A113 Mathematics<br />Problem 15: Making a Strong Case6th Presentation<br />
Central Limit Theorem<br />The Central Limit Theorem states that if the sample size is large (≥ 30),the shape of the histo...
Recap<br />Normal distribution<br />100%<br /><ul><li>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.</...
Forming the hypotheses<br />Manufacturer’s claim<br />The average volume per can is 330 ml.<br />This claim is commonly re...
The average volume per can is less than 330 ml.
This is commonly referred to as the alternative hypothesis, H1.</li></li></ul><li>Testing practically<br /><ul><li>Hypothe...
It is impossible to take all the cans that the manufacturer has produced and find their average volume to prove or disprov...
A more practical way is to take a ‘sample’ and find statistical evidence to decide whether or not to reject the manufactur...
Common terms used<br />Significance level is the probability of making a decision to reject the null hypothesis when the n...
Hypothesis testing  (Group A)<br />By the Central Limit Theorem, the shape of the histogram of the sample means will resem...
Hypothesis testing  (Group A)<br />Average,    = 325.8 ml, Standard deviation, s = 15.2 ml<br />Data collected from 30 can...
Hypothesis testing  (Group A)<br />Average,    = 325.8 ml, Standard deviation, s = 15.2 ml<br />Data collected from 30 can...
Hypothesis testing  (Group A)<br />Average,    = 325.8 ml, Standard deviation, s = 15.2 ml<br />Data collected from 30 can...
Consider testing at a significance level of 5%
Determine the critical value and critical region (Red Zone)</li></ul>Probability = 5%<br />330<br />Critical value: 325.4<...
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A113 6 P P15 Making A Strong Case

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A113 6 P P15 Making A Strong Case

  1. 1. A113 Mathematics<br />Problem 15: Making a Strong Case6th Presentation<br />
  2. 2. Central Limit Theorem<br />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.<br />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.<br />Recap<br />
  3. 3. Recap<br />Normal distribution<br />100%<br /><ul><li>The Normal curve is symmetrical about its mean.
  4. 4. The Normal curve is described by its mean and its standard deviation (or variance).
  5. 5. The area under the Normal distribution curve represents the probability of an event occurring where the total area is 1.</li></ul>Mean<br />
  6. 6. Forming the hypotheses<br />Manufacturer’s claim<br />The average volume per can is 330 ml.<br />This claim is commonly referred to as the null hypothesis, H0.<br />The null hypothesis is presumed true unless we have enough evidence to reject it. <br /><ul><li>Student’s suspicion
  7. 7. The average volume per can is less than 330 ml.
  8. 8. This is commonly referred to as the alternative hypothesis, H1.</li></li></ul><li>Testing practically<br /><ul><li>Hypothesis testing
  9. 9. 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.
  10. 10. A more practical way is to take a ‘sample’ and find statistical evidence to decide whether or not to reject the manufacturer’s claim.</li></li></ul><li>Hypothesis testing<br />First, we assume that the null hypothesis is true.<br />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.<br />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.<br />
  11. 11. Common terms used<br />Significance level is the probability of making a decision to reject the null hypothesis when the null hypothesis is actually true.<br />Critical region is the set of values for which we reject the null hypothesis. <br />Critical value determines the boundary between a decision whether or not to reject the null hypothesis. <br />
  12. 12. Hypothesis testing (Group A)<br />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.<br />
  13. 13. Hypothesis testing (Group A)<br />Average, = 325.8 ml, Standard deviation, s = 15.2 ml<br />Data collected from 30 cans:<br />
  14. 14. Hypothesis testing (Group A)<br />Average, = 325.8 ml, Standard deviation, s = 15.2 ml<br />Data collected from 30 cans:<br /><ul><li>Population mean = 330 ml, estimated standard deviation = </li></ul>330<br />
  15. 15. Hypothesis testing (Group A)<br />Average, = 325.8 ml, Standard deviation, s = 15.2 ml<br />Data collected from 30 cans:<br /><ul><li>Population mean = 330 ml, estimated standard deviation =
  16. 16. Consider testing at a significance level of 5%
  17. 17. Determine the critical value and critical region (Red Zone)</li></ul>Probability = 5%<br />330<br />Critical value: 325.4<br />
  18. 18. Hypothesis testing (Group A)<br />Average, = 325.8 ml, Standard deviation, s = 15.2 ml<br />Data collected from 30 cans:<br /><ul><li>Population mean = 330 ml, estimated standard deviation =
  19. 19. Consider testing at a significance level of 5%
  20. 20. Determine the critical value and critical region (Red Zone)</li></ul>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%. <br />We do not have enough evidence to reject the null hypothesis.<br />Probability = 5%<br />325.8<br />330<br />Critical value: 325.4<br />
  21. 21. Hypothesis testing (Group B)<br />Average, = 327.2 ml, Standard deviation, s = 12.4 ml<br />Data collected from 100 cans:<br /><ul><li>Population mean = 330 ml, estimated standard deviation =
  22. 22. Consider testing at a significance level of 5%
  23. 23. Determine the critical value and critical region</li></ul>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%. <br />We do have enough evidence to reject the null hypothesis.<br />Probability = 5%<br />327.2<br />330<br />Critical value: 328.0<br />
  24. 24. Hypothesis testing (Group B)<br />Average, = 327.2 ml, Standard deviation, s = 12.4 ml<br />Data collected from 100 cans:<br /><ul><li>Population mean = 330 ml, estimated standard deviation =
  25. 25. Consider testing at a significance level of 1%
  26. 26. Determine the critical value and critical region</li></ul>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%. <br />We do not have enough evidence to reject the null hypothesis.<br />Probability = 1%<br />327.2<br />330<br />Critical value: 327.1<br />
  27. 27. Hypothesis testing (Combined data)<br />Average, = 326.9 ml, Standard deviation, s = 13.1 ml<br />Data collected from 130 cans:<br /><ul><li>Population mean = 330 ml, estimated standard deviation =
  28. 28. Consider testing at a significance level of 1%
  29. 29. Determine the critical value and critical region</li></ul>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%. <br />We do have enough evidence to reject the null hypothesis.<br />Probability = 1%<br />326.9<br />330<br />Critical value: 327.3<br />
  30. 30. Establish the null and alternative hypotheses<br />Understand that significance level and critical region determine how confident we are of our decision<br />Determine whether or not to reject the null hypothesis<br />Recognise that a larger sample size is preferred over a smaller sample size in hypothesis testing<br />Learning points<br />
  31. 31. Discussion<br />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.<br />

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