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- Recognize the terms sample statistic and population parameter

- Use confidence intervals to indicate the reliability of estimates

- Know when approximate large sample or exact confidence intervals are appropriate

This topic will cover:

- Sampling distributions

- Point estimates and confidence intervals

- Introduction to hypothesis testing

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- 1. Topic 3: Inferential Statistics 1
- 2. This topic will cover: ◦ Sampling distributions ◦ Point estimates and confidence intervals ◦ Introduction to hypothesis testing
- 3. By the end of this topic students will be able to: ◦ Recognise the terms sample statistic and population parameter ◦ Use confidence intervals to indicate the reliability of estimates ◦ Know when approximate large sample or exact confidence intervals are appropriate
- 4. P(Z < 1) = 0.8413 P( - 1 <Z < 1) = 0.6826 z = 1.0 P(Z > 1) = 0.1587 m P( 0 <Z < 1) = 0.3413 m + s x 𝑧 = 𝑥 − 𝜇 𝜎 0.0 1.0 -1.0 1.0
- 5. P(Z < 1) = 0.8413 z = 1.0m m + s x P( - 1 <Z < 1) = 0.6826 P( 0 <Z < 1) = 0.3413 0.0 1.0 -1.0 1.0 z 0 1 2 3 0.8 0.788 1 0.791 0 0.793 9 0.796 7 0.9 0.815 9 0.818 6 0.821 2 0.823 8 1.0 0.841 3 0.843 8 0.846 1 0.848 5 1.1 0.864 3 0.866 5 0.868 6 0.870 8 z 0 1 2 3 0.8 0.288 1 0.291 0 0.293 9 0.296 7 0.9 0.315 9 0.318 6 0.321 2 0.323 8 1.0 0.341 3 0.343 8 0.346 1 0.348 5 1.1 0.364 3 0.366 5 0.368 6 0.370 8
- 6. 5% z = 1.6449 z = 1.96 5% 5% 2.5% 2.5%2.5% P(Z > 1.6449) = 0.05 1.6449-1.6449 -1.96 1.96 95%
- 7. 5% z = 1.6449 z = 1.96 5% 5% 2.5% 2.5%2.5% 1.6449-1.6449 -1.96 1.96 95% a1 a2 g z 5.0% 10% 1.644 9 2.5% 5% 95% 1.960 0 1.0% 2% 2.326 3 0.5% 1% 99% 2.575 8
- 8. Population Parameters Sample Statistics? ?
- 9. ◦ Population Parameter ◦ mean 𝜇 = 𝑓𝑖 𝑥𝑖 𝑁 ◦ standard deviation 𝜎 = 𝑓𝑖 𝑥𝑖 − 𝜇 2 𝑁 ◦ Sample Statistic ◦ sample mean 𝑥 = 𝑓𝑖 𝑥𝑖 𝑛 ◦ sample standard deviation 𝑠 = 𝑓𝑖 𝑥𝑖 − 𝜇 2 𝑛 − 1
- 10. Population Distribution Distribution of Sample Means m s 𝑥𝜇 𝑥 𝜎𝑥 𝜇 𝑥 = 𝜇 𝜎𝑥 = 𝜎 𝑛
- 11. n > 30 Population Distribution Distribution of Sample Means
- 12. Population Parameters Sample Statistics ?
- 13. 95% z = -1.96 z = 1.96
- 14. 𝑥 = 𝜇 𝑥 − 1.96𝜎 𝑥 𝑥 = 𝜇 𝑥 + 1.96𝜎 𝑥 With probability 0.95 the sample mean 𝑥 lies within 1.96 standard errors of the population mean m 𝜇 = 𝜇 𝑥 95% With probability 0.95 m lies within 1.96 standard errors of the sample mean 𝑥
- 15. a1 a2 g z 5.0% 10% 1.6449 2.5% 5% 95% 1.9600 1.0% 2% 2.3263 0.5% 1% 99% 2.5758 • the g% confidence interval 𝜇− , 𝜇+ = 𝑥 − 𝑧 𝛾 𝑠 𝑛 , 𝑥 + 𝑧 𝛾 𝑠 𝑛
- 16. 𝜇−, 𝜇+ = 𝑥 − 1.96 𝑠 𝑛 , 𝑥 + 1.96 𝑠 𝑛 1 in 20 chance of not containing population mean 19 in 20 chance of containing population mean m
- 17. ◦ A machine produces golf balls. The diameters of a each of a sample of 30 balls is measured. Find the 95% and 99% confidence interval estimates of the mean diameter of balls produced by the machine. 42.83 43.82 43.51 42.64 43.82 43.71 43.37 42.76 43.18 43.22 44.00 42.77 43.00 42.99 42.85 42.75 43.90 43.36 42.81 42.92 43.33 42.78 42.75 43.09 43.72 43.70 43.85 42.91 43.32 43.67 𝑥 = 43.24433 𝑠 = 0.431123 a1 a2 g z 5.0% 10% 1.6449 2.5% 5% 95% 1.9600 1.0% 2% 2.3263 0.5% 1% 99% 2.5758
