HW 7.1-7.3 FA10 1 of 5 http://www.webassign.net/v4cgicricket3@uga/student.pl?c=2010092913... Chris O'Neal STAT 7770 FA10, Fall 2010 Instructor: Chandler Pike WebAssign HW 7.1-7.3 FA10 (Homework) Current Score : 0 / 100 1. Due : Monday, October 4, 2010 05:30 PM EDT 0/8 submissions -/26 points According to a recent Current Population Reports, the population distribution of number of years of education for self-employed individuals in the United States has a mean of 13.3 and a standard deviation of 2.2. (a) Identify the random variable X whose distribution is described here. number of self-employed individuals number of years of education number of educated individuals in the United States sh is ar stu ed d vi y re aC s ou ou rc rs e eH w er as o. co m number of self-employed individuals in the United States (b) Find the mean and standard error of the sampling distribution of Mean = for a random sample of size 90. (1 decimal place) Standard Error = (5 decimal places) (c) Find the mean and standard error of the sampling distribution of Mean = for a random sample of size 399. (1 decimal place) Standard Error = (5 decimal places) (d) Describe the effect of increasing n. The mean increases. The mean stays the same. The mean decreases. The standard error increases. The standard error stays the same. Th The standard error decreases. (e) If a sample of size 399 is selected from this population, what is the probability that the sample average will be less than 13.7? probability = (4 decimal places) (f) Suppose a random sample of 4 self-employed individuals is selected at random from the population described above. The sample data values are: 14, 10, 13, 12. (i) The sample mean is (ii) The sample standard deviation is 2. (2 decimal places) (4 decimal places) -/16 points https://www.coursehero.com/file/6762027/stat-ch-71-hw/ 9/29/2010 9:44 AM HW 7.1-7.3 FA10 2 of 5 http://www.webassign.net/v4cgicricket3@uga/student.pl?c=2010092913... At a university, 61% of the 7400 students are female. The student newspaper reports results of a survey of a random sample of 50 students about various topics involving alcohol abuse, such as participation in binge drinking. They report that their sample contained 26 females. Gender is a categorical variable. To set up a probability distribution where the random variable X represents gender, Let X = 1 for a female and let X = 0 for a male. (b) Show the population distribution of gender at this university. (Use two decimal places.) Gender X Probability Female 1 Male 0 (c) Show the data distribution of gender for this sample. (Use two decimal places.) Gender X Proportion Female 1 Male 0 sh is ar stu ed d vi y re aC s ou ou rc rs e eH w er as o. co m (d) Consider the sampling distribution of the sample proportion of females in a sample of size 50. State its mean and standard error. Mean = Standard error = 3. (2 decimal places) (4 decimal places) -/12 points An exam consists of 50 multiple choice questions. Based on how much you studied, for any given question, you think you have a probability of 0.67 of getting the correct answer. Consider the sampling distribution of the sample proportion of correct questions out of 50. Find the mean and standard error of the sampling distribution of this proportion. Mean = (2 decimal places) Standard error = (5 decimal places) What do you expect for the shape of this sampling distribution? approximately normal because the population is normal approximately normal because n is large skewed left because n is large Th skewed right because p is greater than 0.5 skewed left because p is greater than 0.5 If truly p = 0.67, calculate the probability of getting a sample proportion less than 0.60 (correct answers on less than 60% of the questions ). Probability = (4 decimal places) https://www.coursehero.com/file/6762027/stat-ch-71-hw/ 9/29/2010 9:44 AM HW 7.1-7.3 FA10 3 of 5 4. http://www.webassign.net/v4cgicricket3@uga/student.pl?c=2010092913... -/12 points sh is ar stu ed d vi y re aC s ou ou rc rs e eH w er as o. co m The figure illustrates two sampling distributions for sample proportions when the population proportion p = 0.50. (a) Find the standard error for the sampling distribution of the sample proportion with (i) n = 100, (ii) n = 1000. (i) (ii) (b) Explain why the sample proportion would be very likely (as the figure suggests) to fall (i) between 0.35 and 0.65 when n = 100, and (ii) between 0.45 and 0.55 when n = 1000. The sample proportion is very likely to fall within three standard