Solution Basic Probability

Question 1. Clearly distinguish in your own words among the features of the three approaches to probability. Under what circumstances would each type be more appropriate than the others? Answer 1. There are three approaches of probability which are given below: 1. Classical probability: In this probability, all elements are equally likely. We use this probability when all elements have equal chance of selection and there is no any weightage given to a particular element. This probability is defined as the number of elements in favour divided by total number of elements in the sample space.2. Relative frequency probability: In this probability, each element or class have a particular frequency or weightage, so probability for this element or class is taken as the frequency for this particular class or element divided by the total frequency. This probability used where the elements or classes have different frequencies.3. Subjective Probability: In this probability, the condition is that the sum of all probabilities for all events in the sample space should be one and probability should lies between 0 and 1 for each event. Question 2. A recent road safety study found that in 77% of all accidents the driver was wearing a seatbelt. Accident reports indicated that 92% of those drivers escaped serious injury (defined as hospitalisation or death), but only 63% of the non-belted drivers were so fortunate. What is the probability that a driver who was seriously injured was not wearing seatbelt? (Use the decision tree method to obtain your answer.) Answer 2. The decision tree with relative probabilities is given below: Probability tree diagram Driver with seatbelt (77%) Serious injury (8%) 0.77*0.08 = 0.0616 No serious injury (92%) 0.77*0.92 = 0.7084 Driver without seatbelt (23%) Serious injury (37%) 0.23*0.37 = 0.0851 No serious injury (63%) 0.23*0.63 = 0.1449 Total 1 Probability that a driver who was seriously injured was not wearing seatbelt = 0.0851 Question 3. The aptitude test scores of applicants to a university graduate program are normally distributed with mean 500 and standard deviation 60. If the university wishes to set the cut-off score for graduate admission so that only the top 10 percent of applicants qualify for admission, what is the required cut-off score? What percentage of applicants has test scores within two standard deviations of the mean? Answer 3. We are given mean = 500 and standard deviation = 60 We have to use the formula for cut-off score as given below: X = mean + z*standard deviation Now, we have to find the z value for top 10% applicants by using the normal table. That is, we have to find the z-critical value for lower 90% or probability 0.9 By using z-table or normal table, we have z = 1.28 Now, plug all values in the above formula and calculate the cut-off point X for top 10% applicants given as below: X = 500 + 1.28*60 = 576.8931 (approximately 577 ) Now, we have to find the percentage of applicants who has test scores within two standard deviations of the mean. First, we have to find the two standard deviation limits as below: Lower limit = 500 2*60 = 500 120 = 380 Upper limit = 500 + 2*60 = 500 + 120 = 620 Now, we have to find the area between the test scores 380 and 620. For finding this area, we have to use the z-score formula which is given as below: Z = ( X mean ) / standard deviation First we have to find the z score for x = 380 Z = ( 380 500) / 60 = -2 Now we have to find the z-score for x = 620 Z = ( 620 500) / 60 = 2 Now, we have to find P(-2Z2) = P(Z2) P(Z-2) We have to use z-table or normal table for finding the above probability. We have, P(Z2) = 0.97725 And P(Z-2) = 0.02275 P(-2Z2) = 0.97725 0.02275 = 0.9545 Required answer = 95.45% Question 4. For the following discuss whether a sample or a census would be preferable. Indicate any assumptions you make: An examination of dwellings in a large city to obtain information about the extent of present home insulation and the costs to bring substandard dwellings up to minimum insulation standards. Answer 4. In this case, sample or sampling method would be preferable over a census because in the large city, taking the data for all dwellings is very time consuming and it is also costly. So, by selecting some areas randomly and again selecting some dwellings randomly from selected areas will give us approximate estimate for this study. Question 5. Comment on the veracity of the following statements: a) The Central Limit Theorem is the cornerstone of statistical estimation. b) The overriding factor in determining sample size is the requirement for precision in estimates of population parameters. Answer 5 a) This statement is true because in statistical estimation, there are so many large size data sets for the purpose of estimation and central limit theorem helps us for providing approximate estimates. b) This statement is true, because approximation or margin of accuracy in determining the sample size is based on the precision. Question 6. Explain the difference between sampling error and non-sampling error. Briefly describe three types of non-sampling error. Answer 6. Sampling error is nothing but the error which arises during the data collection process during taking a sample from the given population. The non-sampling error is the error which is not occurred during the time of data collection process during the taking of sample from population. This error may be some bias observation or any other mistakes. Three types of non sampling error are given as below: Processing error: These types of error occurred during the preparation of final data processing. Estimation error: These types of error occurred during the estimation of parameters during the analysis. Respondent error: These type of error occurred if the respondent gives the incorrect information. Question 7. How large a sample is needed so that a 95% confidence interval for the mean number of years of attained education has a margin of error equal to one year? Answer 7. Here we have to use the formula for sample size which is given as below: n = (Z*SD/E)2 We are given, E = 1 And z for 95% confidence interval is given as 1.96 Required sample size = n = (1.96*SD/1)2 = 3.8416*SD2 Question 8. Excessive TV watching has been named as one factor, other than diet, for the increasing proportion of obese Australians. A recent survey asked respondents, On the average day, about how many hours do you personally watch television? The results of the survey are summarised below:a) What do the sample mean and standard deviation suggest about the likely shape of the population distribution? b) How was the standard error obtained? What does it mean? c) Interpret the 95% confidence interval? Answer 8. a) Here we are given a sample mean is 2.98 and sample standard deviation is 2.66, this means, according to the sample mean standard deviation is high. This indicates that the statistical data set values are farther away from the mean. This distribution indicates that the data values have wider curve shape. b) Standard error is obtained by using the following formula: Standard error = sample standard deviation / sqrt(n)Where n is the sample size. Standard error = 2.66 / sqrt(1324) = 0.0731 Standard error indicates that the how precise is the estimate for population parameter. c) We are given a 95% confidence interval as (2.8366, 3.1234), we can interpret this confidence interval as below: We are 95% confident that the population mean is lies between the two values 2.8366 and 3.1234 or we can also say that about there is 95% probability of finding the population mean within the two values 2.8366 and 3.1234.