- 18. ◦ A machine produces golf balls. The diameters of a each of a sample of 30 balls is measured. Find the 95% and 99% confidence interval estimates of the mean diameter of balls produced by the machine. 𝑥 = 43.24433 𝑠 = 0.431123𝑥 − 1.96 𝑠 𝑛 , 𝑥 + 1.96 𝑠 𝑛 = 43.09, 43.40 𝑥 − 2.58 𝑠 𝑛 , 𝑥 + 2.58 𝑠 𝑛 = (43.04, 43.45) a1 a2 g z 5.0% 10% 1.6449 2.5% 5% 95% 1.9600 1.0% 2% 2.3263 0.5% 1% 99% 2.5758
- 19. ◦ Approximate confidence interval for a parameter of population proportion in terms of sample proportion is 𝑝− , 𝑝+ = 𝑝 − 𝑧 𝛾 𝑝 1 − 𝑝 𝑛 , 𝑝 + 𝑧 𝛾 𝑝 1 − 𝑝 𝑛 ◦ Reasonable approximation if np and n(1 - p) are both ≥ 5
- 20. ◦ Taken from a large batch a horticulturist test planted 100 seeds; 10 failed to germinate. Give a 95% CI for the proportion of seeds within the batch that may also fail. 𝑝− , 𝑝+ = 𝑝 − 𝑧 𝛾 𝑝 1 − 𝑝 𝑛 , 𝑝 + 𝑧 𝛾 𝑝 1 − 𝑝 𝑛 𝑝− , 𝑝+ = 10 100 − 1.96 10 100 1 − 10 100 100 , 10 100 + 1.96 10 100 1 − 10 100 100 𝑝−, 𝑝+ = 0.0412, 0.1588
- 21. •If normal model for population then - exact sample confidence interval can be calculated - do not use z percentage points but t percentage points - t percentage points come from a family of distributions called Student’s t-distribution - family because t-distribution depends on degrees of freedom n = (n – 1)
- 22. t(1) t(30) t(4)
- 23. 𝜇−, 𝜇+ = 𝑥 − 𝑡 𝛾 𝑠 𝑛 , 𝑥 + 𝑡 𝛾 𝑠 𝑛 a1 5.00% 2.50% 1.00% 0.50% a2 10.00% 5.00% 2.00% 1.00% g 90.00% 95.00% 98.00% 99.00% n = n - 1 1 6.3138 12.7062 31.8205 63.6567 2 2.9200 4.3027 6.9646 9.9248 3 2.3534 3.1824 4.5407 5.8409 4 2.1318 2.7764 3.7469 4.6041 10 1.8125 2.2281 2.7638 3.1693 100 1.6602 1.9840 2.3642 2.6259 1000 1.6464 1.9623 2.3301 2.5808
- 24. •A random sample of 11 apples is weighed and are found to have a sample mean of 93.25 grams and a sample standard deviation of 15.60 grams. Assuming the apples are a random sample drawn from a normal distribution what is the 95% CI for the mean?
- 25. ◦ A car manufacturer releases a new car and claims that its urban cycle fuel efficiency is 18.5 km per litre. A car enthusiast magazine decides to test this claim. Null hypothesis H0: m = 18.5 Alternative hypothesis is H1: m ≠ 18.5
- 26. 5 test vehicles are obtained. For each vehicle the magazine records the fuel efficiency for an urban cycle. The data, in km/litre are: 18.18, 17.40, 17.21, 18.31, 17.85. 𝑥 = 17.79 𝑠 = 0.4782
- 27. 𝑥 = 17.79 𝑠 = 0.4782 𝜇−, 𝜇+ = 𝑥 − 𝑡 𝛾 𝑠 𝑛 , 𝑥 + 𝑡 𝛾 𝑠 𝑛 = (17.20, 18.38) a1 5.00% 2.50% 1.00% 0.50% a2 10.00% 5.00% 2.00% 1.00% g 90.00% 95.00% 98.00% 99.00% n = n - 1 1 6.3138 12.7062 31.8205 63.6567 2 2.9200 4.3027 6.9646 9.9248 3 2.3534 3.1824 4.5407 5.8409 4 2.1318 2.7764 3.7469 4.6041
- 28. H0: m = 18.5 H1: m ≠ 18.5 The 95% confidence interval, 𝜇− , 𝜇+ = (17.20, 18.38) At the 5% level of significance there is evidence to reject the null hypothesis that the urban cycle fuel efficiency of 18.5 km / litre. 2.5%2.5% 17.20 18.38 95%
- 29. By the end of this topic students will be able to: ◦ Recognise the terms sample statistic and population parameter ◦ Use confidence intervals to indicate the reliability of estimates ◦ Know when approximate large sample and exact confidence intervals are appropriate
- 30. • Dewhurst, F. Quantitative Methods for Business and Management. McGraw-Hill. • Hinton, PR. Statistics Explained Routledge • Oakshot, L. Essential Quantitative Methods for Business, Management and Finance. Palgrave Macmillan.
- 31. Any Questions?

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