July 18, 2012. A Check Day.

              Last Wednesday morning, after entering the gate, I was rushing along the quadrangle just to finish my assignment in Statistics. I find it difficult to answer the last two items of the assignment because last Tuesday Sir Salmorin is the one who taught us about the Confidence Interval for Proportions, Variances, and Standard Deviation and the chi-square distribution and he did not finish discussing the lesson. So I ask the help of my classmates to teach me how to solve.  



             That day is a checked day for all our exercises, quizzes and assignments. We have checked the Exercise no. 6 Central Limit Theorem and Normal approximation to Binomial Distribution and Quiz no. 9 Confidence Interval for the mean- t test and we have also checked one of our assignments. 

 Review of Yesterday’s Lesson

                Confidence Interval for Proportions, Variances, and Standard Deviation

Proportion-represents a part of a whole. It can be expressed as a fraction, decimal or percentage. 12% = 0.12 = 12/100 or 3/25. Proportions can also represent probabilities.

Formula for a specific Confidence Interval for a proportion:

For uniformity and unity of answers, there should be a rounding rule.

Remember: Round off to three decimal places.

Example 1: A survey of 5000 ice cream vendors found that 40% sold a mango flavor. Find the 95% confidence interval of the true proportion of ice cream vendors who sold a mango flavor.

Solution:

0.386   < p < 0.414

Remember to Round off the final answer to three decimal places.

Sample Sizes for Proportions

Confidence Intervals for variances and standard deviation

Chi-square Distribution-similar to the t-distribution, it is a family of curves based on the number if degrees of freedom.

This statistical technique we used to provide answer to such question is based on the chi-square (X²) distribution. The technique was introduced in 1990 by Karl Pearson.

He applied statistics to biological problems of heredity and evolution. From 1893-1912 he wrote 18 papers entitled Mathematical Contribution to the Theory of Evolution which contain his most valuable work. These papers contain contributions to regression analysis, the correlation coefficient and includes the chi-square test of statistical significance (1900). His chi-square test was produced in an attempt to remove the normal distribution from its central position.

Among the uses of chi-square are the following:

•   to test the goodness of fit to a normal curve; that is, to find out whether or not a sample distribution conforms with the hypothetical normal distribution.
•   to find out whether or not an observed proportion is equal to some given ideal or expected proportion.
•   to test the independence of one variable from another variable.

The formula for the confidence intervals for σ² and σ are:

Remember: Rounding Rule:

1.       When using raw data- round off to one more decimal place than the number of decimal places in the original data.

2  2.  When using sample variance or standard deviation- round off to the same number of decimal places as given variance or standard deviation.

By: Tito Nuevacobita Jr. 
III- Gold