Running head: DATA ANALYSIS AND INFERENTIAL STATISTICS ON MALE AND FEMALE BMI Data analysis and inferential statistics using Male and female BMI Part 2 Student Name Date: DATA ANALYSIS AND INFERENTIAL STATISTICS ON MALE AND FEMALE BMI Introduction Inferential statistics involves making a conclusion with regard to some existing statistics in order to determine how far two analyses much or differ. This paper will analyze the inferential statistics based on the dataset on BMI values for male and female. The paper will also construct confidence intervals for means of BMI values for male and female together with the difference of means. DATA ANALYSIS AND INFERENTIAL STATISTICS ON MALE AND FEMALE BMI QUESTION 1: 95% CONFIDENCE FOR THE MEAN OF ALL MALES The BMI data value for the males is as in appendix 1 at the end of this paper. From the table, the male mean BMI is 25.962 and the standard deviation is 3.582 To construct a 95% confidence interval, we use the obtained mean above and the standard deviation Computing = 1- confidence interval/100 = 1- 95/100 = 0.05 Calculating the critical region = 1- 2 = 1- 0.05/2 = 0.975 Degrees of freedom = n -1 = 329257 - 1 = 329256 The critical value is, therefore, a t- score with a cumulative probability of 0.975 and 329256 degrees of freedom. The value is 1.960. Standard error is obtained as SE = S/sqrt of n DATA ANALYSIS AND INFERENTIAL STATISTICS ON MALE AND FEMALE BMI = 3.582 329257 = 0.006242 Margin of error is obtained as: ME = critical value x SE = 1.960 X 0.006242 = 0.01224 Therefore a 95% confidence interval for the male BMI mean is 25.962 0.01224 This implies that the male mean BMI ranges from 25.95 to 25.97 DATA ANALYSIS AND INFERENTIAL STATISTICS ON MALE AND FEMALE BMI QUESTION 1: 95% CONFIDENCE FOR THE MEAN OF ALL FEMALES The BMI data values for the females are as in the table in appendix 2 at the end of this paper. From the table, the female mean BMI is 25.899 and the standard deviation is 6.079 To construct a 95% confidence interval, we use the obtained mean above and the standard deviation Computing = 1- confidence interval/100 = 1- 95/100 = 0.05 Calculating the critical region = 1- 2 = 1- 0.05/2 = 0.975 Degrees of freedom = n -1 = 822886 - 1 = 822885 The critical value is, therefore, a t- score with a cumulative probability of 0.975 and 822885 degrees of freedom. The value is 1.960. Standard error is obtained as SE = S/sqrt of n DATA ANALYSIS AND INFERENTIAL STATISTICS ON MALE AND FEMALE BMI = 6.079 822886 = 0.006701 Margin of error is obtained as: ME = critical value x SE = 1.960 X 0.006701 = 0.01313 Therefore a 95% confidence interval for the male BMI mean is 25.899 0.01313 This implies that the female mean BMI ranges from 25.89 to 25.91 DATA ANALYSIS AND INFERENTIAL STATISTICS ON MALE AND FEMALE BMI QUESTION 3: 95% CONFIDENCE INTERVAL FOR THE DIFFERENCE MALE MEAN AND FEMALE MEAN From the calculation in the tables for question one and two, the summary of the calculations is as shown below: Condition Male n 329257 Mean 25.962 Variance 12.831 Female 822886 25.899 36.954 The 95% confidence interval on the difference of the means is given as: The difference of means = 25.962 - 25.899 = 0.063 To construct a 95% confidence interval, the obtained difference of means above and the variances are used. Computing = 1- confidence interval/100 = 1- 95/100 = 0.05 DATA ANALYSIS AND INFERENTIAL STATISTICS ON MALE AND FEMALE BMI Calculating the probability = 1- 2 = 1- 0.05/2 = 0.975 Standard error is obtained as: SE = sqrt [ s21 / n1 + s22 / n2 ] = 12.831 36.954 + 329257 822886 = 0.009158 Because the sample sizes are large enough, we express the critical value as a z score. The critical value is the Z - score having a cumulative probability equal to 0.975. We obtain this critical value of 1.960 from tables. The margin of error is obtained as: