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PROBABILITY AND STATISTICS

PROBABILITY AND STATISTICS. WEEK 9-10. The sampling distribution of the sample statistics. The sampling distribution of the sample statistics. C onsider a population of N elements from which we can obtain the following distinct data: { 0 , 2 , 4 , 6 , 8 } .

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PROBABILITY AND STATISTICS

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  1. PROBABILITY AND STATISTICS WEEK 9-10 Onur Doğan

  2. The sampling distribution of the sample statistics Onur Doğan

  3. The sampling distribution of the sample statistics Consider a population of N elements from which we can obtain thefollowing distinct data: {0, 2, 4, 6, 8}. • Form samples of size 2 for this population. • Define theirmeansandfigurethe bar chart of themeans. • Define the sampling distribution of the sample rangesandfigure bar chart. Onur Doğan

  4. The Central Limit Theorem The mean is the most commonly used sample statistic and thus it is very important.The central limit theorem is about the sampling distribution of sample means of random samplesof size n. Let us establish what we are interested in when studying thisdistribution: 1) Where is the center? 2) How wide is the dispersion? 3) What are the characteristics of the distribution? The central limit theorem gives us an answer to all these questions. Onur Doğan

  5. The Central Limit Theorem Let µbe the mean andσ the standard deviation of a population variable. If we consider allpossible random sample of size n taken from this population, the samplingdistribution of samplemeans will have the following properties: c) if the population is normally distributed the sampling distribution of the samplemeans is normal; if the population is not normally distributed, the samplingdistribution of the sample means is approximately normal for samples of size 30 or more.The approximation to the normal distribution improves with samples of larger size. Onur Doğan

  6. The Central Limit Theorem Onur Doğan

  7. The Central Limit Theorem Onur Doğan

  8. The Central Limit Theorem Onur Doğan

  9. Example Consider a normal population with µ=100 and σ=25. If we choose arandom sample of size n = 36, what is the probability that the mean value of this sample isbetween 90 and 110? In other words, what is P(90 < x < 110)? Onur Doğan

  10. Example Theaveragemaledrinks 2L of waterwhenactiveoutdoor s(withstandarddeviation of 0,7 L). Youareplanning a fulldaynaturetripfor 50 men andbring 110 L of water. What is theprobabilitythatyouwillrunout? Onur Doğan

  11. Confidence Intervals Onur Doğan

  12. Confidence Interval on the Mean of a Normal Distribution, Variance Known Onur Doğan

  13. Confidence Interval Formula Onur Doğan

  14. Example Suppose that the life length of a light bulb (X; unit: hour) follows the normal distribution N(y, 402). A random sample of n = 30 bulbs is tested and the sample mean is found to be 780 hours. • Construct a 95% two sided confidence interval on the mean life length (µ) of a light bulb. • Find a sample size n to construct a two-sidedconfidence interval on µ with an error = 20 hours from the true mean life length.(use α= 0.05) Onur Doğan

  15. Example • The diameter of a hole (X; unit: in.) for a cable harness is normal with σ2= 0.012. A random sample of n = 10 yields an average diameter of 1,5045 in. • Construct a 99% upper-confidence bound on the mean diameter (p) of the hole. Onur Doğan

  16. Confidence Interval on the Mean of a Normal Distribution,Variance Unknown (t distribution) SmallSample CI? Onur Doğan

  17. Example Yousample 36 applesfromyourfarm’sharvest of over 200.000 apples. Themeanweight of thesample is 112 grams (with a 40 gram sample standart deviation). What is theprobabilitythatthemeanweight of all 200 000 apples is within 100 and 124 grams? Onur Doğan

  18. A Large-Sample Confidence Interval for a Population Proportion • Confidence Interval Formula: • Sample Size Selection Onur Doğan

  19. Example A sample of n = 40 bridges in a city is testedfor metal corrosion, and x = 28 bridgesarefoundcorroded. • Construct a 95% two-sidedconfidenceinterval on theproportion of corrodedbridges (p) in thecounty. • Determine a sample size n to establish a 95%confidence interval onp with an error = 0.05 from the true proportion. Onur Doğan

  20. Example In a teachingdistrictschoolmanagementallowstheacherusingomputer in theirlessons. Fromthe 6000 teachers in district, 250 wererandomlyselectedandaskediftheyfektthatcomputerswere an essentialteachingtoolfortheircalssrom. Of thoseselected, 142 teachersfeltthatcomputerswere an essentialteachintool. • Calculate a 99% confidenceinteralfortheproportion of teacherswhofeltthatcomputersare an essentialteachingtool Onur Doğan

  21. Summary for Confidence Intervals Onur Doğan

  22. Hypothesis Testing Onur Doğan

  23. Test Regions Onur Doğan

  24. Test Errors Onur Doğan

  25. Hypothesis Testing Procedure

  26. Tests on the Mean of a NormalDistribution, VarianceKnown • .

  27. Example For several years, a teacher has recorded his students' grades, and the mean, µfor all these students' grades is 72 and the standard deviation is σ = 12. The current class of36 students has an average x = 75,2 (higher than µ = 72) and the teacher claims that this classis superior to his previous ones. Test the teacher’s claim for the level of significance = α=0,05. Onur Doğan

  28. Relationship Between Hypothesis Test, CI and p-value

  29. Tests on a Population Proportion • .

  30. Example Forthebridgeexample; a specialistclaimsthatmorethanhalf of thebridgesgavebeencorroded in thecity. Test thespecialist’sclaimwith %95 confidence.

  31. Example • Suppose that a factory is producing wheels for airbuses. The manufacturer claims that they produce wheels 3 meters diameter. • The quality control department of the buyer firm investigate a sample from the daily product. They controlled 36 wheels and found that average diamater is 2,92 and s.d. is 0,18. • Test the claim at α=0.05 significance level

  32. Example • According to a recent poll 53% of Americans would vote for the incumbent president. If a random sample of 100 people results in 45% who would vote for the incumbent, test the claim that the actual percentage is 53%. Use a 0.10 significance level.

  33. Example • The national weather service says that the mean daily high temperature for July in İzmir is 42°C. A local weather service wants to test the claim of 42°C because it believes it is different. A sample of mean daily high temperatures for October over the past 31 years yields =44°F and s=3.8°C. Test the claim at α=0.01 significance level.

  34. Example • In a clinical study of an allergy drug, 108 of the 203 subjects reported experiencing significant relief from their symptoms. At the 0.01significance level, test the claim that more than half of all those using the drug experience relief.

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