Probability and Statistics Coursework - 550 Words

Probability and Statistics (Coursework Sample) Content: Probability and StatisticsSubmitted By[Name][Roll Number]University AffiliationSubmitted To[Name]September 2014Part OneConfidence IntervalConfidence interval is used in Statistic to indicate the precision and accuracy of a measurement. Once applied in measurements, it shows the estimation reliability. Thus, identifying the closeness of a measurement to the estimation made (Singh Masuku, 2012).In this case, the teacher seeks to determine the mathematical ability of graduating high school seniors in her state. From a random sample of 15 seniors, she found a;X= 1270 à ¢Mean (Sample Mean)à Ã†â€™ = 160 à ¢Standard Deviationn = 15 à ¢Sample sizeThe requirement here is to obtain an array of scores for the 15 seniors, where 95% will almost certainly fall, considering the Sample Mean of 1270 and the SD of 160.Singh Masuku, (2012) in their statistical tests used the formula;Confidence Interval = x  a/2 (à Ã†â€™/à ¢Ã… ¡n)Where x = Sample Meann = Sample sizeà Ã† ’ = Standard Deviationa = [1 à ¢Ã¢â€š ¬ confidence level/100] à ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬.1- 95/100 = 1 - 0.95 = 0.05Ê = Value of t as indicated in the t table à ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬..[0.05/2 = 2.145]Replacing the values;Confidence Interval = à ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬... [1270  2.145(160/3.873)]= à ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬.. [1270  88.61]= à ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬. [1181.39 to 1358.61]=à ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬ You get the lower and upper intervalsResult: The teacher established that 95% of the scores remained in the range of 1181.39 to 1358.61This deduces that the teacher was 95% confide nt that the mathematical ability for the graduating seniors lay in this range when given the 32-item test.Part TwoThe Central Limit TheoremThis theorem states that,à ¢Ã¢â€š ¬ in statistics, a sampling distribution will tend to normality when an attempt is made to increase the sample size.à ¢Ã¢â€š ¬ Distributions tend to be closer to a normal distribution. The shape of a sample distribution approaches the normal distribution or "bell curve" shape as the sample à ¢Ã¢â€š ¬Ã‹Å"nà ¢Ã¢â€š ¬ increasesà ¢Ã¢â€š ¬ as shown in figure 1.Wooldridge, (2009) gives an illustration of one throwing a die; there is an equal chance that any of the numbers 1 to 6 would be obtained and the result could as illustrated in the left topmost square of Figure 1.Example 1If two dice are thrown, the range of values would be from 2 to 12 i.e., there are 11 possible outcomes instead of just 6 and the shape approaches a cone as depicted in the middle square on the left. This is because the probability of obtai ning the center values is higher than getting the two extreme values 2 and 12.With 6 dice, the range of possible values would be from 6 to 36 or 31 possible outcomes and the shape of the probability distribution now becomes that of a bell as shown in the lower right square of the diagram below. The center values would have more chances of emerging than the values at both ends of the range. In other words, there are more possible combinations of values that would give a sum of 18 from the 6 dice than a sum of 36 or sum of 6. Figure SEQ Figure \* ARABIC 1 Possible OutcomesExample 2Let a random variable X with the random sample say x1, x2à ¢Ã¢â€š ¬ xn, be normally distributed with the expected value  and variance à Ã†â€™2. The sample average will be given by:17208526225500Sn=x1+x2+à ¢Ã¢â€š ¬+xnnAs n approaches infinity, Sn approaches  which therefore implies that: Lim Sn - Â=0 hence the sample mean converges to 0n tending to infinityThe variance of the mean is given by:170624538100081175131750077660533655000Var (Sn) = Var (x1+x2+à ¢Ã¢â€š ¬+xn) à ¢Ã¢â€š ¬... Probability and Statistics Coursework - 550 Words Probability and Statistics (Coursework Sample) Content: Probability and StatisticsSubmitted By[Name][Roll Number]University AffiliationSubmitted To[Name]September 2014Part OneConfidence IntervalConfidence interval is used in Statistic to indicate the precision and accuracy of a measurement. Once applied in measurements, it shows the estimation reliability. Thus, identifying the closeness of a measurement to the estimation made (Singh Masuku, 2012).In this case, the teacher seeks to determine the mathematical ability of graduating high school seniors in her state. From a random sample of 15 seniors, she found a;X= 1270 à ¢Mean (Sample Mean)à Ã†â€™ = 160 à ¢Standard Deviationn = 15 à ¢Sample sizeThe requirement here is to obtain an array of scores for the 15 seniors, where 95% will almost certainly fall, considering the Sample Mean of 1270 and the SD of 160.Singh Masuku, (2012) in their statistical tests used the formula;Confidence Interval = x  a/2 (à Ã†â€™/à ¢Ã… ¡n)Where x = Sample Meann = Sample sizeà Ã† ’ = Standard Deviationa = [1 à ¢Ã¢â€š ¬ confidence level/100] à ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬.1- 95/100 = 1 - 0.95 = 0.05Ê = Value of t as indicated in the t table à ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬..[0.05/2 = 2.145]Replacing the values;Confidence Interval = à ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬... [1270  2.145(160/3.873)]= à ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬.. [1270  88.61]= à ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬. [1181.39 to 1358.61]=à ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬Ãƒ ¢Ã¢â€š ¬ You get the lower and upper intervalsResult: The teacher established that 95% of the scores remained in the range of 1181.39 to 1358.61This deduces that the teacher was 95% confide nt that the mathematical ability for the graduating seniors lay in this range when given the 32-item test.Part TwoThe Central Limit TheoremThis theorem states that,à ¢Ã¢â€š ¬ in statistics, a sampling distribution will tend to normality when an attempt is made to increase the sample size.à ¢Ã¢â€š ¬ Distributions tend to be closer to a normal distribution. The shape of a sample distribution approaches the normal distribution or "bell curve" shape as the sample à ¢Ã¢â€š ¬Ã‹Å"nà ¢Ã¢â€š ¬ increasesà ¢Ã¢â€š ¬ as shown in figure 1.Wooldridge, (2009) gives an illustration of one throwing a die; there is an equal chance that any of the numbers 1 to 6 would be obtained and the result could as illustrated in the left topmost square of Figure 1.Example 1If two dice are thrown, the range of values would be from 2 to 12 i.e., there are 11 possible outcomes instead of just 6 and the shape approaches a cone as depicted in the middle square on the left. This is because the probability of obtai ning the center values is higher than getting the two extreme values 2 and 12.With 6 dice, the range of possible values would be from 6 to 36 or 31 possible outcomes and the shape of the probability distribution now becomes that of a bell as shown in the lower right square of the diagram below. The center values would have more chances of emerging than the values at both ends of the range. In other words, there are more possible combinations of values that would give a sum of 18 from the 6 dice than a sum of 36 or sum of 6. Figure SEQ Figure \* ARABIC 1 Possible OutcomesExample 2Let a random variable X with the random sample say x1, x2à ¢Ã¢â€š ¬ xn, be normally distributed with the expected value  and variance à Ã†â€™2. The sample average will be given by:17208526225500Sn=x1+x2+à ¢Ã¢â€š ¬+xnnAs n approaches infinity, Sn approaches  which therefore implies that: Lim Sn - Â=0 hence the sample mean converges to 0n tending to infinityThe variance of the mean is given by:170624538100081175131750077660533655000Var (Sn) = Var (x1+x2+à ¢Ã¢â€š ¬+xn) à ¢Ã¢â€š ¬...