ME = critical value x SE = 1.960 x 0.009158 = 0.018 The 95% confidence interval for the difference of means is expressed as 0.063 0.018 This implies that the difference of the male means BMI and that of female mean BMI ranges from 0.612 to 0.81 DATA ANALYSIS AND INFERENTIAL STATISTICS ON MALE AND FEMALE BMI QUESTION 4: HYPOTHESIS TESTING WHETHER THERE IS A DIFFERENCE BETWEEN THE MALE BMI MEAN AND FEMALE BMI MEAN 1. Stating null and the alternate hypothesis Since we want to test whether there is a difference between the means, the null and alternative hypotheses are stated as: H0: 1 - 2=0 H1: 1 - 2 0 2. Computing the test statistic Using the 0.05 significance level, the following plan is followed: The standard error is obtained as: SE = sqrt [s21 / n1 + s22 / n2] = 12.831 36.954 + 329257 822886 = 0.009158 The degrees of freedom are obtained as: DF = (s12/n1 + s22/n2)2 / { [ (s12 / n1)2 / (n1 - 1) ] + [ (s22 / n2)2 / (n2 - 1) ] } But DATA ANALYSIS AND INFERENTIAL STATISTICS ON MALE AND FEMALE BMI (s12/n1 + s22/n2)2 = (SE) 2 = 0.00008387 (s12 / n1)2 / (n1 - 1) = 4.6123 X 10 -15 (s22 / n2)2 / (n2 - 1) = 2.451 X 10 -15 Therefore, degrees of freedom are given as: 0.00008387 4.6123 X 1015+2.451 X 1015 = 7 degrees of freedom The t- score is obtained as: t = [( x 1 - x 2 ) - d ] / SE = [(25.962 - 25.899) - 0 ] / 0.009158 = 6.879 3. Obtaining the P- value Since we have a two tailed test, the P-value is the probability that a t statistic having 7 degrees of freedom is more extreme than 6.879. That is: P (t < - 6.879) = 0.0001, and P (t > 6.879) = 0.9999 Thus, the P-value = 0.001 + 0.9999 =1 DATA ANALYSIS AND INFERENTIAL STATISTICS ON MALE AND FEMALE BMI 4. Interpreting the P- value Since the P- Value 1, is greater than the significance level 0.05, we, therefore, accept null that there is no difference between the means of female BMI and that of male BMI. QUESTION 5: FINDING OUT WHETHER RESULTS OF CONFIDENCE INTERVAL ARE DIFFERENT FROM TEST OF HYPOTHESIS Using the results of confidence interval of mean of males, 25.962 0.01224 and that of the mean of female, 25.899 0.01313, this implies that: Mean of males ranges from 25.95 to 25.97 Mean of female ranges from 25.89 to 25.91 This implies that the means are not significantly different as the test of hypothesis depicts. The difference between the means 0.06, is within the range of the difference of the means that was obtained from confidence intervals, 0.063 0.018. The mean is 26 expressed to a whole number. Therefore, results of confidence intervals and that of a test of hypothesis they agree. DATA ANALYSIS AND INFERENTIAL STATISTICS ON MALE AND FEMALE BMI APPENDICES Appendix 1: Table of mean and standard deviation of male BMI DATA ANALYSIS AND INFERENTIAL STATISTICS ON MALE AND FEMALE BMI BMI (X) MALE BMI X MALE (X-mean) (X-mean)2 f*(X-mean)2 23.8 1391 33105.8 -2.162 4.676 6504.46 23.2 2129 49392.8 -2.762 7.631 16246.44 24.6 2489 61229.4 -1.362 1.856 4620.12 26.2 2490 65238 0.238 0.056 140.534 23.5 2738 64343 -2.462 6.064 16602.031 24.5 2988 73206 -1.462 2.139 6390.44 21.5 2989 64263.5 -4.462 19.913 59520.797 31.4 3346 105064.4 5.438 29.567 98931.744 26.4 3606 95198.4 0.438 0.191 690.432 22.7 3607 81878.9 -3.262 10.643 38390.921 27.8 3608 100302.4 1.838 3.377 12183.003 28.1 3610 101441 2.138 4.569 16494.833 25.2 3747 94424.4 -0.762 0.581 2178.129 23.3 4832 112585.6 -2.662 7.089 34251.793 31.9 4839 154364.1 5.938 35.255 170597.677 33.1 5599 185326.9 7.138 50.945 285240.529 33.2 5600 185920 7.238 52.382 293341.553 26.7 5601 149546.7 0.738 0.544 3046.998 26.6 6226 165611.6 0.638 0.406 2530.841 19.9 7190 143081 -6.062 36.753 264254.48 27.1 7192 194903.2 1.138 1.294 9306.92 23.4 7194 168339.6 -2.562 6.566 47236.144 27.0 9073 244971 1.038 1.077 9767.553 21.6 9074 195998.4 -4.362 19.031 172685.435 30.9 10864 335697.6 4.938 24.38 264859.952 