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 𝑚 and velocity 𝑣 is described by the formula:
1
𝐸 = 2 𝑚𝑣 2
Using a balance and light gate apparatus, a student measured the values of 𝑚 and 𝑣.
Quantity
𝑚 / kg
𝑣/s
Measurement
1.85 ± 0.01
8.3 ± 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 ‘simple’ 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, 𝜇
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
𝑇
𝑓 = 2𝐿 √𝜇
You should perform this calculation in the two ways described in the lectures: (i) use the ‘simple’
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
𝑉
determines the resistance 𝑅 = from measurements of potential difference 𝑉 between the ends
𝐼
of the wire and the corresponding current, 𝐼. The estimated uncertainty in each 𝑉 and 𝐼
measurement is 1%. She obtains the following data:
𝑉/V
𝐼/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 Ω. [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 𝜌 = 𝑅𝐴/𝐿. 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 ̴ N( 6.5, 25), find the Z-score when X = x = 5.
(b) If X ̴ N( 𝜇 , 4.32) and P(X < 5) = 0.42, find the mean of X. (c) If X ̴ N(-14, 𝜎 2 ) and P(X > -12) = 0.25, find the standard deviation of X.
(d) If X ̴ N( 𝜇 , 𝜎 2 ), P(X < 21) = 0.29, and P(X > 24) = 0.12, find the mean and standard
deviation of X.
5.* (a) If X ̴ N( 𝜇 , 2.22) and P(X > 51) = 0.33, find the mean of X.
(b) If X ̴ N( 34, 𝜎 2 ) and P(X < 33) = 0.35, find the standard deviation of X. [2] [2] (c) If X ̴ N( 𝜇 , 𝜎 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 – Dubai campus
(r.irfan.1@bham.ac.uk)
Learning Aims
• Understand the nature of uncertainty
• State results with uncertainty
• Understand the relationship between uncertainty and standard deviation
• Standard deviation of the mean
• Propagation of uncertainty (uncertainty in derived quantities)
Random errors in measurement
• 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.
• They would make small random errors.
• This is due to the intrinsic limitation of the measuring equipment. It isn’t
possible to measure the length very precisely with a ruler.
• However, the average of all their measurements should be close to the
true value.
Random errors in measurement
• If we group their measurements and
plot a histogram, this is what we get.
• The blue line shows the normal
distribution for the errors in their
measurements. The normal
distribution is a well-known
mathematical shape.
• The normal distribution is
characterized by its mean value µ
and a measure of width called the
standard deviation (s).
Types of errors
• Systematic – these are due to faults in the measuring instrument or in the
techniques used in the experiment
• These cause readings to differ from the true value by a consistent amount each time
a measurement is made.
• Random – these are associated with unpredictable variations in the
experimental conditions, or a deficiency in defining the quantity being
measured
• 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
• Systematic errors are due to e.g. miscalibration of equipment.
• A result is said to be accurate if it is largely free from systematic error.
• A result is said to be precise if the random errors are small.
Accuracy and precision
• Random uncertainty decreases
the precision of an experiment
• Systematic uncertainty decreases
the accuracy of an experiment

Precision Vs. Accuracy

Errors and Uncertainty
• ‘Error’ is the difference between one
particular measured value and the true
value. It may be positive or negative.
• Measured values tend to be spread
around the true value (unless there is
systematic error).
• ‘Uncertainty’ is a measure of the spread
of these measured values. The measure
of uncertainly is the standard deviation.
It is never negative.
Stating Uncertainties
• The calculated population standard deviation (𝜎𝑥 ) for a set of monovariate
data {𝑥} represents the uncertainty (likely error) in a single measurement.
• Absolute uncertainties are usually reported with the result as ( 𝑥 ± 𝜎𝑥 ) unit.
• For example: 𝑚 = (9.4 ± 0.3) g
• When using scientific notion, the factor of ten multiplier should come after
the significant digits and uncertainty, e.g.
𝑚 = 9.4 ± 0.3 × 10−3 kg
Stating Uncertainties
• The uncertainty in a result may also be given as a percentage e.g.
𝜎𝑥
× 100%
𝑥
• In this case, it is stated as e.g. 𝑚 = 9.4 ± 3%
• But it is less common to state uncertainties in this way.
Comparing values with uncertainties
• An experiment to calculate g at 52o North gives 𝑔 = (9.72 ± 0.05) m𝑠 −2 .
