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Dr. Jaffe / Ms. Wen Advanced Mathematics Homework due Thursday 9/17 NAME: _______________________________________________________ 1. Suppose that a murder occurs in a small town of 10,000 people, and DNA evidence is collected. Suppose also that DNA on everyone in the town is on record. Lastly, suppose that DNA identification is 99.9 percent specific. If we found someone in town whose DNA matches that of the perpetrator of the murder, how likely are you as a juror to believe that he was guilty of murder? To answer this question, convert your percentages into numbers. Justify your answer completely. 99.9% = .999 10,000 * .999 = 9,990 10,000 people in the town 10 share a DNA match 9,990 don't share a DNA match At this point it would be your opinion if you find him guilty or not. If 10 out of every 10,000 will get a positive DNA match, then do you believe beyond a reasonable doubt that he is guilty? 2. Look at every number that you see on the first page of the New York Times. Create a table from 1 to 9 in which you tally all the first digits of each number. Dr. Jaffe / Ms. Wen Advanced Topics in Mathematics Homework due Monday 9/21 NAME: ________________________________________________ The expert witness testifies that there are about 10 million men who could have been the perpetrator. The probability of a randomly select man having a DNA profile that is identical with the trace recovered from the crime scene is approximately .0001 percent. If a man has this DNA profile, it is practically certain that a DNA analysis shows a match. If a man does not have this DNA profile, current DNA technology leads to a reported match with a probability of only .001 percent. A match between the DNA of the defendant and the traces on the victim has been reported. 1) What is the probability that the reported match is a true match, that is, that the person actually has this DNA profile? P(match) = .0001 % .0001% = .000001 .000001 * 10,000,000 = 10 P(match|not DNA) --> P(false positive) = .001% .001% = .00001 .00001 * 9,999,990 = 100 10,000,000 men 10 matches positive test 9,999,990 non matches 100 positive test 9,999,890 negative test 10/110 people have the DNA = 9% chance that it is a true match 2) What is the probability that the person is the source of the trace? There is only a 1/10 chance that if the match is a true match, the person is the source of the trace. 3) Please render your verdict for this case: Guilty or not guilty? (and justify your reasoning) This is your opinion. If the person is a true match, there is only a 1/10 chance they are the source. If the person simply is a reported match, there is only a 1/110 chance. Dr. Jaffe / Ms. Wen Advanced Topics in Mathematics Homework due Tuesday 9/22 NAME: ___________________________________________________________________ 1. Why is it important to consider the way we represent a statistic (a percentage vs. a number)? A statistic can be vague or difficult to follow if we present it as a percentage. When we are dealing with statistics that have conditions (ex. p(match|+ test) ), numbers are far less confusing. When we are dealing with a single statistic and very large numbers, a percentage can be clearer. We want to make sure we are presenting information in a clear way. 2. Do you think statistics play a role in jury verdicts? In an ideal world, absolutely. Unfortunately, the jury is not always given the statistic that is relevant or they do not have enough knowledge to break it down to get an accurate picture. As a result, they often do not play a role in verdicts. 3. Given the following statistics about a test for a rare immunodeficiency as follows: P(child having the immunodeficiency) = .1% P(+ test | child with immunodeficiency) = 92% P(+ test | child without immunodeficiency) = 5% What is the probability that if a child is tested and receives a positive test, they actually have the immunodeficiency? 100,000 children 100 with ID 92 + test 99,900 without ID 8 - test 4,995 + test 94,905 - test There are a total of 5,087 positive tests and only 92 of them are true immunodeficiencies: 92 / 5087 = 1.81% 4. What does the mean tell us about a set of data? It tells us the sum of the numbers divided by the number of numbers (the sum of the distances from each data point below the mean to the mean = the sum of the distances from each data point above the mean to the mean) 5. Give two methods of visually representing data? Bar Graph, Pie Chart, Scatter Plot, Line Graph, Histogram, Stem-and-Leaf Plot, Box-andWhisker Plot, Table, Venn Diagram Dr. Jaffe / Ms. Wen Advanced Topics in Mathematics Homework due Monday 9/28 NAME: ___________________________________________________________ 