28.3 12349 349476.7 2.338 5.464 67477.822 25.5 15515 395632.5 -0.462 0.214 3317.75 24.6 16137 396970.2 -1.362 1.856 29953.748 23.8 16521 393199.8 -2.162 4.676 77253.903 27.4 16523 452730.2 1.438 2.067 34146.558 28.7 16768 481241.6 2.738 7.494 125664.25 26.2 17006 445557.2 0.238 0.056 959.811 26.4 18392 485548.8 0.438 0.191 3521.471 32.1 19017 610445.7 6.138 37.67 716365.941 19.6 19381 379867.6 -6.362 40.481 784552.86 20.7 19635 406444.5 -5.262 27.693 543755.374 26.3 19991 525763.3 0.338 0.114 2278.045 Total -> 329257 8548311.8 - - 4225261.291 Appendix 2: Table of mean and standard deviation of female BMI BMI (X) 19.6 23.8 19.6 29.1 25.2 FEMALE(f) 295 2739 2992 3745 4486 BMI X FEMALE 5782 65188.2 58643.2 108979.5 113047.2 (X-mean) -6.299 -2.099 -6.299 3.201 -0.699 (X-mean)2 39.683 4.408 39.683 10.244 0.489 f*(X-mean)2 11706.516 12072.696 118731.851 38361.917 2194.705 DATA ANALYSIS AND INFERENTIAL STATISTICS ON MALE AND FEMALE BMI 21.4 4488 22.0 4878 27.5 4880 33.5 4881 20.6 4835 29.9 4842 17.7 6225 24.0 8680 28.9 8681 37.7 12348 18.3 14651 19.8 16767 29.8 17765 MAL 29.7 19377 BMI E 31.7 19378 1391 23.8 19382 23.8 2129 44.9 20278 23.2 2489 19.2 21626 24.6 28.7 32233 26.2 2490 28.5 33104 23.5 2738 19.3 33106 24.5 2988 31.0 33334 21.5 2989 25.1 33335 31.4 3346 22.8 34779 3606 26.4 30.9 35035 3607 22.7 26.5 35272 3608 27.8 21.2 35273 3610 28.1 40.6 35505 3747 25.2 21.9 35506 4832 26.0 35507 23.3 4839 23.5 35984 31.9 5599 22.8 35988 33.1 5600 20.7 36115 33.2 5601 20.5 36502 26.7 21.9 38089 26.6 6226 Total -> 7190 82288619.9 7192 7194 9073 9074 10864 12349 15515 16137 16521 16523 16768 17006 18392 19017 19381 19635 19991 20518 21135 32230 27.1 23.4 27.0 21.6 30.9 28.3 25.5 24.6 23.8 27.4 28.7 26.2 26.4 32.1 19.6 20.7 26.3 26.9 25.6 24.2 96043.2 107316 134200 163513.5 99601 144775.8 110182.5 208320 250880.9 465519.6 268113.3 331986.6 529397 575496.9 614282.6 461291.6 910482.2 415219.2 925087.1 943464 638945.8 1033354 836708.5 792961.2 1082581.5 934708 747787.6 1441503 777581.4 923182 845624 820526.4 747580.5 748291 834149.1 21312297.1 -4.499 -3.899 1.601 7.601 -5.299 4.001 -8.199 -1.899 3.001 11.801 -7.599 -6.099 3.901 3.801 5.801 -2.099 19.001 -6.699 2.801 2.601 -6.599 5.101 -0.799 -3.099 5.001 0.601 -4.699 14.701 -3.999 0.101 -2.399 -3.099 -5.199 -5.399 -3.999 - 20.245 15.206 2.562 57.768 28.084 16.004 67.231 3.608 9.003 139.253 57.752 37.203 15.214 14.444 33.646 4.408 361.021 44.883 7.843 6.763 43.553 26.016 0.639 9.607 25.005 0.361 22.085 216.106 15.996 0.01 5.757 9.607 27.034 29.154 15.996 - Appendix 3: Raw data on male BMI 90859.898 74173.561 12501.347 281967.16 135787.106 77493.199 418513.136 31316.753 78157.505 1719494.991 846119.901 623788.137 270281.482 279884.472 651998.933 85430.08 7320779.674 970632.568 252805.506 223877.249 1441858.235 867203.409 21305.223 334108.203 876066.729 12721.092 778999.141 7672846.675 567940.574 358.967 207173.333 345722.592 976344.077 1064182.605 609257.267 30405018.464 DATA ANALYSIS AND INFERENTIAL STATISTICS ON MALE AND FEMALE BMI Appendix 4: Raw data on female BMI FEMAL E 295 2739 2992 3745 4486 BM I 19. 6 23. 8 19. 6 29. 1 25. DATA ANALYSIS AND INFERENTIAL STATISTICS ON MALE AND FEMALE BMI 4488 4878 4880 4881 4835 4842 6225 8680 8681 12348 14651 16767 17765 19377 19378 19382 20278 21626 32233 33104 33106 33334 33335 34779 35035 35272 35273 35505 35506 2 21. 4 22. 0 27. 5 33. 5 20. 6 29. 9 17. 7 24. 0 28. 9 37. 7 18. 3 19. 8 29. 8 29. 7 31. 7 23. 8 44. 9 19. 2 28. 7 28. 5 19. 3 31. 0 25. 1 22. 8 30. 9 26. 5 21. 2 40. 6 21. 9 DATA ANALYSIS AND INFERENTIAL STATISTICS ON MALE AND FEMALE BMI 35507 35984 35988 36115 36502 38089 26. 0 23. 5 22. 8 20. 7 20. 5 21. 