Part 1: Probability Review 1. Suppose that a popular grocery store tells customers the probability they will have to wait in line for at least 3 minutes is 0.25. What does the grocery store mean by "at least"? 2. Return to Question 1. If the grocery store tells customers the probability they will have to wait in line for at most 5 minutes is 0.65, what is meant by "at most"? Did you know that the first digits of numbers in financial records often follow something known as Benford's Law? If you look at a company's financial records and choose one record at random, Benford's Law gives the following probability model for the first digit of that record (and note that the first digit cannot be zero). This is important to understand because faked numbers in invoices, expense account claims, or tax returns often display patterns that aren't present in real records. Please use this information to answer Questions 3 and 4. First digit 2 3 4 6 7 8 9 Probability 0.301 0.176 0.125 0.097 0.079 0.067 0.058 0.051 0.046 3. What is the probability that the first digit of a real financial record is an even number? 4. What is the probability that the first digit of a real financial record is not 2?5. The following probability model was constructed based on the final grades awarded to students in a large biology course. One probability is left out of this model-the probability of receiving a C. Explain what that probability must be and how you know. Grade B D E Probability 0.22 0.31 0.15 0.03 6. At a large university, the probability that a randomly chosen student lives on campus is 0.32. This means that % of the students at this university must live on campus. 7. What is wrong with the following statement? On the OSU campus, the probability is 1.15 that the color of a randomly selected vehicle from the Tuttle Parking Garage is red. Part 2: Sampling Distributions In Chapter 18, you are introduced to the idea of a sampling distribution. The purpose of the next few questions is to help you better understand what a sampling distribution is and how we can use what is known about sampling distributions to answer different kinds of questions. Have you ever wondered what percentage of college students change their major? It turns out that in the population of all college students, 75% change their major.8. Suppose you survey a random sample of n = 40 college students and 26 of these students report having changed their major. What will your sample proportion, or p, be? Please calculate this number below. 9. The number you wrote in response to Question 8 describes a sample. As you learned earlier in the semester, we call a number that describes a sample a 10. As previously mentioned, in the population of all college students, 75% have changed their major. Written as a proportion, this number would be 0.75, and we use the symbol p to denote a population proportion. We would consider the population proportion to be a because it's a number that describes a population. Imagine now that 30 of your STAT 1350 classmates each survey a random sample of n = 40 college students. Each classmate then determines the proportion of the students in the sample who report having changed their major. The graph below-called a dot plot-shows each of these sample proportions. Each proportion appears as a dot above a number line, and the number line gives us different values of our sample proportions. If any dots are stacked in a column, it would mean that more than one sample resulted in that sample proportion. For example, there are four dots stacked in a column at 0.78, and this means that four sample proportions were 0.78. 0.54 0.60 0.66 0.72 0.78 0.84 0.90 Sample proportion 11. Look carefully at the dotplot above. There is variability among the sample proportions. Does it surprise you to see variability among these sample proportions? Please explain why or why not.12. Because the sample proportions above do not systematically overestimate or underestimate the population proportion of 0.75, we'd say there is low A. variability. B. bias. C. undercoverage. D. nonsampling error. E. individuality. What do you think would happen if we were able to survey many more random samples of size n = 40 from the population of college students? In order to better understand how the proportion of students who have changed their major will vary from sample to sample, we have to think about what we would expect to see in the long run, if we could randomly sample many times from this population, compute many sample proportions, and then examine the resulting distribution of these sample proportions. Remember that a distribution shows us all values a variable can take on and how often it can take on different values. A sampling distribution is a collection of the statistics of all possible samples of a particular size taken from a particular population. If we want to use one sample to make an inference about an unknown parameter, we need to understand how our one sample would compare to all other possible samples we could have drawn randomly from the population. To get a sense of what that underlying sampling distribution will look like in this example, we will next examine what would happen if we could survey many random samples college students. We'll focus first on the sample size of n = 40, and then we'll explore what happens as the sample size gets bigger. For anyone who is interested, we used an applet on the Art of Stat site to generate the following graphics. We encourage you to explore this on your own: https://istats.shinyapps.io/SampDist_Prop/. The histogram below shows you what happens when we survey thousands of random samples of size n = 40 from the population of college students and then determine the proportion from each sample who have changed their major. What is being displayed in the histogram is many sample proportions. 