1. What is the purpose of statistics? Statistics can tell us trends in data. They are a numeric analysis of what happened in the past. 2. Add one piece of data to the list below that will significantly affect the mean. Explain why it works without actually calculating the mean. 52, 63, 65, 71, 79, 84, 88 If you add any outlier, that will significantly affect the mean. If you add 1, for example, that will decrease the mean, and if you add 2000, you will significantly increase the mean. 3. What is the difference between parameters and statistics? Parameters are what we use to analyze data collected from a population while statistics are what we use to analyze data collected from a sample. 4. If we want to determine the average salary in the United States, describe who might be in our sample so we can get an accurate answer. Explain. A random sample would be effective. We can select 1000 from each state, for example. 5. When is it possible to have more than one mode? Example: 1,1,2,2,3,3,5,6,6,7 Mode: 1,2,3, and 6 Each of these numbers appears the same number of times and more than any other number in the list 6. What does a scatter plot tell us about our data? Scatter Plot = exact data points and any trends or patterns 7. What is the difference between a bar graph and a histogram? A bar graph can have non-numerical data, and because of this, the bars do not touch. 8. Give one example of a set of data for which we would want to find the mode (the number that occurs most frequently). There are many examples, but two are as follows: We would want to determine the subway station that has the most people who jump the turnstile. We would put more security at that subway station. We would want to determine which republican candidate had the most votes because they will become the Republican nominee in the Presidential election. Dr. Jaffe / Ms. Wen Advanced Topics in Mathematics Homework due Wednesday 9/30 NAME: ___________________________________________________________ 1. Why do we divide by n β 1 when finding the standard deviation for a sample? We divide by n-1 because it makes our sample standard deviation a little bit higher so it is more representative of the population. A sample tends to have a little less variation than a population, which is why we increase it a little bit. 2. When we use 1-Var stats to find standard deviation Sx is always greater than Οx. Why? The calculator does not know if our data is from a sample or a population so one calculation Sx represents the value if it were a sample. The calculator divided by n-1 in this case, resulting in a slightly higher number. 3. Given the following set of data: 12, 23, 42, 51, 66, 68, 69, 73, 85, 89, 93 a) Find the population standard deviation. 25.18 b) Find the sample standard deviation. 26.40 4. Following are the number of calories in a basic hamburger (one meat patty with no cheese) at various fast food restaurants around the country. πππ, πππ, πππ, πππ, πππ, ππππ, πππ, πππ, πππ, πππ, πππ, πππ, ππππ, πππ, ππππ a) Find the mean 679.2 b) Find the sample standard deviation 307.62 c) Create a relative frequency histogram and describe its shape (symmetric, skewed right, skewed left). Explain. Interval Frequency Relative Frequency 200-399 4 4/15 = .27 400-599 1 1/15 = .07 600-799 6 6/15 = .4 800-999 1 1/15 = .07 1000-1199 3 3/15 = .2 The relative frequency histogram will be symmetrical. There will be a mound at 600-799. The relative frequencies of 400-599 and 800-999 are the same, and the relative frequencies of the first and last intervals are almost the same. 5. For the following histogram, describe the shape, and give estimates of the mean and standard deviation of the distributions. Distribution of head circumferences (mm) There will be a variation of answers. The shape is symmetric. We can estimate the mean at about 560 since this is where the mound is. Any answer between 555 and 575 would be acceptable The standard deviation is probably around 20, since about 70% of the data falls between 540 and 580. Any answer between 15 and 25 would be acceptable. Dr. Jaffe / Ms. Wen Advanced Mathematics Homework due Thursday 10/1 NAME: ___________________________________________________________ 1. How do we know when a normal distribution is an appropriate way to represent our data? The relative frequency histogram should look symmetrical with a mound. 