The Central Limit Theorem Essay Example for Free

The Central Limit Theorem Essay The Central Limit Theorem for a Mean state that for a random sample of size n from a population with mean  µ and standard deviation ? , as the sample size increases the distribution of the sample mean approaches a normal distribution with mean  µ and standard deviation . On the other hand, if the population is normal, the distribution of the sample mean is normal regardless of sample size (Doane Seward, 2007). Now, suppose a random sample of size n is taken from a population with mean 100 and standard deviation 10. The sampling error or standard error of mean for samples of n = 1, n = 4, n = 9, n = 16, n = 25, n =100 will be For n = 1, For n = 1, For n = 1, For n = 1, For n = 1, For n = 1, From above it can be seen that as the sample size increases, the sampling error reduces. Further, the histogram (if plotted) of samples means approaches a normal distribution. Therefore, while on cannot get rid of sampling error the results from one’s statistical work can be still useful as the sampling error will be less for larger sample size. References: Doane D. P. Seward L. E. (2007). Applied Statistics in Business and Economics. McGraw-Hill/Irwin: New York DQ2-WK3 What is the difference between a sample and a population? When can the same information (e. g. the age of each of the ten students in our class) be considered both sample data and population data? A sample involves looking only at some items from the population. For example, if a survey is to be taken from student of XYZ College for their choices, than the population will be consist of taking survey of all the students in XYZ College and a sample will be taking survey for only some of the students selected based on sampling method. For small population, there is little (or no) reason to sample. Similarly, if the data are on disk, than 100% of the cases can be examined easily (Doane Seward, 2007). Therefore, the same information can be considered both sample data and population data if the population is small or can be taken easily with no additional cost for analysis purpose. For example, in a class of 30 students, if the average age is to be determined, than the population is entire class that is all 30 students. In this case, the population is small; therefore, there is no need to sample and therefore, the age of all 30 students can be considered both sample data and population data. References: Doane D. P. Seward L. E. (2007). Applied Statistics in Business and Economics. McGraw-Hill/Irwin: New York DQ3-WK3 When would you use ANOVA at your place of employment, in your education, or in politics? Please share the WORDS that would lead you to the null hypothesis for a specific and simple example and, then, show the null and alternative hypothesis in symbols. Analysis of variance (commonly referred to as ANOVA) is used for comparison of more than two means simultaneously and to trace sources of variation to potential explanatory factors (Doane Seward, 2007). For example, if sample data for 15 days of manufacturing defect rates for automotive parts manufactured at four plant locations is collected (or available). Than, ANOVA can be used to answer questions such as are the observed differences in the plants’ sample mean defect rates merely due to random variation? Alternatively, are the observed differences between the plants’ defect rates too great to be attributed to chance? This can be written as null and alternate hypothesis as , i. e. at all four plants mean defect rate are the same. , i. e. at least one mean differs from the other. Now, if the null hypothesis is not rejected than it can be concluded that the observations within each treatment or group actually have a common mean ? (ibid). References: Doane D. P. Seward L. E. (2007). Applied Statistics in Business and Economics. McGraw-Hill/Irwin: New York