9 Laura Reid Part 1 Data Analysis ls Central tendency There are three measures of central tendency mean, median and mode. They are calculated in excel and given below in the table. MEAN MEDIA N MODE 25.74 23.9 19.6 Mean is sum of all observations divided by total number of observations. Here, mean is 25.74 implying that an average BMI of females is 25.74. The median in a series of ordered (say from low to high) values is the value at which there are just as many values less than it as there are greater than it. Here median is 23.9. So BMI of half of the females is more than 23.9 and that of other half is less than 23.9. Mode is the number occurring maximum number of times. Here mode is 19.6 implying BMI of maximum females is 19.6 Laura Reid Part 1 Data Analysis ls Measures of dispersion The three measures of dispersion namely range, standard deviation and variance is given below. range 27.2 standard deviation varianc 38.0142 e 6 Range is the difference between the maximum and minimum value. The variance and the standard deviation are both measures of the spread of the distribution about the mean. The standard deviation measures how concentrated the data are around the mean; the more concentrated, the smaller the standard deviation. Standard deviation measures spread in the same physical unit as the original data. Standard deviation and variance is given in the table above. Laura Reid Part 1 Data Analysis ls Quartiles and Interquartile range The quartiles and Interquartile range is given below Q1 Q2 Q3 IQR = Q3 - Q1 21.07 5 23.9 29.25 8.175 Outlier Any point below Q1 - 1.5*IQR and above Q3 + 1.5*IQR is considered as an outlier. The values of Q1 - 1.5*IQR and Q3 + 1.5*IQR is given below in the table. Q1 - 1.5*IQR Q3 + 1.5*IQR 8.8125 41.512 5 points below 8.8125 points above 41.5125 0 1 Hence there is only one outlier in the data. This is found with the help of conditional formatting tool in excel. The outlier is mentioned below. FEMAL E 20278 BMI 44.9 Laura Reid Part 1 Data Analysis ls Box Plot The Boxplot is an Indicator of Symmetry. If the line is close to the center of the box and the whisker lengths are the same then sample is from a symmetric population. The box plot is given below. The star in blue is the outlier. Here, BMI for females is positively skewed. That is there are very few females with high values of BMI. For a positively skewed data, mean is greater than median which is greater than mode. The same is observed from the table of central tendency. Laura Reid Part 1 Data Analysis ls Histogram The frequency distribution and corresponding histogram is given below. Count of BMI BMI 15-20 20-25 25-30 30-35 35-40 40-45 Grand Total Total 7 14 12 4 1 2 40 16 14 12 10 FREQUENCY 8 6 4 2 0 15-20 20-25 25-30 30-35 35-40 40-45 BMI I observe that distribution of BMI of females is skewed to the right. The same is observed from Box Plot and central tendency. Math215 Statistical Concepts Franklin University What is expected in a Statistical Report? A statistical report contains many elements that are in any formal report. It is in APA format It is written in the third person only (I.e., no \"I,\" \"we,\" or \"you\" used in the paper.) It contains correct grammar and punctuation It has proper formatting for references It contains an introduction, body, conclusion, references, and appendices. In addition a formal statistical report that outlines a study or experiment will have the following parts. Abstract This is limited to one paragraph of about 25 words and is a summary of the most important parts of the paper. A person reading this summary should be able to determine if this is a paper that is relevant to what they are studying or not. Introduction The introduction is a brief statement of the problem or issue that the study is investigating. It should indicate why this is an important problem. Body The following sections are usually labeled. o History This is a section with references and summaries of the previous studies that have been conducted on this