errors of the mean. The sample proportion becomes smaller when n is larger. The sample proportion has to be close to the population proportion. The sample proportion has to be between 0 and 1. (c) Explain how the results in (b) indicate that the sample proportion tends to more precisely estimate the population proportion when the sample size is larger. The larger the sample size the larger the standard error. Th The larger the sample size the smaller the standard error. The smaller the sample size the smaller the standard error. The smaller the sample size the fewer the standard errors needed. https://www.coursehero.com/file/6762027/stat-ch-71-hw/ 9/29/2010 9:44 AM HW 7.1-7.3 FA10 4 of 5 5. http://www.webassign.net/v4cgicricket3@uga/student.pl?c=2010092913... -/13 points A 2007 study suggested that bulimia is practiced by about 1% of college students. A researcher believes the actual population proportion p exceeds 0.01. In a survey that randomly sampled 1500 college students, 27 of them admit to having had problems with bulimia. Does this provide strong evidence that the population proportion actually exceeds 0.01? (a) Assuming p = 0.01, find the mean, standard error, and shape of the sampling distribution of the sample proportion, (i) Mean = . (3 decimal places) (ii) Which expression below gives the standard error? (0.01*0.99/1500) (0.01*0.99/1500) (0.018*0.982/1500) (0.018*0.982/1500) sh is ar stu ed d vi y re aC s ou ou rc rs e eH w er as o. co m (iii) The shape of this distribution is: approximately normal skewed right skewed left (b) Find the probability of observing a sample proportion of standard error = 0.002569) Probability = = 27/1500 =0.018 or larger if actually p = 0.01. (Use 4 decimal places (c) Does this probability indicate that it is likely that the population proportion actually exceeds 0.01? No. If the population proportion was actually p = 0.01, we would be very likely to get a sample proportion as large as 0.018. Th Yes. If the population proportion was actually p = 0.01, we would NOT be very likely to get a sample proportion as large as 0.018. https://www.coursehero.com/file/6762027/stat-ch-71-hw/ 9/29/2010 9:44 AM HW 7.1-7.3 FA10 6. http://www.webassign.net/v4cgicricket3@uga/student.pl?c=2010092913... -/15 points In the past, the Mars candy company has claimed that the color mix for Peanut M & Ms, contains 15% Green. A large statistics lecture class is examining bags containing 105 peanut M & Ms each. (a) If each student had a bag, and the instructor plots a histogram showing the PROPORTION of Green for each student, what would you expect the shape of the histogram to be? skewed left bell-shaped skewed right (b) What would you expect the center of this histogram to be? (3 decimal places) (c) What would you expect the standard error of the proportions to be? (5 decimal places) (d) If we use the normal distribution to model the distribution of the sample proportions, what is the probability that a (4 decimal places) sh is ar stu ed d vi y re aC s ou ou rc rs e eH w er as o. co m randomly selected bag of 105 Peanut M & Ms, will contain more than 20% Green? In another large lecture class, students are examining larger bags of Peanut M & Ms containing more M & Ms. (e) In this class, you would expect the center of the histogram to be larger than smaller than about the same as the center for the histogram found by the other class. (f) In this class, you would expect the standard error of the proportions to be larger than about the same as smaller than the standard error found by the other class. 7. -/3 points Th Typically 5% of tax cases actually go to court. The remainder are settled out of court. A certain district has 80 tax cases and wants to determine the probability that more than 20, or 25%, will eventually go to court? Can a valid probability be found using the normal distribution? Yes. The sample size is greater than 30. Yes. The sample size is large enough. The condition that both np and n(1-p) both be at least 15 is met. No. The sample size is not large enough. The condition that both np and n(1-p) both be at least 15 is not met. 8. -/3 points In the first year of a recent recession, 9% of borrowers did not re-pay loans on time. During that year, a local bank made 200 loans. What is the probability that more than 10% of those loans will not be paid back on time? Probability = (4 decimal places) https://www.coursehero.com/file/6762027/stat-ch-71-hw/ 5 of 5 Powered by TCPDF (www.tcpdf.org) 9/29/2010 9:44 AM