Stats : Exam #1
bar chart
-a chart with bars whose lengths are proportional to quantities -A bar chart compares different categories by using individual bars to represent the tallies for each category. The length of a bar represents theamount, frequency, or percentage of values falling into a category. -categorical data
1. The percentage of the adult population that was married in 1990 was 2. The percentage of the adult population that was married in 1997 was 3.The percentage of the adult population that was married in 2000 was 4.The percentage of the adult population that was married in 2007 was
1 - 59.8 2- 57.9 3- 56.4 4- 53.2 Year: Married Total 1990 113.3 189.5 1997 117.8 203.5 2000 120.5 213.7 2007 124.1 233.3
Match each description with the correct histogram of the data. 1. The age of death of a sample of 19 typical women in the U.S. 2. The yearly tuition for 142 colleges, 85 of which are private and 57 of which are state-supported. 3. The outcomes of rolling a fair die (with six sides) 5000 times.
1- B 2-C 3-A
Name two measures of the variation of a distribution, and state the conditions under which each measure is preferred for measuring the variability of a single data set.
1. The interquartile range is preferred when the data is strongly skewed or has outliers. 2.The standard deviation is preferred when the data is relatively symmetric.
1. What percentage of women are right handed? 2.What percentage of right handed students are females? 3. What percentage of students are right handed? 4.If the percentage of right-handed females remained roughly the same and there were 75 females, how many of them would be right-handed? Total right handed : 7 Total left handed: 4 Total male :2 Total female : 9 Total male and female : 11
1.To find the proportion, divide the number of females who are right-handed by the total number of females in the class. = 66.7% 2. To find the proportion, divide the number of females who are right-handed by the total number of right-handed students in the class.( multiply by 100% to get percent) = 85.7 % 3. To find the proportion, divide the number of students who are right-handed by the total number of students in the class. = 63.6 % 4. Multiply 75 by the proportion 0.667. = 50
Indicate whether the study is an observational study or a controlled experiment: A group of boys is randomly divided into two groups. One group watches violent cartoons for one hour, and the other group watches cartoons without violence for one hour. The boys are then observed to see how many violent actions they take in the next two hours, and the two groups are compared.
A controlled experiment
Left skewed
A density curve where the left side of the distribution extends in a long tail. (Mean < median.)
Data are more than just numbers, because data have _____.
Context
unstacked data
Data stored such that each column represents a variable from a different group. Can only store two variables.
Stacked data
Data values stored in a spreadsheet format. Each row contains data for a single individual. Can store many variables.
Stem Plot
For the number 65.4, the 65 is the stem and the 4 is the leaf. For the number 60, the 6 is the stem and the 0 is the leaf.
According to the ancient Roman architect Vitruvius, a person's armspan (the distance from fingertip to fingertip with the arms stretched wide) is approximately equal to his or her height. For example, people 5 feet tall tend to have an armspan of 5 feet. Explain, then, why the distribution of armspans for a class containing roughly equal numbers of men and women might be bimodal.
Men and women tend to have different heights and therefore different armspans.
If you moved to one of these regions and met 50 people, in which region would you be most likely to meet at least one person living with AIDS?
Most likely to meet one person with aids living in region : F
Indicate whether the study is an observational study or a controlled experiment. A researcher was interested in the effects of exercise on academic performance in elementary school children. She went to the recess area of an elementary school and identified some students who were exercising vigorously and some who were not. The researcher then compared the grades of the exercisers with the grades of those who did not exercise.
Observational study
In your own words, describe to someone who knows only a little statistics how to recognize when an observation is an outlier. What action(s) should be taken with an outlier?
Outliers are observed values far from the main group of data. In a histogram they are separated from the others by space. Outliers must be looked at in closer context to know how to treat them. If they are mistakes, they might be removed or corrected. If they are not mistakes, you might do the analysis twice, once with and once without the outliers.
Which measure of the center (mean or median) is more resistant to outliers, and what does "resistant to outliers" mean?