• Does it agree with the accepted (true) value value of 9.81 m𝑠 −2 ?
• No, it doesn’t.
• The measured value is more than one standard deviation from
the true one.
• In general, two measurements of the same quantity agree if their
distributions of values (represented by their uncertainties) overlap.
Comparing values with uncertainties
• Two experimental groups measured the speed of sound under the same
conditions (sea level, 10000 kPa and 20 oC.)
• One group got 𝑣 = 341 ± 2 m𝑠 −1 ; the other got 𝑣 = (344 ± 2) m𝑠 −1
• Do these two results agree with each other?
• Yes, they do.
• The two distributions of values overlap.
Standard deviation of the mean
• The standard deviation represents the uncertainty in one measurement.
• If a stated result is a mean 𝑥ҧ , then a different (smaller) standard deviation
should be stated with it. This is the standard deviation of the mean and it is
𝜎
given by
.
𝑁
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 ; 𝜎 = 0.377 g; s.d. of mean =
= 0.15 𝑔
6
mean mass of a Digestive biscuit = 14.33 ± 0.15 g
Example 1
• Two groups both make five measurements of the speed of sound under the
same conditions. Their results are as follows:
Group 1 (𝑣 / 𝑚𝑠 −1 ):
342.1, 342.7, 342.9, 343.5, 343.5
Group 2 (𝑣 / 𝑚𝑠 −1 ):
342.8, 343.4, 343.6, 343.6, 344.0
• For each group, find the mean, standard deviation and standard deviation
of the mean. Do the two groups’ results agree?
Example 1
Group 1 :
𝑣1 = 342.94 𝑚𝑠
−1
𝜎1 = 0.53 𝑚𝑠
−1
0.53
s.d. of mean =
= 0.24 𝑚𝑠 −1
5
𝜎2 = 0.39 𝑚𝑠
−1
0.39
s.d. of mean =
= 0.17 𝑚𝑠 −1
5
Group 2:
𝑣2 = 343.48 𝑚𝑠
−1
So 𝑣1 = (342.94 ± 0.24) 𝑚𝑠 −1 and
342.94 + 0.24 = 343.18
𝑣2 = (343.48 ± 0.17) 𝑚𝑠 −1
343.48 – 0.17 = 343.31
• No, they do not agree. The two distributions don’t overlap.
Propagation of Uncertainty
• Galileo (1590) discovered that the distance s travelled by a falling
object is related to its time of fall 𝑡 by this formula:
s = 5𝑡 2
• Suppose we observe an object falling from the top of a building.
We time the fall with a stopwatch and get 5.0 ± 0.2 𝑠
• What is the height of the building?
• What is the uncertainty in the height?
Propagation of Uncertainty
• To answer this question and similar ones, we need some results from
calculus. We will use an approximate formula:
𝛿𝑓
𝑑𝑓
≈
𝛿𝑥
𝑑𝑥
Function, 𝑓
𝑓 = 𝑥2
𝑑𝑓
𝛿𝑥
so 𝛿𝑓 ≈
𝑑𝑥
𝑑𝑓
First derivative,
𝑑𝑥
2𝑥
𝑓 = 𝑥3
3𝑥 2
𝑓 = 𝑥4
4𝑥 3
Propagation of Uncertainty
•
For our problem, s = 5𝑡 2
so
𝑑𝑠
𝛿𝑠 ≈ 𝛿𝑡 = 10𝑡𝛿𝑡
𝑑𝑡
• Hence 𝛿𝑠 = 10 × 5 × 0.2 = 10 𝑚
• We write our final answer as 𝑠 = 125 ± 10 𝑚
Example 2
Find the uncertainty in the volume of a football with measured radius
𝑟 = (20 ± 2) cm.
4
𝑉 = 𝜋𝑟 3 Use the derivative table below to help you.
3
𝑑𝑉
4 2
𝛿𝑉 ≈
𝛿𝑟 = 3 × 𝜋𝑟 𝛿𝑟
𝑑𝑟
3
2
= 4𝜋𝑟 𝛿𝑟
= 4 × 3.141 × 202 × 2
= 10 051 cm3
𝑑𝑓
𝛿𝑓 ≈ 𝛿𝑥
𝑑𝑥
Function, 𝑓
𝑓 = 𝑥2
𝑑𝑓
First derivative,
𝑑𝑥
2𝑥
𝑓 = 𝑥3
3𝑥 2
𝑓 = 𝑥4
4𝑥 3
Propagation of Uncertainty
• Jim is cycling at a furious, record-breaking speed.
