1, 342.7, 342.9, 343.5, 343.5

Group 2 (๐ฃ / ๐๐ โ1 ):

342.8, 343.4, 343.6, 343.6, 344.0

โข For each group, find the mean, standard deviation and standard deviation

of the mean. Do the two groupsโ results agree?

Example 1

Group 1 :

๐ฃ1 = 342.94 ๐๐

โ1

๐1 = 0.53 ๐๐

โ1

0.53

s.d. of mean =

= 0.24 ๐๐ โ1

5

๐2 = 0.39 ๐๐

โ1

0.39

s.d. of mean =

= 0.17 ๐๐ โ1

5

Group 2:

๐ฃ2 = 343.48 ๐๐

โ1

So ๐ฃ1 = (342.94 ยฑ 0.24) ๐๐ โ1 and

342.94 + 0.24 = 343.18

๐ฃ2 = (343.48 ยฑ 0.17) ๐๐ โ1

343.48 โ 0.17 = 343.31

โข No, they do not agree. The two distributions donโt overlap.

Propagation of Uncertainty

โข Galileo (1590) discovered that the distance s travelled by a falling

object is related to its time of fall ๐ก by this formula:

s = 5๐ก 2

โข Suppose we observe an object falling from the top of a building.

We time the fall with a stopwatch and get 5.0 ยฑ 0.2 ๐

โข What is the height of the building?

โข What is the uncertainty in the height?

Propagation of Uncertainty

โข To answer this question and similar ones, we need some results from

calculus. We will use an approximate formula:

๐ฟ๐

๐๐

โ

๐ฟ๐ฅ

๐๐ฅ

Function, ๐

๐ = ๐ฅ2

๐๐

๐ฟ๐ฅ

so ๐ฟ๐ โ

๐๐ฅ

๐๐

First derivative,

๐๐ฅ

2๐ฅ

๐ = ๐ฅ3

3๐ฅ 2

๐ = ๐ฅ4

4๐ฅ 3

Propagation of Uncertainty

โข

For our problem, s = 5๐ก 2

so

๐๐

๐ฟ๐ โ ๐ฟ๐ก = 10๐ก๐ฟ๐ก

๐๐ก

โข Hence ๐ฟ๐ = 10 ร 5 ร 0.2 = 10 ๐

โข We write our final answer as ๐ = 125 ยฑ 10 ๐

Example 2

Find the uncertainty in the volume of a football with measured radius

๐ = (20 ยฑ 2) cm.

4

๐ = ๐๐ 3 Use the derivative table below to help you.

3

๐๐

4 2

๐ฟ๐ โ

๐ฟ๐ = 3 ร ๐๐ ๐ฟ๐

๐๐

3

2

= 4๐๐ ๐ฟ๐

= 4 ร 3.141 ร 202 ร 2

= 10 051 cm3

๐๐

๐ฟ๐ โ ๐ฟ๐ฅ

๐๐ฅ

Function, ๐

๐ = ๐ฅ2

๐๐

First derivative,

๐๐ฅ

2๐ฅ

๐ = ๐ฅ3

3๐ฅ 2

๐ = ๐ฅ4

4๐ฅ 3

Propagation of Uncertainty

โข Jim is cycling at a furious, record-breaking speed.

โข A friend measures the time ๐ก that it takes him to cycle between two fixed

points using a stopwatch. They also measure the distance using a tape.

โข They also estimate the uncertainties in these quantities.

โข They found ๐ก = 4.2 ยฑ 0.1 ๐ and ๐ = 82.0 ยฑ 0.5 ๐

โข What is Jimโs mean speed over this distance and what is the uncertainty

this mean speed?

Propagation of Uncertainty

82.0

(i) Mean speed =

= 19.52 ๐๐ โ1

4.2

(ii) Uncertainty in the mean speed:

The โapproximateโ way (taught at A-level): adding percentage uncertainties

0.50

% uncertainty in ๐ =

ร 100% = 0.61 %

82.0

0.10

% uncertainty in ๐ก =

ร 100% = 2.38 %

4.2

Total % uncertainty = 0.61 + 2.38 = 2.99%

2.99

Absolute uncertainty =

ร 19.52 = 0.58 โ 0.6 ๐๐ โ1

100

Final Answer ยฑ uncertainty: mean speed = (19.5 ยฑ 0.6) ๐๐ โ1

Propagation of Uncertainty

82.0

(i) Mean speed =

= 19.52 ๐๐ โ1

4.2

(ii) Uncertainty in the mean speed:

