CONFIDENCE INTERVALSHow confident can we be in our decisions?

Unit Plan – Week 3-6 (16 lessons)

Distributions – Normal and Otherwise

The Central Limit Theorem

Confidence intervals for the mean

Confidence intervals for the difference between two means

Confidence intervals for proportions

I Can Do…

By the end of this lesson students should be able to:• Observe and understand the distribution of a total.• Observe and understand the distribution of a sample• Understand that the symbol for a sample mean is

called Mu (μ)• Complete the sample stats for two sets of data

(Mean, Med, Mode, Range, SD, …)• Use the standard deviation formula for a population

Lesson One: Means and

Distributions

Distribution of the Total

Distribution of the Total

What kind of total would you expect to get when you throw five dice?A. 16B. 5C. 30D. 22E. 17.5

Simulation

Distribution of the Total

When you throw a die: What do you expect the mean average

to be?

What would the standard deviation be?

Simulation

Distribution of the Total

The Total (T) is the sum of n independent random variables X1 to Xn. Each has the same mean μ and std. dev σ.

If… T = X1 + X2 + … + Xn

Then…E(T) = nμVAR(T) = nσ2

SD(T) =nσProof

Cessna Air: 6 Seater Flights The weights of passengers (W) in a

fully booked plane can be regarded as independent variables each with a mean of 75kg and s.d. of 5kg.

What is the expectedtotal passenger load weight?

Cessna Air: 6 Seater Flights

Passenger Weights (W): Mean (μ) = 75kg & SD (σ) = 5kg

1. Calculate the mean load weight, E(W)E(W) = nμ = 6 x 75 = 450kg

2. Calculate the s.d. of load weights, SD(W)SD(W) = nσ = 6 x 5 = 12.25kg

Distribution of the Mean

Shaking a Die

Outcomes of Shaking a Die and their Probabilities

Calculate: E(X) = 3.5 VAR(X) = 2.91 SD(X) =

1.706

x 1 2 3 4 5 6

P(X=x)

1/61/6

1/61/6

1/61/6

1 2 3 4 5 60

0.08

0.16

Shaking a Die

What if we were to experiment with tossing a die 10 times and recording the mean of these ten throws.

Throw a dice 10 times and record the mean.

Complete the table below for the sample means of your group’s ten dice throws.Sample:

Person 1

Person 2

Person 3

Person 4

x

Shaking a Die

The symbol for each of the sample means is called Mu (μ).

In general, we repeat the same experiment n times. At each trial the random variable is X, with a mean of μ and a std. dev. of σ.

Sample: 1 2 3 4

X

Distribution of the Sample Mean When several different samples are

taken from a population, the results will vary from sample to sample.

These results will have a distribution of their own.

Sample: 1 2 3 4

x

Distribution of the Sample Mean If n (the sample size) is large enough

(>30) the distribution will be approximately normal.

The distribution of the sample mean (X) has a mean of it’s own called Mu (μ) and a standard deviation of

This is also known as the standard error of the sample mean. (It gives an indication of it’s spread)

We will revisit this in another lesson.

Sample Stats and Population Parameters

Sample Statistics

Populations

Parameters

Mean = μ μ

Standard Deviation s = σ

Tiger and Woody

Tiger and Woody are two mates who are very competitive.

They have $20 riding on a game of golf.

Whoever wins, gets the money.

Standard Deviation Formulae

(1)

(2)

By the end of this lesson students should be able to:• Explain the purpose in sampling• Understand that a sample needs to be both

representative and random• Use a graphics calculator to calculate means

and standard deviations for grouped continuous data

• Compare the sample mean and SD with population means and SD.

Lesson Two: Samples and

Populations

Your Height Please…

You need to record your heights to the nearest centimetre. (Here)

We will need to keep males heights and females heights separate.

Calculate μ and σ (Population Parameters)

Take a random sample of 4 from this

Your Height Please…

Take a random sample of 4 from our population.

Calculate x and s Discuss:

Similarities and Differences Why Random? Is it Representative?

Sample Statistics

Populations

Parameters

Mean x μ

Standard Deviation s σ

Repeat three times

Fishing

We are going to model a fish population of 100.

We are going to “Catch” 30 fish from the population, record the lengths (to the nearest 5cm) then “release” them back into the stream.

Your job is to write up and compare the sample mean and sample standard deviation of the sample that you take.

Fishing

Grouped Continuous Data Tablexi fi mi

10- 12.5

15- 17.5

20- 22.5

25- 27.5

30- 32.5

35- 37.5

40- 42.5

45-50 47.5

Midpoint of group

Text Book

Pages 17-18 Sigma Text Book

By the end of this lesson students should be able to:• Use a sample proportion to calculate a C.I. in order

to estimate π given the formula:

Lesson Nine: C.I. for the Population Proportion (π)

Definition Check Point

Confidence Level

Inverse (Normal Value)

Sample Proportion

Sample Size

Population Proportion (most cases this is unknown so we would have to use p as an estimate)

Defective Cubes

We have a company that produces blue beans for commercial use.