2. The histogram below shows the distribution of heights (to the nearest inch) of π, πππ young women. a) Mark the mean on the graph, and mark one deviation above and below the mean. Approximately what proportion of the values in this data set are within one standard deviation of the mean? b) Draw a smooth curve that comes reasonably close to passing through the midpoints of the tops of the bars in the histogram. Describe the shape of the distribution. The curve is not perfect, but you can see that it is symmetrical with a mound in the center. c) What is the width of each bar? What does the height of the bar represent? The width of each bar is 1 inch. d) Shade the area under the curve that represents the proportion of the data within one standard deviation of the mean. 3. Below is a histogram of the top speed of different types of animals. Estimate the mean and sample standard deviation, and justify your reasoning. There will be a variety of answers. The mean is approximately 40 (any answer between 35 and 45 would be acceptable. The histogram is has a mound at 38, but it is slightly skewed to the right so the mean will be pulled a little bit towards the tail. The standard deviation would be about 15 and answer between 10 and 20 would be acceptable. About 70% of our data is between 25 and 55. Dr. Jaffe / Ms. Wen Advanced Mathematics Homework due Tuesday 10/6 NAME: ___________________________________________________________ 1. What is a z-score? A z-score will tell us how many standard deviations away from the mean that piece of data is. 2. In a normal distribution, how do we calculate the percentage of data that falls one standard deviation below the mean to one standard deviation above the mean? We add all the relative frequencies that fall between the lines marking one standard deviation below the mean and one standard deviation above the mean. 3. What is a relative frequency histogram? A relative frequency histogram tells us the percentage of data within each interval. On the x-axis are the intervals, and on the y-axis are the relative frequencies. 4. The prices of the printers in a store have a mean of $πππ and a standard deviation of $ππ. The printer that you eventually choose costs $πππ. a. What is the π score for the price of your printer? z-score = πππππβππππ πππππ πππ π ππππππππ = πππβπππ ππ =π b. How many standard deviations above the mean was the price of your printer? It was two standard deviations above the mean 5. Ashishβs height is ππ inches. The mean height for boys at his school is ππ. π inches, and the standard deviation of the boysβ heights is π. π inches. a. What is the π score for Ashishβs height? (Round your answer to the nearest hundredth.) z-score = πππππβππππ πππππ πππ π ππππππππ = ππβππ.π π.π = βπ. ππ b. What is the meaning of this value? This means Ashish's height is 1.82 standard deviations below the mean. c. Explain how a π score is useful in describing data. A z-score can help us determine the percentage of data within a certain range of standard deviations. If we don't have the relative frequency histogram, we can't just add the relative frequencies, but by knowing the z-score, if the distribution is normal we can use the areas under the normal curve. Dr. Jaffe / Ms. Wen Advanced Mathematics Homework due Wednesday 10/7 NAME: ___________________________________________________________ 1. The distribution of lifetimes of a particular brand of car tires has a mean of ππ, πππ miles and a standard deviation of π, πππ miles. You can use your calculator, but write what you plugged in. a. Assuming that the distribution of lifetimes is approximately normally distributed and rounding your answers to the nearest thousandth, find the probability that a randomly selected tire lasts i. between ππ, πππ and ππ, πππ miles. 55000β51200 z-score for 55,000 = z-score for 65000 = = .46 8200 65000β51200 8200 = 1.68 Normalcdf(.46, 1.68) = .275 ii. less than ππ, πππ miles. z-score for 48000 = πππππβπππππ ππππ = -.39 Normalcdf(-999, -.39) = .348 iii. at least ππ, πππ miles. z-score for 41000 = 41000β51200 8200 = -1.24 Normalcdf(-1.24, 999) = .893 b. Explain the meaning of the probability that you found in part (aβiii). If a large number of tires of this brand were to be randomly selected, then you would expect about ππ. π% of them to last more than ππ, πππ miles. 