新高中數學教育工作坊

1. 新高中數學教育工作坊 羅家儀博士 Dr Agnes Law 點與區間估計的理論與實踐 19th Dec 2009

2. Some notes for teaching

3. Normal distribution Explanation: The name - normal distribution is one where the data is evenly distributed around the mean in a very regular way, which when plotted as a histogram will result in a bell curve. There are a lot of ways of defining "normal distribution" formally, but the simple intuitive idea of it is that in a normal distribution, things tend towards the mean - the closer a value is to the mean, the more you'll see it; and the number of values on either side of the mean at any particular distance are equal. Note: Emphasis on the “continuous” random variable, large sample size

4. How to declare the data follows normal distribution? As the sample size increases, the sampling distribution of the mean, X-bar, can be approximated by a normal distribution with mean µ and standard deviation σ/√n where: µ is the population mean σ is the population standard deviation n is the sample size In other words, if we repeatedly take independent random samples of size n from any population, then when n is large, the distribution of the sample means will approach a normal distribution.

5. How large is large enough? Generally speaking, a sample size of 30 or more is considered to be large enough for the central limit theorem to take effect. The closer the population distribution is to a normal distribution, the fewer samples needed to demonstrate the theorem. Populations that are heavily skewed or have several modes may require larger sample sizes. The central limit theorem is remarkable because it implies that, no matter what the population distribution looks like, the distribution of the sample means will approach a normal distribution. The theorem also allows us to make probability statements about the possible range of values the sample mean may take.

6. Why 30 sample size is used as a cut off point? It is a result of simulation studies involving the Central Limit Theorem.

7. Example All Possible Samples of Size n=2 16 Sample Means 16 Samples Taken with Replacement

8. Comparing the Population with its Sampling Distribution Population N = 4 Sample Means Distribution n = 2 P(X) P(X) .3 .3 .2 .2 .1 .1 X 18 19 20 21 22 23 24 0 0 AB C D (18)(20)(22)(24)

9. Central Limit Theorem the sampling distribution becomes almost normal regardless of shape of population As sample size gets large enough…

10. Another example: s2 is the unbiased estimator of σ2 Unbiased and biased estimator Example: X = µ If the mean value of an estimator equals the true value of the quantity it estimates, then the estimator is called an unbiased estimator. Assume that the sample mean is being used to estimate the mean of a population. Using the Central Limit Theorem, the mean value of the sample means equals the population mean. Therefore, the sample mean is an unbiased estimator of the population mean. If the mean value of an estimator is either less than or greater than the true value of the quantity it estimates, then the estimator is called a biased.

11. Confidence interval Explanation: e.g. C.I. of average sales = $1680≦μ ≦$1800 The explanation would be one with 95% confidence that the mean sales is between $1680 and $1800. The confidence interval states that one is 95% sure that the sample selected is one in which the population mean μ is located within the interval. The validity of this confidence interval estimate depends on the assumption of normality of the force required data. Note: The interval is estimated for population mean but not sample mean.

12. Application in the business and university

13. Use of confidence interval in university Purpose: To find out the range of time for going from home in different districts to the university Method: Ask students to report the time they use for traveling Calculation: To estimate the confidence interval of mean traveling time for different districts which may affect the allocation of hall

14. Use of confidence interval in business Purpose:Telecommunication company would like to know the preferred price of a newly launched mobile phone Method: Conduct random sample surveys to collect the opinions of youngster Calculation: To estimate the confidence interval for preferred average price of the mobile phone

15. Use of confidence interval in business Purpose:Telecommunication company would like to know the proportion of youngster who have interest to buy a newly launched mobile phone Method: Conduct random sample surveys to collect the opinions of youngster Calculation: To estimate the confidence interval for the proportion of youngster who have interest to buy the new mobile phone