The โcorrectโ way: โroot sum squareโ (RSS) method

0.50

% uncertainty in ๐ =

ร 100% = 0.61 %

82.0

0.10

% uncertainty in ๐ก =

ร 100% = 2.38 %

4.2

Total % uncertainty = 0.612 + 2.382 = 2.46 %

2.46

Absolute uncertainty =

ร 19.52 = 0.48 โ 0.5 ๐๐ โ1

100

Final Answer ยฑ uncertainty: mean speed = (19.5 ยฑ 0.5) ๐๐ โ1

Propagation of Uncertainty

Adding % uncertainties: v = (19.5 ยฑ 0.6) ๐๐ โ1

Root Sum Square method: v = (19.5 ยฑ 0.5) ๐๐ โ1

โข The โapproximateโ method of adding percentage uncertainties always

overestimates the total uncertainty.

โข This is because actual errors add like vectors โ they have a sign. The

approximate method ignores the possibility of errors partly cancelling.

Example 3

โข The force on a tennis ball hit by a racket can be calculated using the

equation ๐น = ๐๐ . The measured mass and acceleration are (58ยฑ3) g

and (15ยฑ2) ๐๐ โ2 . Use the RSS method to find the acceleration

together with its uncertainty.

3

% uncertainty in ๐ = ร 100% = 5.17 %

58

2

% uncertainty in ๐ = ร 100% = 13.33 %

15

Total % uncertainty = 5.172 + 13.332 = 14.30 %

14.30

Absolute uncertainty =

ร 870 = 124.41 โ 124 ๐๐ โ1

100

Final Answer ยฑ uncertainty: mean speed = (870 ยฑ 124) ๐๐ โ1

Using fractional uncertainties

โข When calculating total uncertainties with the RSS method, itโs more usual

to use fractional uncertainties rather than the percentage ones, since it

requires fewer steps. We can also combine the result into a single formula.

โข The formula for the absolute uncertainty ๐ฟ ๐๐ in the product of two

quantities (๐ ยฑ ๐ฟ๐) and (๐ ยฑ ๐ฟ๐) can be summarized in this way:

๐ฟ๐ 2 ๐ฟ๐ 2

๐ฟ ๐๐ = ๐๐ ( ) +( )

๐

๐

Using fractional uncertainties

โข Letโs check this formula gives the same result for the uncertainty

in our last problem:

๐ฟ 58 ร 15 = 58 ร 15

3 2

2 2

+

= 124 N

58

15

Using fractional uncertainties

โข What if the calculation involves multiplying more than two quantities?

โข The RSS method still applies โ just square and add all the fractional

uncertainties.

โข For example:

๐ฟ๐ 2 ๐ฟ๐ 2 ๐ฟ๐ 2

๐ฟ ๐๐๐ = ๐๐๐ ( ) +( ) +( )

๐

๐

๐

Formulae with division

โข What if the formula involves dividing not multiplying?

โข You treat this in exactly the same way – square and add all the fractional

uncertainties.

โข For example:

๐ฟ

๐

๐

๐

=

๐

๐ฟ๐ 2

๐ฟ๐ 2

( ) +( )

๐

๐

๐๐

๐๐ ๐ฟ๐ 2

๐ฟ๐ 2

๐ฟ๐ 2

๐ฟ

=

( ) +( ) +( )

๐

๐

๐

๐

๐

Formulae with squares*

โข What if the formula involves squared terms?

โข The fractional uncertainty of the squared term must be multiplied by 2.

โข For example:

๐๐

๐๐

๐ฟ 2 = 2

๐

๐

๐ฟ๐

๐

2

๐ฟ๐

+

๐

2

๐ฟ๐

+2

๐

2

* (We wonโt consider more complex examples with higher or fractional powers in this course)

Formulae with addition or subtraction

โข What if the formula involves adding or subtracting?

โข This is even easier โ in this case you square and add the absolute

uncertainties. You donโt need to use fractional uncertainties.

๐ฟ ๐ยฑ๐ =

๐ฟ๐ 2 + ๐ฟ๐ 2

Example 5

โข The total resistance in a series resistor circuit is given by โฆ

๐
= ๐
1 + ๐
2

โข Find the total resistance and its uncertainty when ๐
1 = 5.0 ยฑ 0.2 ๐ฮฉ

and ๐
2 = 25.0 ยฑ 0.5 ๐ฮฉ.

๐ฟ 5.0 ยฑ 25.0 =

0.2 2 + 0.5 2 = 0.54 ๐ฮฉ

๐
= (30.0 ยฑ 0.5)๐ฮฉ.