The median is more resistant, which indicates that it usually changes less than the mean when comparing data with and without outliers.
Indicate whether the following study is an observational study or a controlled experiment. A researcher was interested in the effect of exercise on memory. She randomly assigned half of a group of students to run up a stairway three times and the other half to rest for an equivalent amount of time. Each student was then asked to memorize a series of random digits. She compared the numbers of digits remembered for the two groups.
This is a controlled experiment. The researcher separates the students into at least two random groups, one of which is the control group. This is essential to conducting a controlled experiment.
Histogram
a bar graph depicting a frequency distribution good for numerical values
pie chart
a chart that shows the relationship of a part to a whole -categorical data
Right Skewed
a distribution with a tail that extends to the right (Mean > median)
Why is this pie chart hard to interpret?
here are so many possible numerical values causing the pie chart to have too many "slices", which makes it difficult to tell which is which.
It was predicted that a country will have an elderly population (65 and older) of 8,840,000 in the year 2050 and that this will be 23.3% of the population. What is the total predicted population of this country in 2050?
1. To find the total number of people in the country, first convert the percentage of elderly into its decimal equivalent. (23.3%/100%) = 0.233 2.The problem states that 23.323.3% of the unknown total population (call it x) is equal to 8,840,000. Set up an equation as described and solve for x. 0.233x= 8,840,000. (Divide both sides by 0.233) 8,840,000 X= 0.233 Predicted pop in 2050 = 37,940,000
Two sections of statistics are offered, the first at 8 a.m. and the second at 10 a.m. The 8 a.m. section has 25 women, and the 10 a.m. section has 15 women. A student claims this is evidence that women prefer earlier statistics classes than men do. What information is missing that might contradict this claim?
The percentage of female students in the two classes is unknown. There may be more females in the 8 a.m. because there are more students in the 8 a.m. class than the 10 a.m. class. This claim could be true only if the classes were the same size.
In a recent competition, do you think the standard deviation of the running times for all men who ran the 100-meter race would be larger or smaller than the standard deviation of the running times for the men's marathon? Explain.
The standard deviation for the 100-meter event would be less. All the runners come to the finish line within a few seconds of each other. In the marathon, the runners can be quite widely spread after running that long distance.
A study concludes that the use of pesticides is associated with the development of Parkinson's disease, a neurological disease that causes people to shake. The study reported that exposure to bug killers and weed killers is"associated with" an increase of 33% to 80% in the chances of getting Parkinson's. Does this study show that pesticides cause Parkinson's disease? Why or why not?
The study does not show that pesticides cause Parkinson's disease. This was an observational study because researchers could not have deliberately exposed people to pesticides. Observational studies cannot conclude causation.
A study compared the rate of pneumonia before and after a vaccine was introduced. In the study, annual hospitalization rates were estimated from any cause using a database. Average annual rates of pneumonia-related hospitalizations before and after introduction of the vaccine were used to estimate annual declines in pneumonia-related hospitalizations. The annual rate of pneumonia-related hospitalizations among children of various age groups significantly declined relative to expected rates before introduction of the vaccine. Does this show that pneumonia vaccine caused the decrease in pneumonia that occurred? Explain.
The study does not show that the vaccine caused the decrease in pneumonia. This is an observational study because the children were not randomly assigned by the researchers. It is possible that confounding variables (other advances in medicine, for instance) would affect the rates of pneumonia.
The histogram shows the ages of 25 CEOs listed on a certain website. Based on the distribution, what is the approximate mean age of the CEOs in this dataset? Write a sentence in context (using words in thequestion) interpreting the estimated mean.
The typical CEO is between 56 and 60.
Is the variable Eye ColorEye Color numerical or categorical? Explain.
The variable is categoricalcategorical because the values are categoriescategories.