2. Suppose that a particular medical procedure has a cost that is approximately normally distributed with a mean of $ππ, πππ and a standard deviation of $π, πππ. For a randomly selected patient, find the probabilities of the following events. (Round your answers to the nearest thousandth.) a. The procedure costs between $ππ, πππ and $ππ, πππ. z-score for 18000 = z-score for 22000 = 18000β19800 2900 22000β19800 2900 = -.621 = .759 Normalcdf(-.621, 22000) = .509 b. The procedure costs less than $ππ, πππ. z-score for 15000 = 15000β19800 2900 =-1.655 Normalcdf(-999, -1.655) = .049 c. The procedure costs more than $ππ, πππ. z-score for 17250 = 17250β19800 2900 = -.879 Normalcdf(-.879, 999) = .810 3. A farmer has πππ female adult sheep. The sheep have recently been weighed, and the results are shown in the table below. πππ to < πππ π Weight (pounds) Frequency πππ to < πππ ππ πππ to < πππ πππ πππ to < πππ πππ πππ to < πππ πππ πππ to < πππ ππ πππ to < πππ π a. Construct a histogram that displays these results. b. Looking at the histogram, do you think a normal distribution would be an appropriate model for this distribution? Yes. The histogram is approximately symmetric and mound shaped. c. The weights of the πππ sheep have mean πππ. ππ pounds and standard deviation ππ. ππ pounds. For a normal distribution with this mean and standard deviation, what is the probability that a randomly selected sheep has a weight of at least πππ pounds? (Round your answer to the nearest thousandth.) Using Normalcdf(πππ, πππ, πππ. ππ, ππ. ππ), you get π·(β₯ πππ) = π. πππ. 4. State if the following is an observational study, survey, or experiment, and give a reason for your answer. Ken wants to compare how many hours a week that sixth graders spend doing mathematics homework to how many hours a week that eleventh graders spend doing mathematics homework. He randomly selects ten sixth graders and ten eleventh graders and records how many hours each student spent on mathematics homework in a certain week. This is a survey. He is selecting a group and asking questions 5. Data from a random sample of ππ students in a school district showed a positive relationship between reading score on a standardized reading exam and shoe size. Can it be concluded that having bigger feet causes one to have a higher reading score? Explain your answer. No. The bigger feet and higher reading score are only correlated. They both increase as someone gets older. Use the following scenarios for Problems 6β8. A. Researchers want to determine if there is a relationship between whether or not a woman smoked during pregnancy and the birth weight of her baby. Researchers examined records for the past five years at a large hospital. B. A large high school wants to know the proportion of students who currently use illegal drugs. Uniformed police officers asked a random sample of πππ students about their drug use. C. A company develops a new dog food. The company wants to know if dogs would prefer its new food over the competitionβs dog food. One hundred dogs, who were food-deprived overnight, were given equal amounts of the two dog foods; the new food vs. competitorβs food. The proportion of dogs preferring the new food was recorded. 6. Which scenario above describes an experiment? Explain why. Scenario C is an experiment because we have groups that are receiving treatments (dogs with new food & dogs with old food), and there were responses (preference for which food) 7. Which scenario describes a survey? Will the results of the survey be accurate? Why or why not? Scenario B is a survey. A group of students is being asked questions. The survey is not going to be accurate because students are unlikely to reveal to a police officer drug use. 8. The remaining scenario is an observational study. Is it possible to perform an experiment to determine if a relationship exists? Why or why not? We can perform an experiment by creating groups with different treatments (women smoke during pregnancy & women do not smoke during pregnancy. The response would be whether or not the size of the baby changes. However, this experiment would be unethical. We cannot have a treatment group in which we may be hurting a human being. Dr. Jaffe / Ms. Wen Advanced Mathematics Homework due Thursday 10/8 NAME: ___________________________________________________________ 1. A salesman kept track of the gas mileage for his car over a ππ-week span. The mileages (miles per gallon rounded to the nearest whole number) were ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ, ππ. a) Construct a relative frequency table Interval 23 24 25 26 27 28 29 Frequency 1 3 5 7 5 2 2 Relative Frequency .04 .12 .2 .28 .2 .08 .08 b) Estimate the mean. Justify your answer. The data is almost symmetrical, so the mean would be at the mound, which is 26 c) Estimate the standard deviation. Justify your answer. I would estimate the standard deviation to be 1. Sixty Eight percent of the data falls between 25 and 27. 