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A113 Mathematics

Problem 15: Making a Strong Case

6th Presentation

• Central Limit Theorem– The Central Limit Theorem states that if the sample

size is large (≥ 30), the shape of the histogram of the sample means will resemble a “bell-shaped curve”, also known as the Normal distribution curve.

– Hence, we will be able to use the Normal distribution curve to estimate the chance that a sample mean falls within a certain range of values.

Recap

– The Normal curve is symmetrical about its mean.– The Normal curve is described by its mean and its standard

deviation (or variance).– The area under the Normal distribution curve represents the

probability of an event occurring where the total area is 1.

• Normal distribution

Recap

Mean

100%

• Manufacturer’s claim– The average volume per can is 330 ml.– This claim is commonly referred to as the null

hypothesis, H0.– The null hypothesis is presumed true unless we have

enough evidence to reject it.

• Student’s suspicion– The average volume per can is less than 330 ml.– This is commonly referred to as the alternative

hypothesis, H1.

Forming the hypotheses

Testing practically

• Hypothesis testing

– It is impossible to take all the cans that the manufacturer has produced and find their average volume to prove or disprove the manufacturer’s claim.

– A more practical way is to take a ‘sample’ and find statistical evidence to decide whether or not to reject the manufacturer’s claim.

• First, we assume that the null hypothesis is true.

• If our experiment produces a result (e.g. a sample mean) which is highly unlikely, we conclude that the null hypothesis is probably not correct. In this case, we have reason to reject the null hypothesis.

• Otherwise, we are unable to reject the null hypothesis. But this does not mean that we are 100% sure that the null hypothesis is true.

Hypothesis testing

• Significance level is the probability of making a decision to reject the null hypothesis when the null hypothesis is actually true.

• Critical region is the set of values for which we reject the null hypothesis.

• Critical value determines the boundary between a decision whether or not to reject the null hypothesis.

Common terms used

Hypothesis testing (Group A)

By the Central Limit Theorem, the shape of the histogram of the sample means will resemble a “bell-shaped curve”, which is known as Normal Distribution, as shown below.

Hypothesis testing (Group A)

xData collected from 30 cans: Average, = 325.8 ml, Standard deviation, s = 15.2 ml

x

Hypothesis testing (Group A)

Data collected from 30 cans:

– Population mean = 330 ml, estimated standard deviation = 3015.2

330

Average, = 325.8 ml, Standard deviation, s = 15.2 ml

x

Average, = 325.8 ml, Standard deviation, s = 15.2 ml

Hypothesis testing (Group A)

Data collected from 30 cans:

– Population mean = 330 ml, estimated standard deviation =

330

– Consider testing at a significance level of 5%

x

Probability = 5%

30

15.2

– Determine the critical value and critical region (Red Zone)

Critical value: 325.4

Hypothesis testing (Group A)

330

Probability = 5%

Critical value: 325.4

325.8

Since 325.8 does not lie in the critical region, it is likely to obtain the average (for 30 cans) of 325.8 ml at a significance level of 5%.

We do not have enough evidence to reject the null hypothesis.

Since 325.8 does not lie in the critical region, it is likely to obtain the average (for 30 cans) of 325.8 ml at a significance level of 5%.

We do not have enough evidence to reject the null hypothesis.

Data collected from 30 cans:

– Population mean = 330 ml, estimated standard deviation = – Consider testing at a significance level of 5%

30

15.2

– Determine the critical value and critical region (Red Zone)

Average, = 325.8 ml, Standard deviation, s = 15.2 ml

x

Hypothesis testing (Group B)

Data collected from 100 cans: Average, = 327.2 ml, Standard deviation, s = 12.4 ml

– Population mean = 330 ml, estimated standard deviation = 100

12.4

330

– Consider testing at a significance level of 5%

Probability = 5%

Critical value: 328.0

– Determine the critical value and critical region

327.2

Since 327.2 lies in the critical region, it is highly unlikely to obtain the average (for 100 cans) of 327.2 ml at a significance level of 5%.

We do have enough evidence to reject the null hypothesis.

Since 327.2 lies in the critical region, it is highly unlikely to obtain the average (for 100 cans) of 327.2 ml at a significance level of 5%.

We do have enough evidence to reject the null hypothesis.

x

Hypothesis testing (Group B)

Data collected from 100 cans: Average, = 327.2 ml, Standard deviation, s = 12.4 ml

– Population mean = 330 ml, estimated standard deviation =

330

– Consider testing at a significance level of 1%

Probability = 1%

Critical value: 327.1

– Determine the critical value and critical region

327.2

Since 327.2 does not lie in the critical region, it is likely to obtain the average (for 100 cans) of 327.2 ml at a significance level of 1%.

We do not have enough evidence to reject the null hypothesis.

Since 327.2 does not lie in the critical region, it is likely to obtain the average (for 100 cans) of 327.2 ml at a significance level of 1%.

We do not have enough evidence to reject the null hypothesis.

x

100

12.4

Hypothesis testing (Combined data)

Data collected from 130 cans: Average, = 326.9 ml, Standard deviation, s = 13.1 ml

– Population mean = 330 ml, estimated standard deviation = 130

13.1

330

– Consider testing at a significance level of 1%

Probability = 1%

Critical value: 327.3

– Determine the critical value and critical region

326.9

Since 326.9 lies in the critical region, it is highly unlikely to obtain the average (for 130 cans) of 326.9 ml at a significance level of 1%.

We do have enough evidence to reject the null hypothesis.

Since 326.9 lies in the critical region, it is highly unlikely to obtain the average (for 130 cans) of 326.9 ml at a significance level of 1%.

We do have enough evidence to reject the null hypothesis.

x

• Establish the null and alternative hypotheses• Understand that significance level and critical

region determine how confident we are of our decision

• Determine whether or not to reject the null hypothesis

• Recognise that a larger sample size is preferred over a smaller sample size in hypothesis testing

Learning points

Discussion

If you are one of the students conducting the study to check the manufacturer’s claim, what would be your ideal sample size? State any assumptions that you have made.