Apc310
Question 1 Section A 1a
It may not be possible or practical to analyse an entire population, instead a sample of the population may
be used to predict or infer something about the population. Inferences may be point estimates which estimate a single parameter or interval estimates which represent a range of values likely to contain the parameter, known as confidence intervals. The width of the confidence interval gives us some idea about how uncertain we are about the unknown parameter. The narrower the range of the confidence interval the more accurate the estimate will be.
1b The central limit theorem, ‘whatever the distribution of the population, the mean of a large sample has an (approximate) normal distribution’. (Swift and Pitt p 498). The calculation of the confidence interval is therefore based on a standardised normal distribution with mean 0 and variance 1. Each sample mean is a single observation of a random variable designated as x . ¯ The mean value of the sample means is an unbiased estimator of the actual population mean μ and the standard deviation of the sample means is the equivalent of .
The 99% confidence interval is formed by the range which is 2.58 standard deviations either side of the mean. The stated confidence interval 82,636 to 87,364 is a range of 4,728 is the equivalent of 2.58 x 2 = 5.16 standard deviations where one standard deviation is
It is known that the population variance is 56,250,000 and therefore the population standard deviation σ is the square root of this number i.e. 7,500. Therefore, 4,728 = 5.16 . Solving for the unknown = (5.16 x 7500)/4728 = 8.185
Therefore n = 67 and the mean of the population is 82,636 + (4728/2) = 85,000.
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1c (1) Hypothesis testing is used to determine the probability that a specified hypothesis is true. The assumption in this case that the mean salary of the population is 85,000, is called the null hypothesis (H0). The alternative hypothesis (H1) is a claim to be
It may not be possible or practical to analyse an entire population, instead a sample of the population may
be used to predict or infer something about the population. Inferences may be point estimates which estimate a single parameter or interval estimates which represent a range of values likely to contain the parameter, known as confidence intervals. The width of the confidence interval gives us some idea about how uncertain we are about the unknown parameter. The narrower the range of the confidence interval the more accurate the estimate will be.
1b The central limit theorem, ‘whatever the distribution of the population, the mean of a large sample has an (approximate) normal distribution’. (Swift and Pitt p 498). The calculation of the confidence interval is therefore based on a standardised normal distribution with mean 0 and variance 1. Each sample mean is a single observation of a random variable designated as x . ¯ The mean value of the sample means is an unbiased estimator of the actual population mean μ and the standard deviation of the sample means is the equivalent of .
The 99% confidence interval is formed by the range which is 2.58 standard deviations either side of the mean. The stated confidence interval 82,636 to 87,364 is a range of 4,728 is the equivalent of 2.58 x 2 = 5.16 standard deviations where one standard deviation is
It is known that the population variance is 56,250,000 and therefore the population standard deviation σ is the square root of this number i.e. 7,500. Therefore, 4,728 = 5.16 . Solving for the unknown = (5.16 x 7500)/4728 = 8.185
Therefore n = 67 and the mean of the population is 82,636 + (4728/2) = 85,000.
1|Page
1c (1) Hypothesis testing is used to determine the probability that a specified hypothesis is true. The assumption in this case that the mean salary of the population is 85,000, is called the null hypothesis (H0). The alternative hypothesis (H1) is a claim to be
References: Field, M., Keller, L. (1998) Project Management. Open University Piff, S., Swift, L. (2006) Quantitative Methods for Business, Management and Finance. Palgrave 8|Page Miaoulis, George., R. D. Michener. (1976. An Introduction to Sampling. Dubuque, Iowa: Kendall/Hunt Publishing Company. The statistics glossary accessed 15th August 2012. http://www.stats.gla.ac.uk/steps/glossary/confidence_intervals.html. 9|Page Table 1 10 | P a g e Calculations Lease excluding promotion Lease low profit Lease mid low profit Lease mid high profit Lease high profit Total Lease including promotion Lease low profit Lease mid low profit Lease mid high profit Lease high profit Total Promotion Lease low profit promote low profit Lease low profit promote high profit Total Don 't lease Don 't lease low profit Don 't lease high profit Total Note. Values shown in thousands 0.9 x 750 0.1 x 780 675 78 753 0.6 x 40 0.4 x -20 24 -8 16 0.5 x 716 0.3 x 800 0.15 x 900 0.05 x 1,000 358 240 135 50 783 0.5 x 700 0.3 x 800 0.15 x 900 0.05 x 1,000 350 240 135 50 775 11 | P a g e