The Standard Error of the Sample - Assignment Example

BCO105 – Problem Solving: Short Answer Questions by Course: Tutor: University: Charles Darwin University Department: Deadline: Question 1 a) When looking at a sample size, it is essential to consider the standard error. Standard error plays the role of indicating reliability of mean. When the standard error is small, it means that the sample mean accurately reflects population mean. Standard error is given by , hence by increasing the sample size, standard error would decrease (Carter, et al. 2011). Depending on the type of sample, some samples would obtain a mean that is the same as the population mean. Other samples would give a mean that is very close to population mean (above or below) while others will give a mean that is further away from population mean (above or below). For a normal distribution, we can be 95% confident that population mean is within 1.96 standard errors of mean. Using this analysis, it is true for normally distributed data that the sample mean is as likely to lie above the population mean as below it when sample data is large. b) The case of trees in a timber tract: i. Sample mean used in estimating population mean: The value of parameter being estimated is , the population mean. The estimator is the sample mean. The sampling distribution of the estimator refers to the probability related to a statistic when a sample of four was randomly drawn from the population (Johnson & Kuby, 2011). The value of the estimate is 8.28 as calculated below. sample of 4 trees 8.76 7.93 7.88 8.55 sum 33.12 mean 8.28 The magnitude of sample error is: ii. Sample median and the sample midrange: Sample median and sample midrange: Ordered sample 7.88 7.93 8.55 8.76 In order to determine whether mean, median, or midrange has the smallest sampling error, it is important to compare their values with . Through comparison, it is evident that midrange has the smallest sampling error given that its value is close to . The estimator might not necessarily have the lowest sampling error for another sample because midrange is affected by extremely high or low values. Question 2 a) Indicating if a correct decision, a Type I error or a Type II error was made: i. H0: μ = 1.5 litres. The decision was to not reject H0 and μ is actually 1.5 litres. This is a correct decision since the decision accepts true hypothesis. ii. H0: μ = 1.5 litres. The decision was to reject H0 and μ is actually 1.5 litres. This is a Type I error since a true null hypothesis has been rejected. iii. H0: μ = 1.5 litres. The decision was to reject H0 and μ is actually 1.6 litres. This is a correct decision given that a false null hypothesis has been rejected. b) Gallop survey: i. Null and alternate hypotheses: ii. Significance level is . The critical values are and . Decision rule is to reject the and accept the if the . If , we will fail to reject the null hypothesis. iii. Test statistic: iv. The P-value calculated as: is greater than 0.05 hence we do not reject null hypothesis. Question 3 a) While correlation is used to determine existence of linear relationship between variables, regression analysis describes the nature of the relationship i.e. whether linear or nonlinear, positive or negative (Weiers, 2010). The strength of linear relationship between variables will be deduced using correlation coefficients. b) Regression analysis of the relationship between number of houses sold and interest rates: By conducting regression analysis, the output is as tabulated below. The resulting regression equation is: The regression equation shows that number of houses sold has an inverse relationship with interest rate. This means that when interest rate is increased by 1 point, number of houses sold declines by 8.13. The intercept of 127.33 indicates that approximately 127 houses are sold when interest rate is zero. The inverse relationship between number of houses sold and interest rate is further validated in the scatter plot below with a downward sloping trend line. One of the limitations in this analysis is the use of a line of best fit to develop the regression equation. This means that this analysis is more about estimation. In order to obtain accurate results, additional data should be included in the analysis. This will serve the purpose of reducing errors, which are apparent in this analysis. c) Houses to be sold when interest rate is 5% p.a. is calculated as follows: Reservations about prediction are explained by finding the prediction interval as follows: Step 1: Finding the values of: Step 2: Finding the value of Step 3: Obtaining From regression equation, Step 4: Solving for Lower limit: Upper limit Hence prediction interval: From the calculations above, we can be sure that the stated interval contains the actual value of y. Question 4 a) An estimator can be viewed as a statistic that is applied in estimating a parameter (Devore, 2015). As an example, sample mean is classified as an estimator because it is used to explain the population. On the other hand, an estimate indicates the value of a quantity that is unknown based on data that has been observed. Briefly, an estimate is the value that results from a calculation and serves the purpose of indicating the value of the population. One of the properties of a reliable estimator is that it should be unbiased (Kennedy, 2003). It means that the estimator should have an expected value that is equal to the parameter being estimated. The second property is that the estimator be efficient. It means that it should have relatively small variance (Kothari, 2004). Thirdly, a good estimator should be sufficient hence, it should utilize information from the sample as much as possible. Finally, a good estimator should be consistent. This means that it should approach value of the population with an increasing sample. b) The sample mean for the airline reservation records of a random 100 days is as follows: Number of No-Shows (x) Number of days (f) fx 0 20 0 -1.5 2.25 1 37 37 -0.5 0.25 2 23 46 0.5 0.25 3 15 45 1.5 2.25 4 4 16 2.5 6.25 5 0 0 3.5 12.25 6 1 6 4.5 20.25 Sum of (fx) 150 sum of (f) 100 Mean, xbar 1.5 43.75 0.4375 0.661438 Constructing 99 percent confidence interval for μ: A different random sample would have provided a different confidence interval. When sample is increased, sampling error is reduced effectively affecting confidence interval. Conversely, a lower random sample increases sampling error and confidence interval. c) Sample size required to estimate population mean within 0.2 of a SD with 99% confidence: Question 5 a) A normal distribution is a continuous, symmetric, and bell-shaped distribution of a variable (Berenson, 2012). It is symmetrical and has the same mean, median, and mode located at the center of the distribution. The distribution is unimodal and continuous without gaps. It is important to note further that the curve never touches x-axis. The two parameters that determine its location and shape are standard deviation and mean respectively (Bluman, 2001). b) Family of normal probability distribution: i. The distribution is characterized by mean and standard deviation i.e. . For each pair , it corresponds to another probability distribution within the family of normal distribution (Mian and Mohammed, 2002) ii. The expressions for the standardized normal variable Z are as follows: c) NT Trucking Company: i. Proportion of trucks expected to travel between 80,000 and 120,000 kms per year ii. Percentage of trucks expected to travel either below 60,000 or above 140,000 kms per year iii. Kilometres that will be travelled by at least 80 percent of the trucks: Reference List Berenson, M, David, L, Judith, W, Nicola, J, O'Brien, M 2012, Business Statistics: Concepts and Applications, Pearson Higher Education, Upper Saddle River, N.J. Bluman, AG 2001, Elementary Statistics: A Step by Step Approach, McGraw Hill, Boston. Carter, H, Griffiths, W & Lim, G 2011, Principles of Econometrics, John Wiley & Sons, New York. Devore, J 2015, Probability and Statistics for Engineering and the Sciences, Cengage Learning, Mason, OH. Kennedy, P 2003, A Guide to Econometrics, MIT Press, Cambridge, MA. Kothari, CR 2004, Research Methodology: Methods and Techniques, New Age International, New Delhi. Mian, MA & Mohammed, AM 2002, Project Economics, and Decision Analysis: Probabilistic models, PennWell Books, Tulsa. Weiers, R 2010, Introduction to Business Statistics, Cengage Learning, Mason, OH. Johnson, R & Kuby, P 2011, STAT 2, Cengage Learning, Mason, OH. Read More