Regression Analysis - Assignment Example

Short Answer Questions by Course: Tutor: University: Charles Darwin University Department: Deadline: Question 1 a) It is true that when the sample size is large, the sample mean is likely to lie above the population mean as below it. A research by Bryman and Cramer (2012), some samples will have a sample mean that is the same as population mean. In the same vein, normally distributed population will have samples that are normally distributed. If sample is large, sample mean will be the same as or close to population mean. However, other samples will deviate from the population mean. b) Mean growth rate: Sample of 4 trees measures: i. Sample mean applied in estimating population mean: The value of the parameter being estimated is population mean of 8.38 cms. The sample mean is the estimator in this particular context. Sampling distribution of the estimator captures the chance or rather the probability of sample mean when the mentioned samples are drawn from the population in a random manner. The estimate is obtained as follows: Heiman (2011) gives sampling error as: ii. Sample median and midrange: The values of sample median and midrange are obtained after arranging the sample data from lowest to highest values as shown below: 7.88 7.93 8.55 8.76 In order to obtain the median, we use the values at the middle after organising the data. The above midrange is obtained by finding the average of the lowest and highest values. Midrange has the lowest sampling error since it has a value that is near population mean of 8.38. The estimator would not have the smallest sampling error for another sample because midrange will fluctuate depending to the lowest and highest values. Question 2 a) Decisions in hypothesis testing: i. The decision was correct because it is the acceptance of a true hypothesis. ii. This is a Type I error given that a true null hypothesis was rejected. iii. The decision is correct because of the rejection of a false null hypothesis. b) The mean charitable contribution according to a survey conducted by Gallop was . Sample of 2012 tax return was used on hypothesis testing: i. Null and alternate hypotheses are as follows: ii. The critical value and rejection region of the test statistic using is as follows: The critical values are +1.961 and -1.961 for a two tailed test and alpha of 0.05 with a degree of freedom of 2011. Decision rule is to reject null hypothesis and accept the alternate hypothesis if the calculated t is less than or equal to - .Otherwise we will fail to reject the null hypothesis. iii. Sample of 200 tax returns shows a sample mean of $1160 and SD of 840. The value of test statistic is: We will use t-test since the value of population standard deviation is unknown. The first step is to find the standard error: iv. The corresponding area for . Given that P-value is greater than 0.05, we fail to reject null hypothesis. Question 3 a) Regression analysis tells us whether relationship between variables is linear or nonlinear. Besides, regression analysis gives information on whether the relationship is positive or negative (Thompson, 2012). On the other hand, correlation determines whether there is a linear relationship between variables and the strength of such relationship. b) The regression analysis conducted below seeks to explain the relationship that exists between number of houses sold and interest rate. SUMMARY OUTPUT Regression Statistics Multiple R 0.902286456 R Square 0.814120849 Adjusted R Square 0.790885955 Standard Error 6.081936454 Observations 10 ANOVA   df SS MS F Significance F Regression 1 1296.080392 1296.08 35.03872 0.000354 Residual 8 295.9196082 36.98995 Total 9 1592         Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Lower 95.0% Upper 95.0% Intercept 127.3364619 17.05934719 7.464322 7.17E-05 87.99754 166.6754 87.99754 166.6754 Interest Rate (X) -8.130993675 1.373629254 -5.91935 0.000354 -11.2986 -4.9634 -11.2986 -4.9634 From the regression output above, the regression equation is: Starting with summary regression output, number of houses sold declines with an increase in interest rate. A 1% in increase in interest rate leads to an 8.13% decrease in house sales. At an interest rate of zero, 127 houses are sold. The downward sloping line plot above summarizes the relationship between number of houses sold and interest rate. The reservation is that variables in the equation will change when a large sample is included in the study (Black, 2011). c) At an interest rate of 5%, number of houses sold are: Reservation surrounding the forecasted value of 86.6814 is determined by finding the prediction interval as indicated below (Siegel, 2011): First, we calculate Secondly, we use the predicted value of y when interest rate is 5% The third step is to find given The fourth step is to solve for It means that the predicted value of y lies between . Question 4 a) An estimator is a statistic like the one sample mean used in estimation of a parameter. An estimate is the value of the unknown obtained using a sample. A sample mean sales of 25 represents an estimate for the population number of customers. Kothri (2004) outlined the following properties of a good estimator: Unbiased: The expected value of the estimator should have an expected value that is equal to the estimated parameter. Efficient: The estimator should have small variance. Consistent: The estimator should approach the population parameter when sample is being increased. d) Constructing 99% confidence interval for population mean: Shows (x) days (f) 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 Constructing 99 percent confidence interval for μ: A different random sample of 100 days would not give the same confidence interval because standard error will increase or decrease depending on size of the sample. e) A sample size that is required to estimate the population mean to within 1/5 of SD with 99% confidence is: Question 5 a) A normal distribution is characterized by its symmetrical nature hence the mean, median, and mode are the same (Sharma, 2012). Secondly, the distribution is unimodal i.e. has a single mode, which effectively gives it a bell-shaped character. Finally, the distribution of data is continuous. In a normal distribution, mean and standard deviation are used to describe the location and shape of the distribution. i. The probability distribution is a normal distribution with means of 20, 0, and 5 and the standard deviation of 9, 9, and 1. ii. The expressions for the standardized normal variable Z are as follows: a) NT Trucking Company determined that on an annual basis the distance travelled per truck is normally distributed: 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. The number of kilometers that will be travelled by at least 80 percent of the trucks: Reference List Black, K 2011, Business Statistics: For Contemporary Decision Making, John Wiley & Sons, Somerset, NJ. Bryman, A, Cramer, D 2012, Quantitative Data Analysis with IBM SPSS 17, 18 & 19: A Guide for Social Scientists, Routledge, Hove. Heiman, G 2011, Behavioral Sciences STAT, Cengange Learning, Mason, OH. Kothri, CR 2004, Research Methodology: Methods and Techniques, New Age International, New Delhi. Sharma, JK 2012, Business Statistics, Pearson Education, New Delhi. Siegel, A 2011, Practical Business Statistics, Academic Press, Burlington, MA. Thompson, S 2012, Sampling, John Wiley and Sons, New York. Read More