A study was conducted to see whether participants would ignore a sign that said, "Elevator may stick between floors. Use the stairs." Those who used the stairs were said to be compliant, and those who used the elevator were said to be noncompliant. There were three possible situations, two of which involved confederates. A confederate is a person who is secretly working with the experimenter. In the first situation, there was no confederate. In the second situation, there was a compliant confederate (one who used the stairs), and in the third situation, there was a noncompliant confederate (one who used the elevator). The subjects tended to imitate the confederates. What more do you need to know about the study to determine whether the presence or absence of a confederate causes a change in the compliance of subjects?
-Identify whether there was random assignment to groups. Without random assignment there is the possibility of bias, so we cannot infer causation. Your answer is correct. -Identify the sample size of the study. Without enough participants to observe the full range of variability in subjects we cannot control for other relevant factors, so we cannot infer causation.
Typical value found on a histogram
A typical value can be found by finding the center of the distribution the typical number of sleep hours for these men is 7.0-7.5
A :A statistics class is made up of 171 men and 222 women. What percentage of the class is male?. B: A different class has 278 students, and 62.2% of them are men. How many men are in the class? C: A different class is made up of 54% women and has 272 women in it. What is the total number of students in the class?
A 1. To find the percentage of the class that is male, first find the total number of students in the class. 2.Next find the proportion of men in the statistics class by dividing the number of men in the class by the total number of students in the class. 3.To convert the decimal proportion found in the previous step to a percentage, multiply the number by 100%. Ex: 0.436X100% = 43.6 B. 1. To find the number of men in the class, first convert the percentage 62.262.2% to its decimal equivalent. Ex: 62.2% / 100% = 0.622 B. 2. Now multiply the proportion by the total number of people in the class to find the number of men in the class. there are 173 men in the 278 C. 1. To find the total number of people in the class, first convert the percentage of women into its decimal equivalent. ( 60%/100%)= 0.54 C. 2 : The question can be interpreted as 60% of some unknown number (call it x) is equal to 27. Set up an equation as described and solve for x. 0.54x = 27 X= 27/0.54. (divide both sides by 0.54) = 50
Pareto graph
A bar graph where the bars are arranged in descending order. - good for categorical
Stemplot
A graphical representation of a quantitative data set. Leading values of each data point are presented as stems and second digits are given as leaves.
The mean birth weight for children born in a certain country at full term (after 40 weeks) is 3470 grams (about 7.7 pounds). Suppose the standard deviation is 600 grams and the shape of the distribution is symmetric and unimodal. Complete parts a and b below. A.Find the range of birth weights (in grams) of children born from one standard deviation below the mean to one standard deviation above the mean.
A. To find the top value, add one standard deviation to the mean, (bar over X) + s =3470+600 = 4070 To find the bottom value, subtract one standard deviation from the mean, (bar over X) - s = 3470-600 = 2870
A study looked at the effects of light on female mice. Fifty mice were randomly assigned to a regimen of 12 hours of light and 12 hours of dark (LD), while another fifty mice were assigned to 24 hours of light (LL). Researchers observed the mice for two years. Three of the LD mice and 10 of the LL mice developed tumors. The accompanying table summarizes the data. Complete parts a through c. A. Determine the percentage of mice that developed tumors from each group (LL and LD). Compare them and comment. B.Was this a controlled experiment or an observational study? How do you know? C. Can we conclude that light for 24 hours a day causes an increase in tumors in mice? Why or why not?
A. In the LD mice, 6% developed tumors. In the LL mice, 20% developed tumors. The LD mice developed tumors at a lower rate than the LL mice. B. This was a controlled experiment because there were two groups that were randomly assigned by the researchers C. Because it was a controlled experiment, it can be concluded that light for 24 hours a day causes an increase in tumors in mice.
Refer to the accompanying graph, which shows the time spent on a typical day talking on the cell phone for some men and women. Each person was asked to choose the one of four intervals that best fitted the amount of time they spent on the phone (for example, "0 to 4 hours" or "12 or more hours"). Complete parts (a) through (d) below. B. Is the graph a bar chart or a histogram? Which would be the better choice for these data? C.If you had the actual number of hours for each person, rather than just an interval, what type of graph should you use to display the distribution of the actual numbers of hours?