Review

โข Uncertainty can be stated in the form:

๐๐ฅ

(๐ฅ ยฑ ๐๐ฅ ) ๐ข๐๐๐ก or ๐ฅ ยฑ

ร 100 %

๐ฅ

โข ๐๐ฅ always has the same units as ๐ฅ

โข If the result is a mean ๐ฅาง then the stated uncertainty should be the

๐๐ฅ

standard deviation of the mean, i.e. (๐ฅ ยฑ ) ๐ข๐๐๐ก

๐

โข When comparing two measurements of the same quantity with their own

uncertainties, look for an overlap in the distributions. Overlap means that

they agree.

Uncertainty propagation formulae

Further reading

Note that this is one part of the course where we deviate significantly from how

this topic is treated at U.K. A-Level.

For a very short introduction, see R. Hogan, University of Reading (2006), โHow

to combine errorsโ http://www.met.rdg.ac.uk/~swrhgnrj/combining_errors.pdf

โข Next Lecture: Applying theory to practical situations and practice questions

We have finished the content for the data handling part of the course. In the

remaining Tuesday lectures, we will be revising and attempting lots of exam-style

practice problems concerning data handling. Please look on Canvas at

Stats & Data Sample Exam Questions.pdf Questions 1-5

Week 8

Foundation Year

Stats & Data

Lecture 8 Probability

The Normal Distribution

A. Tomlinson (Edgbaston), S. Howard (Edgbaston), R. Irfan (Dubai)

Delivered by: Rimsha Irfan โ Dubai campus

(r.irfan.1@bham.ac.uk)

Learning Outcomes

โข Define the normal distribution.

โข Calculate probabilities using the normal cumulative

distribution (Normal CD).

โข Calculate regions in the normal distribution when the

probability is known (Inverse Normal).

โข Apply the normal distribution to solve problems in

inferential statistics.

2

Continuous Variables

โข The binomial distribution is appropriate for modelling discrete

outcomes (numbers of successes / failures).

โข How many times do we roll a 5?

โข How many aces do we draw from a deck?

โข But a lot of real-world data is continuous.

โข A continuous variable can take any value within a domain (3, 3.1, ฯ, โฆ)

โข A discrete variable can only take an integer value within a domain.

3

Examples of Continuous Variables

โข Heights of plant stems in a growth experiment.

โข Time taken to run 100 m.

โข Mass of CO2 in car exhaust gases.

โข Lifetimes of AA cells.

โข With enough data, and when only random differences are present

between them, all of these are normally distributed.

4

Histograms

โข Histograms display the distribution of continuous data in

classes.

โข We can introduce a relative frequency density which is the

frequency density divided by the total frequency.

โข Letโs take some (theoretically-generated) data from a plant

stem growth experiment.

5

If we decrease the bin size,

then it starts to look like a

smooth curve can be plotted

over the distribution.

๏ฏ Class size 3 โ doesnโt look like a smooth distribution yet.

Normal Distributions

6

Normal Distributions

๏ฏ Class size 2 โ beginning to look smoother.

7

Normal Distributions

๏ฏ Class size 1 โ looking smoother still.

8

Normal Distributions

๏ฏ If we sample some data from this

distribution, we can think about this as

a probability density.

๏ฏ The standard deviation, ๐, on either

side of the mean covers ~68.2% of the

data.

๏ฏ Two standard deviations cover 95.4%

of the data.

๏ฏ Three standard deviations cover 99.6%

of the data.

9

Normal Distributions

โข Normally-distributed data are symmetric about the mean, ๐, so the mode and median

both equal the mean.

โข A normal distribution which represents a probability density function (PDF) is

normalized โ the area under the curve is 1.

10

Normal Distributions

โข A normal distribution probability density function (PDF) is described by:

1 ๐ฅโ๐ 2

1

โ

๐ 2 ๐

๐ 2๐

โข Unlike other distributions, โโ < ๐ฅ < +โ .
โข Normal Distributions are very common in the real world.
โข The Central Limit Theorem (beyond this course) tells us that, if we
sample lots of data, they eventually start to approximate a normal
distribution even if they arenโt normally distributed.
โข In Lecture 10, weโll see how the binomial distribution does this.
11
Notation and Probabilities
โข For a normally distributed random variable X, we write:
๐~๐(๐, ๐ 2 )
โข ๐ 2 is the variance.
โข The probability of getting a precise value from a normal distribution is
zero. P>0 only for a range of values.

โข There is a Normal PD function on the fx-991EX calculator which gives

the value of the PDF at a specific value of ๐ฅ โ but frankly, it isnโt very

useful.

โข The Normal CD (cumulative distribution) gives the probability for a

range of ๐ฅ values, which is useful.

12

Example 1

โข IQ (Intelligence Quotient) is a measure of an individualโs cognitive

ability. IQ is normally distributed across a population with a mean of

100 and standard deviation of 15.

