The heights of 16-18 year old young adults varies between males and females Essays

The heights of 16-18 year old young adults varies between males and females Essays The heights of 16-18 year old young adults varies between males and females Essay The heights of 16-18 year old young adults varies between males and females Essay ? à ¯Ã‚ ¿Ã‚ ½) then n ~ N ( à ¯Ã‚ ¿Ã‚ ½ , ? à ¯Ã‚ ¿Ã‚ ½ ) provided that n is n sufficiently large. (A good rule of thumb is n ? 30) By using the Central Limit Theorem it enables me to make a prediction about the distribution of the sample mean even if I dont know the parent population. Providing the sample is large enough I am able to be confident that the mean of the sample is close to the population mean. Heights of 16-18 year old females x (inches) x^2 F xF x^2F 5ft 60 3600 1 60 3600 5ft 1inc 61 3721 5 305 18605 5ft 2inc 62 3844 8 496 30752 5ft 3inc 63 3969 10 630 39690 5ft 4inc 64 4096 5 320 20480 5ft 5inc 65 4225 4 260 16900 5ft 6inc 66 4356 6 396 26136 5ft 7inc 67 4489 3 201 13467 5ft 8inc 68 4624 2 136 9248 5ft 9inc 69 4761 2 138 9522 5ft 10inc 70 4900 1 70 4900 5ft 11inc 71 5041 1 71 5041 6ft 72 5184 1 72 5184 6ft 1inc 73 5329 1 73 5329 Total 50 3228 208854 Mean 64.56 Variance 9.0864 Heights of 16-18 year old males x (inches) x^2 F xF x^2F 5ft 2inc 62 3844 1 62 3844 5ft 3inc 63 3969 0 0 0 5ft 4inc 64 4096 2 128 8192 5ft 5inc 65 4225 2 130 8450 5ft 6inc 66 4356 2 132 8712 5ft 7inc 67 4489 3 201 13467 5ft 8inc 68 4624 3 204 13872 5ft 9inc 69 4761 6 414 28566 5ft 10inc 70 4900 4 280 19600 5ft 11inc 71 5041 7 497 35287 6ft 72 5184 10 720 51840 6ft 1inc 73 5329 6 438 31974 6ft 2inc 74 5476 2 148 10952 6ft 3inc 75 5625 1 75 5625 6ft 4inc 76 5776 1 76 5776 Total 50 3505 246157 Mean 70.1 Variance 9.13 A 99% confidence interval for the height of females aged 16-18 = 64.56 Sà ¯Ã‚ ¿Ã‚ ½ = 9.0864 n = 50 ? (0.995) = 2.5758 ?à ¯Ã‚ ¿Ã‚ ½ n-1 ( n ) Sà ¯Ã‚ ¿Ã‚ ½ n-1 = 50 x 9.0864 = 9.272 49 à ¯Ã‚ ¿Ã‚ ½ = à ¯Ã‚ ¿Ã‚ ½ x ? n-1 V n à ¯Ã‚ ¿Ã‚ ½ = 64.56 à ¯Ã‚ ¿Ã‚ ½ 2.5758 x 9.272à ¯Ã‚ ¿Ã‚ ½ V 50 à ¯Ã‚ ¿Ã‚ ½ = 64.56 à ¯Ã‚ ¿Ã‚ ½ 2.5758 x V 9.272 7.071 à ¯Ã‚ ¿Ã‚ ½ = 64.56 à ¯Ã‚ ¿Ã‚ ½ 2.5758 x 3.045 7.071 à ¯Ã‚ ¿Ã‚ ½ = 64.56 à ¯Ã‚ ¿Ã‚ ½ 2.5758 x 0.431 à ¯Ã‚ ¿Ã‚ ½ = 64.56 à ¯Ã‚ ¿Ã‚ ½ 1.110 à ¯Ã‚ ¿Ã‚ ½ is in the interval [ 63.45 , 65.67 ] 63.45 x 65.67 A 90% confidence interval for the height of females aged 16-18 = 64.56 Sà ¯Ã‚ ¿Ã‚ ½ = 9.0864 n = 50 ? (0.95) = 1.6449 ?à ¯Ã‚ ¿Ã‚ ½ n-1 ( n ) Sà ¯Ã‚ ¿Ã‚ ½ n-1 = 50 x 9.0864 = 9.272 49 à ¯Ã‚ ¿Ã‚ ½ = à ¯Ã‚ ¿Ã‚ ½ x ? n-1 V n à ¯Ã‚ ¿Ã‚ ½ = 64.56 à ¯Ã‚ ¿Ã‚ ½ 1.6449 x 9.272à ¯Ã‚ ¿Ã‚ ½ V 50 à ¯Ã‚ ¿Ã‚ ½ = 64.56 à ¯Ã‚ ¿Ã‚ ½ 1.6449 x V 9.272 7.071 à ¯Ã‚ ¿Ã‚ ½ = 64.56 à ¯Ã‚ ¿Ã‚ ½ 1.6449 x 3.045 7.071 à ¯Ã‚ ¿Ã‚ ½ = 64.56 à ¯Ã‚ ¿Ã‚ ½ 1.6449 x 0.431 à ¯Ã‚ ¿Ã‚ ½ = 64.56 à ¯Ã‚ ¿Ã‚ ½ 0.709 à ¯Ã‚ ¿Ã‚ ½ is in the interval [ 63.85 , 65.27 ] 63.85 x 65.27 A 99% confidence interval for the height of males aged 16-18 = 70.1 Sà ¯Ã‚ ¿Ã‚ ½ = 9.13 n = 50 ? (0.995) = 2.5758 ?à ¯Ã‚ ¿Ã‚ ½ n-1 ( n ) Sà ¯Ã‚ ¿Ã‚ ½ n-1 = 50 x 9.13 = 9.316 49 à ¯Ã‚ ¿Ã‚ ½ = à ¯Ã‚ ¿Ã‚ ½ x ? n-1 V n à ¯Ã‚ ¿Ã‚ ½ = 70.1 à ¯Ã‚ ¿Ã‚ ½ 2.5758 x 9.316à ¯Ã‚ ¿Ã‚ ½ V 50 à ¯Ã‚ ¿Ã‚ ½ = 70.1 à ¯Ã‚ ¿Ã‚ ½ 2.5758 x V 9.316 7.071 à ¯Ã‚ ¿Ã‚ ½ = 70.1 à ¯Ã‚ ¿Ã‚ ½ 2.5758 x 3.052 7.071 à ¯Ã‚ ¿Ã‚ ½ = 70.1 à ¯Ã‚ ¿Ã‚ ½ 2.5758 x 0.432 à ¯Ã‚ ¿Ã‚ ½ = 70.1 à ¯Ã‚ ¿Ã‚ ½ 1.113 à ¯Ã‚ ¿Ã‚ ½ is in the interval [ 68.99 , 71.21 ] 68.99 x 71.21 A 90% confidence interval for the height of males aged 16-18 = 70.1 Sà ¯Ã‚ ¿Ã‚ ½ = 9.13 n = 50 ? (0.95) = 1.6449 ?