particular topic. These are often studies that investigate related issues. o Methods This section delineates the process that the researcher went through. It should include how the units for the subjects were selected (sampling method), how the data was collected, what procedures (tests) were used to analyze the data. If hypotheses tests are used this section would contain the original statement of the hypotheses, requirements check. For a confidence interval, this would specify the parameter being estimated and the requirements. o Results This is a statement of the results of the tests. If hypothesis tests are run it would include the hypotheses, test statistics, p-values, and decision for each. It describes how the results can be interpreted. If confidence intervals are included it contains the point estimates and error terms as well as the intervals and an 1 260314 Math215 Statistical Concepts Franklin University interpretation of the interval. It may contain tables or graphical displays illustrating the results. As always, all graphs and tables must have titles, all scales labeled, and must be accompanied by an explanation, even if the explanation is quite obvious. o Discussion This is a discussion of the difficulties encountered, an interpretation of the results, implications of the results, how the results can be used, and an explanation of why these are important results. It also contains suggestions for future studies on the topic or related topic. Conclusion This is a quick summary of the most important findings in the study. Bibliography and Source Appendices The appendix should contain any raw data that was used or technical supplements. Some notes: In this course you are expected to follow the outline above; however, you may omit the abstract and the history sections. Any graphs or figures used should be labeled (usually with consecutive numbers-Graph 1, Graph 2,..., Figure 1, Figure 2,...) and captioned, for example, Graph 1: Histogram distribution of body weight The graphs or figures should be included in the body of the paper (and not in appendices). When referring to these figures, you should refer to \"Graph 1\" rather than \"the histogram on page 3\". Do not assume that your reader knows all about your favorite statistical software package. Your choice of \"statistics calculator\" should be as unobtrusive as possible. For example, you should write \"the parameters were fitted by least squares'' rather than \"the parameters were fitted using the TI-84 Calculator.\" 2 260314 Boxplot 3 2 1 10 15 20 25 30 35 40 45 50 FEMALE 295 2739 2992 3745 4486 4488 4878 4880 4881 4835 4842 6225 8680 8681 12348 14651 16767 17765 19377 19378 19382 20278 21626 32233 33104 33106 33334 33335 34779 35035 35272 35273 35505 35506 35507 35984 35988 36115 36502 38089 BMI 19.6 23.8 19.6 29.1 25.2 21.4 22.0 27.5 33.5 20.6 29.9 17.7 24.0 28.9 37.7 18.3 19.8 29.8 29.7 31.7 23.8 44.9 19.2 28.7 28.5 19.3 31.0 25.1 22.8 30.9 26.5 21.2 40.6 21.9 26.0 23.5 22.8 20.7 20.5 21.9 Notes: Age in years HT: height in inches WT: weight in pounds WAIST: waist circumference in cm PULSE: pulse rate in beats per minute MEAN MEDIAN MODE 25.74 23.9 19.6 range 27.2 standard deviation variance 38.01426 Q1 Q2 Q3 IQR 21.075 23.9 29.25 8.175 Q1 - 1.5*IQR Q3 + 1.5*IQR 8.8125 41.5125 points below p[oints above 0 1 Count of BMI BMI 15-20 20-25 25-30 30-35 35-40 40-45 Grand Total SYS: systolic blood pressure in mmHg DIAS: diastolic blood pressure in mmHg CHOL: cholesterol in mg BMI: body mass index LEG: upper leg length in cm ELBOW: elbow breadth in cm WRIST: wrist breadth in cm ARM: arm circumference in cm Credit: U.S. Department of Health and Human Services. National Center for Health Statistics Third National Health and Nutrition Examination Survey. ***delete the first column of ID numbers*** Total 7 14 12 4 1 2 40