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# Confidence Intervals And The T Distribution

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### Confidence Intervals And The T Distribution

1. 1. Statistics 3 Confidence Intervals and the t - Distribution - Lesson 1 - <ul><li>Key Learning Points/Vocabulary: </li></ul><ul><li>The concept of a confidence intervals (notes). </li></ul><ul><li>Calculating a 95% confidence interval for a sample drawn from a normal population of known variance. </li></ul><ul><li>Calculating any confidence interval. </li></ul>
2. 2. Lesson 1 - Example Question I The heights of the 1320 lower school students at Poole High School are normally distributed with mean μ and with a standard deviation of 10cm. A sample of size 25 is taken and the mean height of the sample is found to be 161cm. a.) Find the 95% confidence interval for the height of the students. b.) If 200 samples of size 25 are taken with a 95% confidence interval being calculated for each sample, find the expected number of intervals that do not contain μ , the population mean.
3. 3. Lesson 1 - Example Question II The masses of sweets produced by a machine are normally distributed with a standard deviation of 0.5 grams. A sample of 50 sweets has a mean mass of 15.21 grams. a.) Find a 99% confidence interval for μ , the mean mass of all sweets produced by the machine correct to 2dp. b.) The manufacturer of the machine claims that is produces sweets with a mean mass of 15 grams, state whether the confidence interval supports this claim. Source: Page 48 of Statistics 3 by Jane Miller
4. 4. Commonly used z – values for Confidence Intervals 2.576 99% 2.326 98% 1.96 95% 1.645 90% z Confidence Interval
5. 5. Generalisation: Sample from a Normal Population A 100(1 – α )% confidence interval of the population mean for a sample of size n taken from a normal population with variance σ 2 is given by where x is the sample mean and the value of z is such that Ф (z) = 1 – ½ α .
6. 6. Practice Questions Statistics 3 and 4 by Jane Miller Page 50, Exercise 3A Question 1 onwards
7. 7. Statistics 3 Confidence Intervals and the t - Distribution - Lesson 2 - <ul><li>Key Learning Points/Vocabulary: </li></ul><ul><li>The central limit theorem (S2) </li></ul><ul><li>Unbiased estimate of the population variance (S2). </li></ul><ul><li>Calculating the confidence interval for a large sample. </li></ul>
8. 8. Unbiased Estimate of the Population Variance Given a sample of size n (n large) from a population of which the variance is unknown, we estimate the population variance s 2 as detailed below:
9. 9. Generalisation: Large Sample for any Population Given a large sample (n>30) from any population, a 100(1 – α )% confidence interval of the population mean is given by where x is the sample mean and the value of z is such that Ф (z) = 1 – ½ α .
10. 10. Lesson 2 - Example Question On 1 st September, 100 new light bulbs were installed in a building, together with a device that detailed for how long each light bulb was used. By 1 st March, all 100 light bulbs had failed. The data for the recorded lifetimes, t (in hours of use), are summarised by Σ t = 10500 and Σ t 2 = 1712500. Assuming that the bulbs constituted a random sample, obtain a symmetric 99% confidence interval for the mean lifetime of the light bulbs, giving your answer correct to the nearest hour. Source: Page 48 of Statistics 3 by Jane Miller
11. 11. Practice Questions Statistics 3 and 4 by Jane Miller Page 50, Exercise 3B Question 1 onwards
12. 12. Statistics 3 Confidence Intervals and the t - Distribution - Lesson 3 - <ul><li>Key Learning Points/Vocabulary: </li></ul><ul><li>Expectation and variance of the binomial distribution. </li></ul><ul><li>Conditions for normal approximation to the binomial. </li></ul><ul><li>Calculating the approximate confidence interval for a population proportion from a large sample. </li></ul>
13. 13. Lesson 3 - Example Question You are the manufacturer of tin openers to be used specifically by left handed people. A random sample of 500 people finds that 60 of them are left handed. What is the 95% confidence interval for this estimate of the proportion of people who are left handed? Source: Page 55 of Statistics 3 by Jane Miller
14. 14. Generalisation: Confidence Interval for a Proportion Given a large random sample of size n from a population in which a proportion of members p has a particular attribute, the approximate confidence interval is given by:
15. 15. Practice Questions Statistics 3 and 4 by Jane Miller Page 50, Exercise 3C Question 3 onwards
16. 16. Statistics 3 Confidence Intervals and the t - Distribution - Lesson 4 - <ul><li>Key Learning Points/Vocabulary: </li></ul><ul><li>Use flow chart to help decide when to use either the t or z distribution. </li></ul><ul><li>The t – distribution. </li></ul><ul><li>Calculating the confidence interval for a small sample drawn from a normal population of unknown variance. </li></ul>
17. 17. Source: http:// en.wikipedia.org/wiki/File:Student_densite_best.JPG
18. 18. Lesson 4 - Example Question Ten university physics students independently conducted experiments to determine the value of g . They obtained the following results: 9.812 9.807 9.804 9.805 9.812 9.808 9.807 9.814 9.809 9.807 Calculate the 95% confidence limits for g , stating any assumptions made. Source: Page 105 of Statistics 2 by M E M Jones
19. 20. Theory For a random sample from a normal population with mean μ , the variable has a t distribution with ν degrees of freedom, where ν = n – 1. That is,
20. 21. Generalisation: t-distribution Given a sample from a normal population of unknown variance, a 100(1 – α )% confidence interval for the population mean is given by where x is the sample mean and the value of t is such that P(T ≤ t) = 1 – ½ α for ν = n – 1 degrees of freedom.
21. 22. Practice Questions Statistics 3 and 4 by Jane Miller Page 62, Exercise 3D Question 2 onwards
22. 23. Statistics 3 Confidence Intervals and the t - Distribution - Lesson 5 - <ul><li>Key Learning Points/Vocabulary: </li></ul><ul><li>Hypothesis test on the population mean for a small sample from a normal population. </li></ul><ul><li>Shortened name: t-Test. </li></ul>
23. 24. Lesson 5 - Example Question The weights of eggs laid by a hen when fed on ordinary corn are known to be normally distributed with a mean of 32kg. When a hen was fed on a diet of vitamin enriched corn a random sample of 10 eggs was weighed and the following results (in grams) were recorded: 31, 33, 34, 35, 35, 36, 32, 31, 36, 37 Test, using a 5% significance level, the claim that the new diet has increased the mean weight of eggs laid by the hen by more than 1g. Source: Page 152 or Statistics2 by MEM Junes
24. 25. Practice Questions Statistics 3 and 4 by Jane Miller Page 67, Exercise 3E Question 1 onwards
25. 26. Statistics 3 Confidence Intervals and the t - Distribution - Lesson 6 - <ul><li>Key Learning Points/Vocabulary: </li></ul><ul><li>Mixed questions on Confidence Intervals and the t-Distribution. </li></ul><ul><li>Mind map to summarise key learning points. </li></ul>
26. 27. Practice Questions Statistics 3 and 4 by Jane Miller Page 687, Miscellaneous Exercise 3 Questions 1, 3, 6 and 9 (first part only)