à ¯Ã‚ ¿Ã‚ ½ n-1 ( n ) Sà ¯Ã‚ ¿Ã‚ ½ n-1 = 50 x 9.13 = 9.316 49 à ¯Ã‚ ¿Ã‚ ½ = à ¯Ã‚ ¿Ã‚ ½ x ? n-1 V n à ¯Ã‚ ¿Ã‚ ½ = 70.1 à ¯Ã‚ ¿Ã‚ ½ 1.6449 x 9.316à ¯Ã‚ ¿Ã‚ ½ V 50 à ¯Ã‚ ¿Ã‚ ½ = 70.1 à ¯Ã‚ ¿Ã‚ ½ 1.6449 x V 9.316 7.071 à ¯Ã‚ ¿Ã‚ ½ = 70.1 à ¯Ã‚ ¿Ã‚ ½ 1.6449 x 3.052 7.071 à ¯Ã‚ ¿Ã‚ ½ = 70.1 à ¯Ã‚ ¿Ã‚ ½ 1.6449 x 0.432 à ¯Ã‚ ¿Ã‚ ½ = 70.1 à ¯Ã‚ ¿Ã‚ ½ 0.711 à ¯Ã‚ ¿Ã‚ ½ is in the interval [ 69.39 , 70.81 ] 69.39 x 70.81 Comparing confidence intervals Below I have presented my confidence intervals graphically: This one shows the confidence intervals of 90% and 99% for the heights of females aged 16-18 years old. This one shows the confidence intervals of 90% and 99% for the heights of males aged 16-18 years old. As you can see by these to diagrams the bigger the confidence the more confident I am that the population will lie between the two values. However the smaller the confidence the less confident I am that the two values will lie between them two values. When you compare both the male and female confidences graphically you can see that the female heights are concentrated down the left side of the scale whereas the male heights are situated on the right side of the scale. This represents that males are taller than females. Conclusion When I relate my evidence that I have obtained from doing confidence intervals and working out the mean and variance values of both populations to my hypothesis, the distribution of the heights of males is bigger than the heights of females. I calculated two levels of confidence for both males and females; 90% and 99%. When I collected my data it wasnt random, however I did try and make sure that when I collected my data I ensured that I wasnt biased, I did this by not paying must attention to the persons height and the majority of people I asked were sitting down at a table, also I didnt collect my data at a modelling studio where the majority of the population would be over 6ft, If I had time I would have extended my problem by widening the age of males and females as I only managed to obtain the heights if males and females aged 16-18. I could have increased this by including younger and older populations. I could have concentrated my data collection on children aged 10-15 to compare the difference in height of all 10 year olds up to the age of 15. I could have also compared the difference in the heights of boys and girls as the age increases from age 10-15 to see if girls increase in height more gradually than boys do as they might shoot up at a certain age, or maybe it occurs the other way round. If I decided to keep to the data I had collected I could have adapted it by seeing if that the taller a person is, the bigger their shoe size would be and the shorter a person is, the smaller their shoe size would be. Formulae and definitions Variance Sà ¯Ã‚ ¿Ã‚ ½ Mean Standard error s.e. = ? s.e. Vn Unbiased estimate ( n-1 ) Sà ¯Ã‚ ¿Ã‚ ½ = ? n-1 n Confidence intervals C.I. à ¯Ã‚ ¿Ã‚ ½ = à ¯Ã‚ ¿Ã‚ ½ x ? n-1 V n