6000 4000 Frequency 2000- 0.0 0.2 0.3 Sample Proportion p 13. How would you describe the shape of the distribution that is displayed in the above histogram?It turns out that the mean, or the average, of all the sample proportions in the histogram above is 0.75. You might recall that 0.75 is p, or the population proportion. All of the sample proportions in the distribution average to a value that is equal to the claimed population proportion. The standard deviation of the above distribution of sample proportions is 0.0685. We explain in our Chapter 18 coverage that the formula you would use to determine the standard deviation of the sampling distribution is: P(1-P Here, we know our sample size is n = 40 and the claimed population proportion is p = 0.75, so the standard deviation of the sampling distribution is: (0.75) (1 - 0.75) 0.1875 40 40 VO.0046875 ~ 0.0685 What happens to the distribution of sample proportions when we increase our sample size? In other words, how does sample size affect the resulting sampling distribution? Below, we have attempted to compare three sampling distributions. Each distribution is based on taking samples of a particular size from the population of college students and then computing the proportion of students who have changed their major in each sample. We start with a smaller sample of size n = 40 and then increase the sample size, first to n = 80 and then to n = 120. Sample Distribution of Sample Proportions Mean and Standard size Deviation 71 = 40 6000- 4000 Mean = 0.75 Frequent Standard deviation = 0.0685 0.0 02 03 04 Sample Proportion p 71 = 80 5000 Mean = 0.75 4020- Frequency 2000 Standard deviation = 0.0483 0.0 0.1 02 03 Sample Proportion p 7 = 120 6000 Mean = 0.75 400 Frequency 2000 Standard deviation = 0.0394 0.0 0.1 0.2 0.3 0.4 05 0.6 Sample Proportion pAs you look at the three distributions above, we hope you are focusing on a few important things: 0 Notice that regardless of the chosen sample sizes, each of the resulting sampling distributions has a symmetric, bell shape. Each ofthese distributions is approximately Normal in shape. Each distribution, regardless of sample size, is centered at a mean of0.75. This is the value of our claimed population proportion. As sample size increases, the overall range of sample proportions, as you can see as you look along the x-axis in each distribution, gets smaller, and the standard deviation, or the variability of the sample proportions, gets smaller with an increase in sample size. . Why do you think the standard deviation of the sampling distribution gets smaller as the sample size gets bigger? . Look again at the graphs on the previous page. Without performing any calculations, explain which one of the following would be more likely to happen, and why. A. Surveying a random sample ofn 120 college students and observing that the proportion who have changed their major is 0.80 or higher. B. Surveying a random sample of n 40 college students and observing that the proportion who have changed their major is 0.80 or higher. 16. Because the distribution of sample proportions based on samples of size n = 120 is approximately Normal, with a mean of 0.75 and a standard deviation of 0.0394, we can apply the Empirical Rule. A. Approximately 68% of the sample proportions are between what two values? B. Approximately 95% of the sample proportions are between what two values? C. Approximately 99.7% of the sample proportions are between what two values? 17. Think about some of the examples we worked through in our lecture coverage of Chapter 18 that involved converting a sample statistic to a z-score and then using Table B. You can even find a similar example in your textbook, in Chapter 18 (see Example 5 in Chapter 18). Based on what you learned from that coverage, what is the probability of surveying a random sample of n = 80 college students and finding the proportion who have changed their major to be 0.65 or less? As you answer this, remember the sampling distribution for samples of size n = 80 has a mean of 0.75 and a standard deviation of 0.0483. Please show your work below as you attempt to answer this question. 18. Return to the information shared in Question 17. What is the probability of surveying a random sample of n = 80 college students and finding the proportion who have changed their major to be 0.78 or higher? Again, please show your work below as you attempt to answer this