A. One variable is GENDER which is CATEGORICAL variable since it describes QUALITIES. The other variable is TIME RANGE which is a CATEGORICAL variable since it describes QUALITIES B.The graph is a bar chart. It is the better choice since both variables are categorical. C. Two histograms should be used since the time would be a numerical value
A group of overweight people are asked to participate in a weight loss program. Participants are allowed to choose whether they want to go on a vegetarian diet or follow a traditional low-calorie diet that includes some meat. Half of the people choose the vegetarian diet, and half choose to be in the control group and continue to eat meat. Suppose that there is greater weight loss in the vegetarian group. Complete parts (a) and (b) below. A. Suggest a plausible confounding variable that would prevent us from concluding that the weight loss was due to the lack of meat in the diet. Explain why it is a confounding variable. B. Explain a better way to do the experiment that is likely to remove the influence of confounding variables.
A. People who are not prepared to change their diet very much (such as by excluding meat) might also not change other factors that affect weight, such as how much exercise they get. B. The experiment would be improved if some subjects were randomly assigned to eat meat and the remaining subjects to consume a vegetarian diet.
The data were collected from a statistics class. The column head gives the variable, and each of the rows represents a student in the class. Find the frequency, proportion, and percentage of women.
A. The frequency of women in the class is 6 B. The proportion of women in the class is 6/11 C. The percentage of women in the class is 54.5%
A. Interpret the mean. Choose the correct answer below. B. Interpret the standard deviation. Choose the correct answer below.
A. The mean is the typical or average number of vacation days B. The standard deviation measures how many vacation days more or less than the mean a country usually has
The idea of sending delinquents to "Scared Straight" programs has appeared recently in several media programs. In a 1983 study, each male delinquent in the study (all aged 14-18) was randomly assigned to either Scared Straight or no treatment. The males who were assigned to Scared Straight went to a prison, where they heard prisoners talk about their bad experiences there. Then the males in both the experimental and the control group were observed for 12 months to see whether they were rearrested. Complete parts (a) and (b) below. A.a. Report the rearrest rate for the Scared Straight group and for the No Treatment group, and state which is higher. B. This experiment was done in the hope of showing that Scared Straight would cause a lower arrest rate. Did the study show that? Explain.
A. The rearrest rate for the Scared Straight group is 81.1%. the rearrest rate for the No Treatment group is 67.3%. The rearrest rate for the Scared Straight group is higher than the rearrest rate of the No Treatment group. B.No. The study does not show that Scared Straight causes a lower arrest rate, because the rearrest rate in the Scared Straight group was higher than in the No Treatment group.
A doctor reported on a study that treated children who had sleep apnea, which interferes with breathing while a child is asleep. In the study, 464 children, 5 to 9 years of age, were randomly assigned to either surgery or to be under constant watch for a certain period of time. The study found that there were significantly greater improvements in behavioral, quality-of-life, and sleep study findings in the group that had surgery than the group assigned to constant watch. Complete parts (a) and (b) below. A. Was the study a controlled experiment or an observational study? Explain how you know. B. Assuming that the study was properly conducted, can we conclude that the early surgery caused the improvements? Explain.
A. The study was a controlled experiment because the children were randomly assigned to either surgery or constant watch. This is essential to conducting a controlled experiment. B. We can conclude that the early surgery caused the improvements because it was a randomized controlled experiment.
The accompanying histograms show the circumferences of heads for a group of men and a group of women. Complete parts (a) through (e) below A.If you were describing the men's heads in terms of shape, center, and spread, without comparing them to the women's heads, would you use the mean and standard deviation or the median and interquartile range? Why? B.If you were describing the women's heads in terms of shape, center, and spread, without comparing them to the men's heads, would you use the mean and standard deviation or the median and interquartile range? Why? C.If you were comparing the two groups, what measures would you use, and why? D. In which of the two graphs would the mean and median be close together, and why? E.In which of the two graphs would the mean and median be farther apart, and which would be larger?