2. The USDA document described in Example 1 also states that the average cost of food for a ππβππ year old female (again, on the βModerate-costβ plan) is $πππ. ππ per month. Assume that the monthly food cost for a ππβππ year old female is approximately normally distributed with mean $πππ. ππ and standard deviation $ππ. ππ. Find the probability that the monthly food cost for a randomly selected ππβππ year old female is i. less than $πππ. Use z-scores z-score for 225 = 225β215.20 14.85 =-.66 Normalcdf(-999,-.66) = .7454 ii. between $πππ and $πππ. Use z-scores z-score for 190 = z-score for 220 = 190β215.2 14.85 =-1.697 220β215.2 14.85 =.323 Normalcdf(-1.697, .323) = .582 iii. more than $πππ. Do not use z-scores Normalcdf(250, 999, 215.20, 14.85) = .0096 iv. between $180 and $205. Do not use z-scores Normalcdf(180, 250, 215.20, 14.85) = .237 Dr. Jaffe / Ms. Wen Advanced Mathematics Homework due Tuesday 10/13 NAME: ___________________________________________________________ 1. A student conducted a simulation of ππ coin flips. Below is a dot plot of the sampling distribution of the proportion of heads. This sampling distribution has a mean of π. ππ and a standard deviation of π. ππ. a. Describe the shape of the distribution. The shape is irregular, perhaps a little skewed to the left. b. Describe what would have happened to the mean and the standard deviation of the sampling distribution of the sample proportions if the student had flipped a coin ππ times, calculated the proportion of heads, and then repeated this process for a total of ππ times. The more times we flip the coin for each sample, the more accurate our proportion; therefore, the sample distribution would become more symmetrical. The mound would be at .5, which would be the mean. The standard deviation would get smaller. 2. What effect does increasing the sample size have on the standard deviation of the sampling distribution? The standard deviation decreases as we increase the size of each sample. Our sample proportion for each sample becomes more accurate so the standard deviation will get smaller. Our proportions get closer and closer to the mean. 3. The same student flipped the coin πππ times, calculated the proportion of heads, and repeated this process a total of ππ times. Below is the sampling distribution of sample proportions of heads. The mean and standard deviation of the sampling distribution is π. πππ and π. πππ. Do you think this was a fair coin? Why or why not? This is not a fair coin. If the coin were fair, then there would be an equal chance of getting heads or tails. This would mean that the mound of the sample proportions would be at .5, which is not the case here. 4. The durations of high school baseball games are approximately normally distributed with mean πππ minutes and standard deviation ππ minutes. Suppose also that the durations of high school softball games are approximately normally distributed with a mean of ππ minutes and the same standard deviation, ππ minutes. Is it more likely that a high school baseball game will last between πππ and πππ minutes or that a high school softball game will last between πππ and πππ minutes? Answer this question without doing any calculations and justify your reasoning! 70% of the games will last within one standard deviation of the mean. That would mean 70% of the softball games last between 94 and 116 minutes, and 70% of the basketball games would last between 84 and 106 minutes. This would mean it is more likely that the softball games would last between 100 and 110 minutes because the bulk of the data surrounds the mean. 5. What does a sample proportion tell us? A sample proportion tells us the percentage of successful outcomes we get in a sample. 6. How is it possible to get different sample proportions? Explain. Each sample results in different data. Not all samples are uniform. You may be testing different subjects or the probability of getting a particular outcome is only 50%, for example, so you will not get the same result each time. Dr. Jaffe / Ms. Wen Advanced Mathematics Homework due Wednesday 10/14 NAME: ___________________________________________________________ 1. What is a margin of error? A margin of error is our wiggle room. It creates an interval so we can generalize to a population. 2. The following intervals were plausible population proportions for a given sample. Find the margin of error in each case. a. from π. ππ to π. ππ We can find the margin of error in three different ways: center β lowest proportion highest proportion β center (highest proportion + lowest proportion) / 2 We will find the margin of error using the center so we can review how to find the center too. Center = .35+ .65 2 = .5 Margin of error = .65 - .5 = .15 a. from π. ππ to π. ππ Center = .72+ .78 2 = .75 Margin of error = .78 - .75 = .03 b. from π. ππ to π. ππ Center = .84+ .95 2 = .895 Margin of error = .95 - .895 = .055 c. from π. ππ to π. ππ Center = .47+ .57 2 = .52 Margin of error = .57 - .52 = .05 3. Decide if each of the following statements is true or false. Explain your reasoning in each case. a. The smaller the sample size, the smaller the