โข If a person is selected at random, what is the probability that their IQ

score is greater than 120?

You can plot this e.g. using the University of Iowa

applet; see the Normal distribution tool link on

Canvas:

Example 1

โข If a person is selected at random, what is the probability that their IQ

score is greater than 120?

โข X~N(100,152)

โข The highlighted area

tells us the probability.

With the applet itโs easy to answer

this question. Just enter ๐ฅ = 120

and choose P(X > ๐ฅ) = from the

drop-down menu.

Example 1

โข ๐~๐(100, 152 )

โข Calculators perform numerical

calculations so they donโt understand

โ.

โข Enter a very large number for the

upper limit (instead of โ).

โข You should get the same answer:

P X > 120 = 0.09121

Example 1 (contd.)

Try this with your

calculator!

โข Try these two questions. Use your calculator.

a) If a person is selected at random, what is the probability

that their IQ score is less than 100?

b) If a person is selected at random, what is the probability

that their IQ score is between 85 and 115?

16

Example 1 (contd.)

(a) If a person is selected at random, what is the probability that their IQ score is

less than 100?

โข ๐~๐(100, 152 )

โข P X < 120 = 0.5
โข Since the distribution is symmetric about the mean, we should expect this value.
17
Example 1 (contd.)
(b) If a person is selected at random, what is the
probability that their IQ score is between 85 and 115?
โข ๐~๐(100, 152 )
โข P 85 > ๐ > 115 = 0.6827

โข Itโs easier to use the calculator than the Applet in this case, since the Applet doesnโt allow

you to enter arbitrary lower and upper limits.

18

Example 1 (contd.)

(b) If a person is selected at random, what is the

probability that their IQ score is between 85 and 115?

โข You have to do the calculation in two steps with the Applet.

โข P 85 > ๐ > 115 = 0.8413 โ 0.1587 = 0.6826

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What if we know the probability?

โข You may be asked to calculate the value of ๐ฅ such that:

๐ ๐ ๐ฅ = ๐?

Example 4 (With the Applet)

โข What if the question asks for the value of ๐ฅ such that ๐ ๐ > ๐ฅ = ๐?

โข The Applet does this immediately. For example, if we want ๐ ๐ > ๐ฅ = 0.1 for

the same distribution, then just choose P(X > ๐ฅ) = from the drop-down

menu.

24

Example 4 (with the fx-991EX)

โข For the random variable ๐~๐(15,22 ), find the value of x which

satisfies ๐ ๐ > ๐ฅ = 0.1

โข Use the fact that:

โข ๐ ๐ >๐ฅ =1โ๐ ๐ ๐ฅ = 0.40

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Example 5

(i) ๐ก ~๐(28, 2.52 )

With the calculator:

๐ ๐ก > 32 = 1 โ ๐ ๐ก < 32 = 1 โ 0.9452 = 0.0548
Or with the Applet:
27
Example 5
(ii) ๐ก ~๐(28, 2.52 )
If ๐ ๐ก > ๐ฅ = 0.40 then ๐ ๐ก < ๐ฅ = 1 โ 0.40 = 0.60
Inverse Normal function on fx-991EX with Area = 0.6 : ๐ก =28.63 h
OR use Applet with ๐ X > ๐ฅ = 0.40

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Review

โข For a normally distributed random variable X, ๐~๐ ๐, ๐ 2 where ๐ is the population mean and ๐ 2 is

the variance. โโ < ๐ < +โ
โข A normal distribution is a probability density function (PDF). This means that probabilities are
calculated cumulatively as areas under the curve for given ranges of ๐.
โข Use the Normal CD function to do this on a calculator.
โข To sum from one end of the distribution, use a very large number instead of โ because most calculators
donโt understand โ.
โข The inverse normal function returns the value of ๐ฅ such that ๐ ๐ < ๐ฅ = ๐, where ๐ is a known
constant.
โข Use e.g. the relationship ๐ ๐ > ๐ฅ = 1 โ ๐ ๐ < ๐ฅ to find the value of ๐ฅ when ๐ ๐ > ๐ฅ = ๐.

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Further Reading

โข Normal distribution, see CGP Edexcel A-Level Mathematics (2021),

pp 174 -175.

โข OpenStax Introductory Statistics Chapter 6. pp 365-366, pp 371-378.

Next Lecture

The Standard Normal Distribution, see e.g.

โข CGP Edexcel A-Level Mathematics (2021), pp 176 -177.

โข OpenStax Introductory Statistics Chapter 6, pp 366 โ 371.

https://www.mathsisfun.com/data/standard-normal-distribution.html

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