We conduct regular quality control tests of our product.

Occasionally the odd yellow bean is produced much to our annoyance!

Defective Cubes

Collect eight (8) quality control tests. Record the proportion of yellow beans with a φ:0.0 0.1 0.2 0.3 0.4 0.5 0.6

0.7 0.8 0.9

12345678

Formulae to RememberSituation Standard

ErrorIndividual Continuous Data

Distribution of Sample Means

Distribution of Sample Proportions

Distribution of Difference of Two Means

Defective Cubes

Calculate the 95% confidence level for the population of yellow beans:

0.0 0.1 0.2 0.3 0.4 0.5 0.60.7 0.8 0.9

12345678

Your Turn…

Have a go at the questions on page 86 and 87 of the W.O.N.

or

Exercise 14.3 Page 232 Sigma Text Book

Defective Cubes

Here’s our data for the eight (8) quality control tests:

0.0 0.1 0.2 0.3 0.4 0.5 0.60.7 0.8 0.9

12345678

How would this data change if we knew that the actual population proportion of yellow beans was 0.25?

By the end of this lesson students should be able to:• Calculate the sample size needed for a given C.I.

width for a mean and a proportion.

Lesson Ten: Sample Size (n)

Starter…

A Golf ball has a bounce which is normally distributed and σ = 3.6cm. If a sample of 100 balls is tested and x = 82cm, find a 95% C.I.

Find the total width of the C.I. What sample size would be needed if

a C.I. was to have a total width of 1cm?

Starter – Qu. 1

A Golf ball has a bounce which is normally distributed and σ = 3.6cm. If a sample of 100 balls is tested and x = 82cm, find a 95% C.I.

81.2982.71

Starter – Qu. 1

Find the total width of the C.I.

81.2982.71 1.42 cm

The Margin of Error

Starter…

What sample size would be needed if a C.I. was to have a total width of 1cm?

Two Cases

There are two cases for proportion: π is given π is unknown

Two Cases – π is Given

Research has shown that 42% of households have SkyTV.

How large a sample is needed to have a 99% confidence that the sample proportion is within 4% of the true percentage?

Two Cases – π is Unknown

An opinion poll is to be conducted. What is the minimum size needed so that the margin of error is no greater than 3% for a 95% C.I.

Use π = 0.5 (gives largest C.I.)

Your Turn

Mount Albert By-election…

Your Turn

Complete the Worksheet: Calculating Sample

Sizeor

W.O.N. Pages 94-96or

Sigma Text Ex.. Page ..

By the end of this lesson students should be able to:• Calculate the difference of two means needed when

comparing two populations.

Lesson Eleven: Difference of Two Means

C.I. for the Difference between two means A frequent problem in statistics is to

determine whether two populations are: Similar, or Whether there is a significant

difference (and unlikely to be the same)

Who Drives Faster?

In a recent trial, 16 girls and 12 boys took part in a drag strip simulator to refute/confirm this statement:

“Girls can drive faster than Boys”

The results were as follows…

Who Drives Faster?

Females Males xF = 165.7 km/h

SF = 45 N = 16

xM = 160 km/h

SF = 25 N = 12

Is there or is there not a significant difference between these?

Calculate a 95% confidence level for both of these and graph it.

Who Drives Faster?

Boys

Girls

Who Drives Faster?

How do we decide whether small differences are significant or not?

We have a small unreliable sample (if we are testing the idea that the

population of girls can drive faster than

boys.)

We need a formula to test this properly…

Distribution of the Difference of Two Means We are estimating the difference

between two population means (a parameter)

μ1 – μ2

The logical statistic to use is x1 – x2

The C.I. for the Difference A confidence interval for a parameter

usually takes the form:

so…

How…

(Sample value) ± (Confidence Level) × (SD of sample value)

Who Drives Faster?

Use the above formula to calculate the difference between the sample means

Who Drives Faster?

If there is no underlying difference between the speeds that girls and boys can drive at then:

μF – μM = 0 (Zero)

So all we have to do is check whether zero is included in the 95% C.I.

Who Drives Faster?

Our Conclusion:

Since _______ lies within the confidence

interval, there is insufficient evidence to

conclude that

_________________________.

(on a drag strip simulator)

Significant Differences

It IS possible for even smaller differences to be significant if the sample is large enough.

Consider…Females Males

xF = 165.7 km/h

SF = 45 N = 20 000

xM = 163.2 km/h

SF = 25 N = 18500

Formulae to RememberSituation Standard

ErrorIndividual Continuous Data

Distribution of Sample Means

Distribution of Sample Proportions

Distribution of Difference of Two Means

Your Turn

Legal Drinking Age…