margin of error. This is false. When your sample size is small, your results are less accurate. This means your standard deviation is likely to be higher and your range is greater. Since margin of error is based on the range, the margin of error will also increase. b. If the margin of error is π. ππ and the observed proportion of red chips is π. ππ, then the true population proportion is likely to be between π. ππ and π. ππ. This is false. If the margin of error is .05 and the proportion is .35, we would describe the true population proportion to be .35 ± .05, which is between .03 and .04 not .04 and .05. Dr. Jaffe / Ms. Wen Advanced Mathematics Homework due Thursday 10/15 NAME: ___________________________________________________________ 1. Using the formula for finding margin of error with a 95% confidence interval, answer the following questions: a. If the proportion of females at Union High School is π. π, what is the standard deviation of the distribution of the sample proportions of females for random samples of size ππ? Round your answer to three decimal places. The standard deviation is β π(1βπ) . The sample proportion is .4, and the sample size is 50 π β b. The proportion of males at Union High School is π. π. What is the standard deviation of the distribution of the sample proportions of males for samples of size ππ? Round your answer to three decimal places. The standard deviation is β π(1βπ) π . The sample proportion is .6, and the sample size is 50 β c. . 4(.6) = .069 50 . 6(.4) = .069 50 Think about the graphs of the two distributions in parts (a) and (b). You do not need to graph the distributions, but how do you think they would be different. Discuss the shape and spread. The standard deviation is the same, and the data is the opposite. If in part a, we get a proportion of .4, in part b we get an answer of 1-.4 because the people who arenβt girls are boys. This means the shape will be the same. The only change is that the graph is simply shifted. In part a, our mound will be at .4, and in part b, our mound will be at .6. 2. A newspaper in a large city asked πππ women the following: βDo you use organic food products (such as milk, meats, vegetables, etc.)?β 280 women answered βyes.β Compute the margin of error. What does this mean? p = 280/500 = .56 n = 500 Margin of error = 2 * β π(1βπ) π = 2 *β .56(.44) 500 = .044 This means that 95% of our data falls within the interval .56 ± .044 3. A newspaper in New York took a random sample of πππ registered voters from New York City and found that πππ favored a certain candidate for governor of the state. A second newspaper polled ππππ registered voters in upstate New York and found that πππ people favored this candidate. Why do you think the results of the two polls were different? You are asking a different number of people, and a different group of people so the answers will not be the same: different people favor different candidates. 4. The school newspaper at a large high school reported that πππ out of πππ randomly selected students favor assigned parking spaces. Compute the margin of error. What does your answer mean? p = 120/200 = .6 n = 200 Margin of error = 2 * β π(1βπ) π = 2 *β .6(.4) 200 = .069 This means that 95% of our data falls within the interval .6 ± .069 5. Why do we say that the proportion ± margin of error is a 95% confidence interval? We say this because the margin of error is equal to two standard deviations, and 95% of our data falls within two standard deviations of our mean. Dr. Jaffe / Ms. Wen Advanced Mathematics Homework due Monday 10/19 NAME: ___________________________________________________________ 1. Describe the difference between a sample distribution and a simulated sample distribution. In a regular sample distribution, we are working with sample proportions, and in a simulated sample distribution, we are working with sample means. 2. Explain the difference between the sample mean and the mean of a sampling distribution. The mean of the sampling distributions takes the mean of each sample, and then finds the mean of all that data. 3. At the beginning of the school year, school districts implemented a new physical fitness program. A student project involves monitoring how long it takes eleventh graders to run a mile. The following data were taken mid-year. a. What is the estimate of the population mean time it currently takes eleventh graders to run a mile based on the following data (minutes) from a random sample of ten students? 