• A friend measures the time 𝑡 that it takes him to cycle between two fixed
points using a stopwatch. They also measure the distance using a tape.
• They also estimate the uncertainties in these quantities.
• They found 𝑡 = 4.2 ± 0.1 𝑠 and 𝑑 = 82.0 ± 0.5 𝑚
• What is Jim’s mean speed over this distance and what is the uncertainty
this mean speed?
Propagation of Uncertainty
82.0
(i) Mean speed =
= 19.52 𝑚𝑠 −1
4.2
(ii) Uncertainty in the mean speed:
The ‘approximate’ way (taught at A-level): adding percentage uncertainties
0.50
% uncertainty in 𝑑 =
× 100% = 0.61 %
82.0
0.10
% uncertainty in 𝑡 =
× 100% = 2.38 %
4.2
Total % uncertainty = 0.61 + 2.38 = 2.99%
2.99
Absolute uncertainty =
× 19.52 = 0.58 ≈ 0.6 𝑚𝑠 −1
100
Final Answer ± uncertainty: mean speed = (19.5 ± 0.6) 𝑚𝑠 −1
Propagation of Uncertainty
82.0
(i) Mean speed =
= 19.52 𝑚𝑠 −1
4.2
(ii) Uncertainty in the mean speed:
The ‘correct’ way: ‘root sum square’ (RSS) method
0.50
% uncertainty in 𝑑 =
× 100% = 0.61 %
82.0
0.10
% uncertainty in 𝑡 =
× 100% = 2.38 %
4.2
Total % uncertainty = 0.612 + 2.382 = 2.46 %
2.46
Absolute uncertainty =
× 19.52 = 0.48 ≈ 0.5 𝑚𝑠 −1
100
Final Answer ± uncertainty: mean speed = (19.5 ± 0.5) 𝑚𝑠 −1
Propagation of Uncertainty
Adding % uncertainties: v = (19.5 ± 0.6) 𝑚𝑠 −1
Root Sum Square method: v = (19.5 ± 0.5) 𝑚𝑠 −1
• The ‘approximate’ method of adding percentage uncertainties always
overestimates the total uncertainty.
• This is because actual errors add like vectors – they have a sign. The
approximate method ignores the possibility of errors partly cancelling.
Example 3
• The force on a tennis ball hit by a racket can be calculated using the
equation 𝐹 = 𝑚𝑎 . The measured mass and acceleration are (58±3) g
and (15±2) 𝑚𝑠 −2 . Use the RSS method to find the acceleration
together with its uncertainty.
3
% uncertainty in 𝑚 = × 100% = 5.17 %
58
2
% uncertainty in 𝑎 = × 100% = 13.33 %
15
Total % uncertainty = 5.172 + 13.332 = 14.30 %
14.30
Absolute uncertainty =
× 870 = 124.41 ≈ 124 𝑚𝑠 −1
100
Final Answer ± uncertainty: mean speed = (870 ± 124) 𝑚𝑠 −1
Using fractional uncertainties
• When calculating total uncertainties with the RSS method, it’s 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.
• The formula for the absolute uncertainty 𝛿 𝑎𝑏 in the product of two
quantities (𝑎 ± 𝛿𝑎) and (𝑏 ± 𝛿𝑏) can be summarized in this way:
𝛿𝑎 2 𝛿𝑏 2
𝛿 𝑎𝑏 = 𝑎𝑏 ( ) +( )
𝑎
𝑏
Using fractional uncertainties
• Let’s check this formula gives the same result for the uncertainty
in our last problem:
𝛿 58 × 15 = 58 × 15
3 2
2 2
+
= 124 N
58
15
Using fractional uncertainties
• What if the calculation involves multiplying more than two quantities?
• The RSS method still applies – just square and add all the fractional
uncertainties.
• For example:
𝛿𝑎 2 𝛿𝑏 2 𝛿𝑐 2
𝛿 𝑎𝑏𝑐 = 𝑎𝑏𝑐 ( ) +( ) +( )
𝑎
𝑏
𝑐
Formulae with division
• What if the formula involves dividing not multiplying?
• You treat this in exactly the same way – square and add all the fractional
uncertainties.
