TOPIC 4 – Topic Exercises2

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STATISTICS

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Statistics

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Apr 3, 2024

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Example 1 A statistics class has a total of 45 students. A sample of 10 students is to be selected from that population. The name of each student is written on a piece of paper and deposited in a box. After they have been thoroughly mixed, the first selection is made by drawing a piece of paper from the box without looking at it. This process is repeated until the sample of 10 students is chosen. Random Sampling is being used. Example 2 A statistics class has a total of 45 students. A sample of 10 students is to be selected from that population. For this population, we would select every 4th (45/10) student. Random sampling is used in the selection of the first name. Then, select every 4 th name on the list thereafter. Systematic Sampling is being used. Example 3 Suppose we want to study the advertising expenditures for the 352 largest companies in the U.S to determine whether firms with high returns on equity (a measure of profitability) spent more of each sale on advertising than firms with a low return. Say 50 firms are needed for intensive study. To make sure that the sample is a fair representation of population, the companies are grouped on percent return on equity and a sample proportional to the relative size of the group is randomly selected. Stratified sampling was used to allow researchers to draw conclusions about the relationship between return on equity and advertising expenditures while making sure that companies with different profitability levels are adequately represented. Example 4 Suppose you want to determine the views of residents in Selangor about state environmental protection policies. Cluster sampling can be used by subdividing the state into districts, selecting at random say 4 districts (e.g. Gombak District, Klang District, Petaling District and Kuala Selangor District), then take samples of the residents in each of these districts and interviewing them. Cluster sampling is the most effective. You can divide the states into districts and randomly select a subset of them. The obtain a representative sample that captures regional diversity and ensures the findings are applicable to the whole population. This way will simply data collection, minimize cost and facilitate a more in-depth analysis of the residents perspective. 1

Example 5 Your firm has won a contract to do a study for an air travel industry. The industry would like to know the passengers’ thoughts on air travel factors such as ticket purchase, services, safety, comfort, economic growth, and security. The editors have given you their passengers’ database and 20 questions they would like to ask. You know that it is too expensive to contact all the readers, so you need to determine a way to contact a representative sample of the entire readership population. a. What sampling technique would you use to select the sample for the study? Justify your chosen sampling technique. I would use Stratified random because it would ensure representation from different subgroups and provide a comprehensive view of the air travel industry’s readership. b. When the Resource Systems Group did a similar study, they used an Internet survey. They sent out 1000 invitations to participate in the survey and received 621 completed surveys. What type of sampling method do they use? Identify the flaws in their approach. They used the convenience sampling method. The flaws would be that the survey is sent through the internet, and it could lead to bias results., lack of representation and generalizability for the broader readership population. Example 6 The population is the weight of six watermelons (in kg) displayed in a fruit carnival "guess the weight" game booth. You are asked to guess the average weight of the six watermelons by taking a random sample without replacement from the population. Watermelon A B C D E F Weight (in kg) 9 6 7 4 5 8 a. What is the population mean? 6.5 KG b. What is the sampling distribution of the sample mean for a sample of size 2? 15 c.What is the mean of the sampling distribution of sample means? 6.5 KG d. What observation can be made about the population and the sampling distribution of the sample means? The sampling distribution is less variable than the population itself. The larger the samples the more the sample means will be closer to the population means. Example 7 A gas supplier maintains a team of engineers who are available to deal with leaks reported by customers. Most reported leaks can be dealt with quickly, but some require a long time. The time taken to deal with reported leaks is found to have a mean of 65 minutes and a standard deviation of 60 minutes. If the times may be modelled by a normal distribution, estimate the probability that a. it will take more than 185 minutes to deal with a reported leak. 0.02275 b. it will take between 50 minutes and 125 minutes to deal with a reported leak. 0.440 c.the average time to deal with a random sample of 90 reported leaks is less than 70 minutes. 0.785 Example 8 The average amount of time that people spend going through airport security for flights at a busy airport is 21 minutes with a standard deviation of 4.2 minutes. a. What is the probability that the average wait for 40 people is more than 22 minutes? 0.0661 2

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