6.5, 8.4, 8.1, 6.8, 8.4, 7.7, 9.1, 7.1, 9.4, 7.5 The mean is 7.9 The standard deviation is .9568 .9568 The margin of error =2* = .604 β10 The population mean time is 7.9 ± .604 b. The students doing the project collected 50 random samples of 10 students each and calculated the sample means. The standard deviation of their distribution of 50 sample means was 0.6 min. Based on this standard deviation, what is the margin of error for their sample mean estimate? Explain your answer. The margin of error is 2 β π βπ =2* .6 β50 = .170 This means that the margin of error is .17 seconds. It is double the standard deviation so we include 95% of our data. 4. It is well known that astronauts increase their height in space missions because of the lack of gravity. A question is whether or not we increase height here on Earth when we are put into a situation where the effect of gravity is minimized. In particular, do people grow taller when confined to a bed? A study was done in which the heights of six men were taken before and after they were confined to bed for three full days. a. The before-after differences in height measurements (mm) for the six men were: ππ. π ππ. π ππ. π ππ. π ππ. π ππ. π. Assuming that the men in this study are representative of the population of all men, what is an estimate of the population mean increase in height after three full days in bed? An estimate for the population mean is the mean of all these values = 14.15 b. Calculate the margin of error associated with your estimate of the population mean from part (a). Round your answer to three decimal places. The margin of error is 2 β π βπ =2* .9397 β6 = .767 c. Based on your sample mean and the margin of error from parts (a) and (b), what are plausible values for the population mean height increase for all men who stay in bed for three full days? We can describe the population mean as 14.15 ± .767 5. A new brand of hot dog claims to have a lower sodium content than the leading brand. a. A random sample of ten of these new hot dogs results in the following sodium measurements (mg): πππ πππ πππ πππ πππ πππ πππ πππ πππ πππ. Estimate the population mean sodium content of this new brand of hot dog based on the ten sampled measurements. An estimate for the population mean is the mean of all these values = 314.5 b. Calculate the margin of error associated with your estimate of the population mean from part (a). Round your answer to three decimal places. The margin of error is 2 β π βπ =2* 29.44 β10 = 18.619 c. The mean sodium content of the leading brand of hot dogs is known to be πππ mg. Based on the sample mean and the value of the margin of error for the new brand, is a mean sodium content of πππ mg a plausible value for the mean sodium content of the new brand? Comment on whether you think the new brand of hot dog has a lower sodium content on average than the leading brand. We can describe the population mean as 314.5 ± .18.619. 350 is outside of the estimated population mean so it is not a plausible sodium content. The new brand of hot dog has a lower sodium content than the leading brand. d. Another random sample of ππ new brand hot dogs is taken. Should this larger sample of hot dogs produce a more accurate estimate of the population mean sodium content than the sample of size ππ? Explain your answer by appealing to the formula for margin of error It would yield a more accurate estimate. The margin of error would be smaller because you are dividing by a larger number Dr. Jaffe / Ms. Wen Advanced Mathematics Homework due Tuesday 10/20 NAME: ___________________________________________________________ For Problems 1 and 2, identify (i) the subjects, (ii) the treatments, and (iii) the response variable for each experiment. 1. A botanist was interested in determining the effects of watering (three days a week or daily) on the heat rating of jalapeño peppers. The botanist wanted to know which watering schedule would produce the highest heat rating in the peppers. He conducted an experiment, randomly assigning each watering schedule to half of twelve plots that had similar soil and full sun. The average final heat rating for the peppers grown in each plot was recorded at the end of the growing season. Subjects: jalapeño peppers (these are the things that are receiving treatments) Treatments: the amount of watering β three days a week or daily (this is what is changing) Responses: the heat rating (this is what we are measuring) 2. A manufacturer advertises that its new plastic cake pan bakes cakes more evenly. A consumer group wants to carry out an experiment to see if the plastic cake pans do bake more evenly than standard metal cake pans. Twenty cake mixes (same brand and type) are randomly assigned to either the plastic pan or the metal pan. All of the cakes are baked in the same oven. The rating scale was then used to rate the evenness of each cake. Subjects: cake mixes (these are the things