A. Without comparing the men's data set to the women's data set, the MEAN AND STANDARD DEVIATION are preferred measures of shape, center, and spread for the men's data set because the distribution is ROUGHLY SYMMETRIC B.Without comparing the women's data set to the men's data set, the MEDIAN AND INTERQUARTILE RANGE are preferred measures of shape, center, and spread for the women's data set because the distribution is ASYMMETRIC DUE TO OUTLIERS C. The MEDIAN AND INTERQUARTILE RANGE are preferred measures of shape, center, and spread for comparing the two groups. The distribution of THE WOMENS set of data is ASYMMETRIC DUE TO OUTLIERS meaning that the MEDIAN AND INTERQUARTILE RANGE are appropriate measures for comparison D.The mean and median would be closer together in the MENS data set than the WOMENS data set, because the distribution of the data is ROUGHLY SYMMETRIC E.The mean and median would be farther apart in the WOMENS data set than the MENS because the distribution of the data is ASYMMETRICAL DUE TO OUTLIERS the mean would be LARGER than the median because the larger outliers INCREASES the value of the MEAN
The accompanying relative frequency histogram shows the number of hours of sleep (Sleep Hours) reported as experienced "last night" for 2626 college men. Complete parts (a) and (b) below. A. About how many men had 6 or fewer hours of sleep? B. The graph is bimodal. What are the two modes?
A. about 5 men had 6 or fewer hours of sleep B. 7.0, 8.0 calculate the total proportion of men who got 6 or less hours of sleep by adding the proportion of men who got 5.5 hours of sleep to the proportion of men who got 6.0 hours of sleep. (0.04+0.15=0.19) Multiply the total proportion by the total number of men to determine the number of men who got 6 hours of sleep or less. (0.19X26)
Because this was an observational study, it only shows an association; it does not show that the tutoring worked. It could be that more motivated students attended the tutoring and that was what caused the higher grades. A. The doctor is concerned that if his most severely depressed patients do not receive the antidepressants, they will get much worse. He therefore decides that the most severe patients will be assigned to receive the antidepressants. Explain why this will affect his ability to determine which approach works best. B.What advice would you give the doctor to improve his study? C. The doctor asks you whether it is acceptable for him to know which treatment each patient receives and to evaluate them himself at the end of the study to rate their improvement. Explain why this practice will affect his ability to determine which approach works best. D.What improvements to the plan in part (c) would you recommend?
A. if the doctor decides on the treatment, this could introduce bias. B. The doctor should randomly assign the patients to the different treatments. C.If the doctor is aware of the treatment each patient receives, that might influence his opinion about the effectiveness of the treatment. D. To prevent bias, the experiment should be double-blind. Neither the patients nor the doctor evaluating the patients should know whether each patient received medication
Name two measures of the center of a distribution, and state the conditions under which each is preferred for describing the typical value of a single data set. A. What are two measures of the center of a distribution? B.Under what conditions is the median preferred? C.Under what conditions is the mean preferred
A.median and mean B.The median is preferred when the data is strongly skewed or has outliers. C.The mean is preferred when the data is relatively symmetric.
Mounds of distribution
A one-mound distribution is called unimodal, a two-mound distribution is called a bimodal distribution, and a distribution that has more than two modes is referred to as multimodal. Notice that the modes do not have to be the same height.
Two drugs were tested to see whether they helped women with breast cancer. Of 1060 women, about half were randomly assigned to drug A and the other half were assigned to drug B. After 77 months, 473 out of 539, and 426 out of 521 women assigned to drugs A and B, respectively, were alive. Complete parts (a) and (b) below. A. The survival rate for drug A is 87.8%, the survival rate for drug B would be 81.8%. The survival rate for drug A is HIGHER than drug B B. Was this a controlled experiment or an observational study? Explain why. From studies like these, can we conclude a cause-and-effect relationship between the drug type and the survival percentage? Why or why not?