• For example:
𝛿
𝑎
𝑏
𝑎
=
𝑏
𝛿𝑎 2
𝛿𝑏 2
( ) +( )
𝑎
𝑏
𝑎𝑏
𝑎𝑏 𝛿𝑎 2
𝛿𝑏 2
𝛿𝑐 2
𝛿
=
( ) +( ) +( )
𝑐
𝑐
𝑎
𝑏
𝑐
Formulae with squares*
• What if the formula involves squared terms?
• The fractional uncertainty of the squared term must be multiplied by 2.
• For example:
𝑎𝑏
𝑎𝑏
𝛿 2 = 2
𝑐
𝑐
𝛿𝑎
𝑎
2
𝛿𝑏
+
𝑏
2
𝛿𝑐
+2
𝑐
2
* (We won’t consider more complex examples with higher or fractional powers in this course)
Formulae with addition or subtraction
• What if the formula involves adding or subtracting?
• This is even easier – in this case you square and add the absolute
uncertainties. You don’t need to use fractional uncertainties.
𝛿 𝑎±𝑏 =
𝛿𝑎 2 + 𝛿𝑏 2
Example 5
• The total resistance in a series resistor circuit is given by …
𝑅 = 𝑅1 + 𝑅2
• Find the total resistance and its uncertainty when 𝑅1 = 5.0 ± 0.2 𝑘Ω
and 𝑅2 = 25.0 ± 0.5 𝑘Ω.
𝛿 5.0 ± 25.0 =
0.2 2 + 0.5 2 = 0.54 𝑘Ω
𝑅 = (30.0 ± 0.5)𝑘Ω.
Review
• Uncertainty can be stated in the form:
𝜎𝑥
(𝑥 ± 𝜎𝑥 ) 𝑢𝑛𝑖𝑡 or 𝑥 ±
× 100 %
𝑥
• 𝜎𝑥 always has the same units as 𝑥
• If the result is a mean 𝑥ҧ then the stated uncertainty should be the
𝜎𝑥
standard deviation of the mean, i.e. (𝑥 ± ) 𝑢𝑛𝑖𝑡
𝑁
• 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), ‘How
to combine errors’ http://www.met.rdg.ac.uk/~swrhgnrj/combining_errors.pdf
• 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 – Dubai campus
(r.irfan.1@bham.ac.uk)
Learning Outcomes
• Define the normal distribution.
• Calculate probabilities using the normal cumulative
distribution (Normal CD).
• Calculate regions in the normal distribution when the
probability is known (Inverse Normal).
• Apply the normal distribution to solve problems in
inferential statistics.
2
Continuous Variables
• The binomial distribution is appropriate for modelling discrete
outcomes (numbers of successes / failures).
• How many times do we roll a 5?
• How many aces do we draw from a deck?
• But a lot of real-world data is continuous.
• A continuous variable can take any value within a domain (3, 3.1, π, …)
• A discrete variable can only take an integer value within a domain.
3
Examples of Continuous Variables
• Heights of plant stems in a growth experiment.
• Time taken to run 100 m.
• Mass of CO2 in car exhaust gases.
• Lifetimes of AA cells.
• With enough data, and when only random differences are present
between them, all of these are normally distributed.
4
Histograms
• Histograms display the distribution of continuous data in
classes.
• We can introduce a relative frequency density which is the
frequency density divided by the total frequency.
• Let’s 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.
 Class size 3 – doesn’t look like a smooth distribution yet.
Normal Distributions
6
Normal Distributions
 Class size 2 – beginning to look smoother.
7
Normal Distributions
 Class size 1 – looking smoother still.
8
Normal Distributions
 If we sample some data from this
distribution, we can think about this as
a probability density.
 The standard deviation, 𝜎, on either
side of the mean covers ~68.2% of the
data.
 Two standard deviations cover 95.4%
of the data.
 Three standard deviations cover 99.6%
of the data.
9
Normal Distributions
• Normally-distributed data are symmetric about the mean, 𝜇, so the mode and median
both equal the mean.
• A normal distribution which represents a probability density function (PDF) is
normalized – the area under the curve is 1.
10
Normal Distributions
• A normal distribution probability density function (PDF) is described by:
1 𝑥−𝜇 2
1
−
𝑒 2 𝜎
𝜎 2𝜋
• Unlike other distributions, −∞ < 𝑥 < +∞ . • Normal Distributions are very common in the real world. • 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 aren’t normally distributed. • In Lecture 10, we’ll see how the binomial distribution does this. 11 Notation and Probabilities • For a normally distributed random variable X, we write: 𝑋~𝑁(𝜇, 𝜎 2 ) • 𝜎 2 is the variance. • The probability of getting a precise value from a normal distribution is zero. P>0 only for a range of values.