that are receiving treatments) Treatments: the kind of pan β plastic or metal (this is what is changing) Responses: the evenness of the cake (this is what we are measuring) 3. In one high school, there are eight math classes during 2 nd period. The number of students in each 2nd period math class is recorded below. ππ ππ ππ ππ ππ ππ ππ ππ This data set is randomly divided into two equal size groups, and the group means are computed. a) Will the two groups means be the same? Why or why not? They will not necessarily be the same because we are randomly dividing the numbers into two groups. One group could potentially get the 4 highest numbers, while the other could get the four lowest numbers. The random division into two groups process is repeated many times to create a distribution of group mean class size. b) What is the center of the distribution of group mean class size? The center of the distribution would be the mean of the single list, which is 25.5 c) What is the largest possible range of the distribution of group mean class size? This would occur if one group included the four largest numbers and the other included the four smallest numbers. In this case, the mean of the group with the four largest numbers is 28.75, and the mean of the group with the four smallest numbers is 22.25. The range is 28.75 β 22.25 = 6.5 d) What possible values for the mean class size are more likely to happen than others? Explain why you chose these values. The answers may vary here, but they should definitely fall within the range 22.25 to 28.75. 24-27 is right around the mean, so that would be a reasonable range where the most values occur. 4. Why do we want to use random assignment in an experiment? Random assignment allows us to make cause and effect conclusions. If the assignment is not random, it is possible that the people in one group are too similar and yet different from the people in the second group so we donβt know if the treatment caused a difference or it was something else about the people in group A. 5. Why canβt we use random assignment in an observational study? Random assignment is when we randomly assign our subjects to different treatments, but in an observational study there are no treatments. Dr. Jaffe / Ms. Wen Advanced Mathematics Homework due Wednesday 10/21 NAME: ___________________________________________________________ 1. Group A: π dieters lost an average of π pounds. Group B: π non-dieters lost an average of π pounds over the same time period. Calculate and interpret "Diff" = the mean of Group A minus the mean of Group B Μ Μ Μ π₯ π₯π΅ = 8 β 2 = 6 π΄ β Μ Μ Μ The diff value is positive so this means the treatment (diet) in group A caused greater weight loss. 2. Group A: ππ students were on average π. π seconds faster in their πππ meter run times after following a new training regimen. Group B: ππ students were on average π. π seconds slower in their πππ meter run times after not following any new training regimens. Calculate and interpret "Diff" = the mean of Group A minus the mean of Group B. Μ Μ Μ π₯ π₯π΅ = 0.4 - - 0.2 = .6 π΄ β Μ Μ Μ The diff value is positive so this means that the new training regimen was effective 3. a) Using the randomization distribution shown in the Exit Ticket, what is the probability of obtaining a "Diff" value of βπ. π or less? There are 29 values that are -0.6 or less. There are a total of 100 values. 29/100 = .29. b) Would a "Diff" value of βπ. π or less be considered a βstatistically significant differenceβ? Why or why not? It would not be statistically significant because it occurs 29% of the time, which is not less than 5%. c) Using the randomization distribution shown in the Exit Ticket, what is the probability of obtaining a "Diff" value of βπ. π or less? There are 6 values that are -1.2 or less. So the probability of getting -1.2 or less is 6/100 = .06. d) Would a "Diff" value of βπ. π or less be considered a βstatistically significant differenceβ? Why or why not? It would not be considered statistically significant because it occurs 6%. In order to be statistically significant, it must occur less than 5% of the time. 3. Describe the purpose of randomization testing. Randomization testing allows us to determine if the treatment has some effect. We want to make sure that any differences we see between the two groups are happening because of the treatment and not as a result of chance. 4. How do we determine if a diff value between two groups in an experiment is statistically significant? A diff value is statistically significant if it would occur less than 5% of the time if we were to divide the data into two groups at random instead of by who received the treatment and who did not.
