Statistical Sampling Assignment

Questions: 1) In Yoon (2005), determine the type of sample that was selected. 2) In Huh et al. (2004) determine the type of sample that was selected. 3) In Metzger (2006) determine the type of sample that was selected. 4) In Matsunga and Todd (2009) determine the type of sample that was selected From the Japanese population From the American population 5) In Conway and Rubin (1991) determine the type of sample that was selected. 6) In Neuwirth and Frederick (2004) determine the type of sample selected. 7) In Diener and Woody (1981) determine the type of sample selected in Study 3. 8) For the NCA data set (nca modified.sav) consider the 1,001 participants as the population. Take a simple random sample of size 30 and record the proportion of females in the sample. Repeat this 50 times to obtain 50 values for the sample proportion of females. Start SPSS and enter these 50 values in the first column. Construct a histogram for these 50 values. What is the approximate shape of this histogram? (You may need to use your imagination here depending on the results of your simulations.) Approximately where is the histogram centered? Answer: 1. Simple random sample2. Simple random sample3. Simple random sample4. Stratified random sample5. Cluster sample6. Simple random sample7. Purposive sample 8. Introduction: It was observed that most of the sampling distributions follow an approximately normal distribution. For this assignment, we have to check with the given data set. We have given a data set containing different variables. One of the variables in this data set is given as sex. For this variable, we are given a sample size of 1001 observations. By using some samples from these observations, we have to find the histogram for the proportion of females in these data sets and again we have to check the shape and structure of this histogram. We are interesting in checking whether it follows an approximately normal distribution or not. Procedure: First of all we have to take a random sample of size 30 from the variable sex given in the data set. We have to repeat this process about 50 times. That is, we have to select a total 50 samples of size 30. Then we have to find the proportion of females in each sample. Then we have to find the descriptive statistics for these 50 proportions. Also we have to draw the histogram for these 50 proportions of females. We have to check the shape of the histogram whether it follows an approximate normal distribution or not. We have to check at which value the data is centred. Let us see this step by step by using SPSS. Data analysis: First of all, let us see some information about the total observations given for the variable sex in this data set. We are given a total observations as 1001 and we found that there are total number of males is 478 and total number of females is 523. That is, proportion for female in the total observations is given as 52.2% or 0.522. The proportion for the male is given as 47.8% or 0.478. The SPSS output is given below: Statistics sex N Valid 1001 Missing 0 sex Frequency Percent Valid Percent Cumulative Percent Valid Male 478 47.8 47.8 47.8 Female 523 52.2 52.2 100.0 Total 1001 100.0 100.0 Now, we have to take the 50 samples of size 30 from the given data for the variable sex. Then we have to calculate the proportions of females for each sample. The proportions of these 50 samples of size 30 are given in the following table: Sample No. Proportion of female Sample No. Proportion of female 1 40.95502326 26 51.27777853 2 58.71984001 27 53.23472705 3 53.6938884 28 49.83418127 4 54.35901276 29 49.95743763 5 51.69683661 30 52.51152629 6 50.47701603 31 53.62696924 7 49.61262808 32 48.48401742 8 53.92512653 33 57.60656333 9 48.40706528 34 54.1941655 10 46.76494334 35 48.35132368 11 53.87762596 36 52.26253738 12 49.2199217 37 50.52487109 13 49.62147471 38 49.7573443 14 56.08128077 39 49.90728761 15 52.32568856 40 54.76303163 16 52.61949497 41 53.42068393 17 44.83826512 42 48.35487342 18 53.48641247 43 57.04045955 19 49.53616485 44 54.01028057 20 52.13452921 45 51.80268777 21 53.21964813 46 49.76943762 22 52.99027824 47 51.36415712 23 54.97215734 48 49.95485869 24 53.93664378 49 54.95175683 25 49.64889798 50 48.74594296 Now, we have to see some descriptive statistics for these 50 proportions of females. Descriptive statistics are given below: Descriptive Statistics N Minimum Sum Mean Std. Deviation Variance Female proportion 50 42.40 2605.08 52.1016 3.90992 15.287 Valid N (list wise) 50 The average proportion for female is observed as 52.10% or 0.521 with standard deviation of 3.91%. Now, let us see some other descriptive statistics for these proportions of females given in the following table: Descriptive Statistics N Range Maximum Mean Skewness Kurtosis Statistic Statistic Statistic Std. Error Statistic Std. Error Statistic Std. Error Female proportion 50 18.13 60.53 .55295 -.088 .337 -.198 .662 Valid N (list wise) 50 The histogram for the female proportions is given as below: This histogram follows an approximate normal distribution and the centre of this histogram is located at the value 52%. That is, we get the sampling proportion for female same as the population proportion of females. Let us see the p-plot for normality for the female proportion which is given as below: This p-plot suggests that the sampling distribution follows an approximately normal distribution. Conclusion: It is concluded that the sampling distribution of the female proportions centred at the value of female population proportion. Also, it was observed that the sampling distribution follows an approximately normal distribution with the mean approximate equal to the population proportion. References: 1. David Freedman, Robert Pisani, Roger Purves, Statistics, 3rd ed., W. W. Norton Company, 1997.2. Morris H. DeGroot, Mark J. Schervish Probability and Statistics, 3rd ed., Addison Wesley, 2001.3. Leonard J. Savage, The Foundations of Statistics, 2nd ed., Dover Publications, Inc. New York, 1972.4. George Casella, Roger L. Berger, Statistical Inference, 2nd ed., Duxbury Press, 2001.5. David R. Cox, D. V. Hinkley, Theoretical Statistics, Chapman Hall/CRC, 1979.6. Peter J. Bickel, Kjell A. Doksum, Mathematical Statistics, Volume 1, Basic Ideas and Selected Topics, 2rd ed. Prentice Hall, 2001.7. T. S. Ferguson, Mathematical Statistics: A Decision Theoretic Approach, Academic Press, Inc., New York, 19678. Harald Cramr, Mathematical Methods of Statistics, Princeton, 1946