• There is a Normal PD function on the fx-991EX calculator which gives
the value of the PDF at a specific value of 𝑥 – but frankly, it isn’t very
useful.
• The Normal CD (cumulative distribution) gives the probability for a
range of 𝑥 values, which is useful.
12
Example 1
• IQ (Intelligence Quotient) is a measure of an individual’s cognitive
ability. IQ is normally distributed across a population with a mean of
100 and standard deviation of 15.
• 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
• If a person is selected at random, what is the probability that their IQ
score is greater than 120?
• X~N(100,152)
• The highlighted area
tells us the probability.
With the applet it’s easy to answer
this question. Just enter 𝑥 = 120
and choose P(X > 𝑥) = from the
drop-down menu.
Example 1
• 𝑋~𝑁(100, 152 )
• Calculators perform numerical
calculations so they don’t understand
∞.
• Enter a very large number for the
upper limit (instead of ∞).
• You should get the same answer:
P X > 120 = 0.09121
Example 1 (contd.)
Try this with your
calculator!
• 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?
• 𝑋~𝑁(100, 152 )
• P X < 120 = 0.5 • 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? • 𝑋~𝑁(100, 152 ) • P 85 > 𝑋 > 115 = 0.6827
• It’s easier to use the calculator than the Applet in this case, since the Applet doesn’t 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?
• You have to do the calculation in two steps with the Applet.
• P 85 > 𝑋 > 115 = 0.8413 − 0.1587 = 0.6826
19
What if we know the probability?
• You may be asked to calculate the value of 𝑥 such that:
𝑃 𝑋 𝑥 = 𝑝?
Example 4 (With the Applet)
• What if the question asks for the value of 𝑥 such that 𝑃 𝑋 > 𝑥 = 𝑝?
• The Applet does this immediately. For example, if we want 𝑃 𝑋 > 𝑥 = 0.1 for
the same distribution, then just choose P(X > 𝑥) = from the drop-down
menu.
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Example 4 (with the fx-991EX)
• For the random variable 𝑋~𝑁(15,22 ), find the value of x which
satisfies 𝑃 𝑋 > 𝑥 = 0.1
• Use the fact that:
• 𝑃 𝑋 >𝑥 =1−𝑃 𝑋 𝑥 = 0.40
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Example 5
(i) 𝑡 ~𝑁(28, 2.52 )
With the calculator:
𝑃 𝑡 > 32 = 1 − 𝑃 𝑡 < 32 = 1 − 0.9452 = 0.0548 Or with the Applet: 27 Example 5 (ii) 𝑡 ~𝑁(28, 2.52 ) If 𝑃 𝑡 > 𝑥 = 0.40 then 𝑃 𝑡 < 𝑥 = 1 − 0.40 = 0.60 Inverse Normal function on fx-991EX with Area = 0.6 : 𝑡 =28.63 h OR use Applet with 𝑃 X > 𝑥 = 0.40
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Review
• For a normally distributed random variable X, 𝑋~𝑁 𝜇, 𝜎 2 where 𝜇 is the population mean and 𝜎 2 is
the variance. −∞ < 𝑋 < +∞ • 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 𝑋. • Use the Normal CD function to do this on a calculator. • To sum from one end of the distribution, use a very large number instead of ∞ because most calculators don’t understand ∞. • The inverse normal function returns the value of 𝑥 such that 𝑃 𝑋 < 𝑥 = 𝑝, where 𝑝 is a known constant. • Use e.g. the relationship 𝑃 𝑋 > 𝑥 = 1 − 𝑃 𝑋 < 𝑥 to find the value of 𝑥 when 𝑃 𝑋 > 𝑥 = 𝑝.
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Further Reading
• Normal distribution, see CGP Edexcel A-Level Mathematics (2021),
pp 174 -175.
• OpenStax Introductory Statistics Chapter 6. pp 365-366, pp 371-378.
Next Lecture
The Standard Normal Distribution, see e.g.
• CGP Edexcel A-Level Mathematics (2021), pp 176 -177.
• OpenStax Introductory Statistics Chapter 6, pp 366 – 371.
https://www.mathsisfun.com/data/standard-normal-distribution.html
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