B: This is a controlled experiment because the researchers randomly determined the groups. We can conclude a cause-and-effect relationship because the women were randomly assigned to the treatment and controlgroups, which controls for other variables
A group of educators want to determine how effective tutoring is in raising students' grades in a math class, so they arrange free tutoring for those who want it. Then they compare final exam grades for the group that took advantage of the tutoring and the group that did not. Suppose the group participating in the tutoring tended to receive higher grades on the exam. Does that show that the tutoring worked? If not, explain why not and suggest a confounding variable.
Because this was an observational study, it only shows an association; it does not show that the tutoring worked. It could be that more motivated students attended the tutoring and that was what caused the higher grades.
When you are comparing two sets of data, and one set is strongly skewed and the other is symmetric, which measures of the center and variation should you choose for the comparison?
The medians and interquartile ranges
The histogram shows frequencies for the ages of 25 randomly selected CEOs. Convert this histogram to one showing relative frequencies by relabeling the vertical axis with the appropriate relative frequencies. Note that the new labels for the vertical axis are the only thing that will change.
D
The histograms contain data with a range of 1 to 6. Which group would have the larger standard deviation, group A or group B? Why?
Group A has a larger standard deviation. There are more observations far from the mean, which contribute to a larger standard deviation.
The five-number summary for a distribution of final exam scores is 35,75,79,90,96. Explain why it is not possible to draw a boxplot based on this information. (Hint: What more do you need to know?)
It isn't possible to draw a box plot based on the five-number summary because the box plot must mark all potential outliers. Since the minimum is lower than the left limit, there may be other unknown outliers between the minimum and the left limit and so the box plot cannot mark all potential outliers
In 1994, major league baseball players went on strike. At the time, the average salary was $1,049,589, and the median salary was $337,500. If you were representing the owners, which summary would you use to convince the public that a strike was not needed? If you were a player, which would you use? Why was there such a large discrepancy between the mean and median salaries? Explain.
If you were representing the owners, you would use the AVERAGE salary to convince the public that a strike was not needed. If you were a player, you would use the MEDIAN to convince the public that a strike was needed. The average and median salaries differ so greatly because THE DISTRIBUTION OF SALARIES IS SKEWED RIGHT
Is this easier to determine with the pie chart or with the bar chart? Explain. Choose the correct answer below.
It is easier to use a bar chart because you can compare values using the heights of the bars, whereas it can be hard to determine which of two slices in a pie chart is larger when they are close in size and not adjacent to each other.
If you moved to one of these regions and met 50 people, in which region would you be least likely to meet at least one person living with AIDS?
Least likely in region D
In 2011, the mean rate of violent crime (per 100,000 people) for 24 particular states was 431 The standard deviation was 172. Assume that the distribution of violent crime rates is unimodal and symmetric. Complete parts (a) through (d) below. A. The percentage of data that should occur within 2 standard deviations of the mean would be = 95% (see figure B) Between which two values would you expect to find about 95% of the violent crime rates? (X-2s and X+2s) = 87,775 B.If one of these states had a violent crime rate of 438crimes per 100,000 people, would this be consideredunusual? Explain.
Recall that the Empirical Rule states that if a distribution of data is bell-shaped, then the following is true. Fig A. Approximately 68% of the observations fall within one standard deviation of the mean, that is, between, x(bar)-s and x(bar)+s ( denoted Xbar +/- S) Fig B. Approximately 95% of the observations fall within two standard deviations of the mean (X bar +/- 2s) Fig C. All or nearly all observations fall within three standard deviations of the mean (Xbar +/- 3s) B. Take the term "unusual" to be defined such that the data value in question lies more than two standard deviations from the mean. Use the Empirical Rule to find out within which range of standard deviations the given values lie.
standard deviation
Recall that the standard deviation is a measurement of how spread out the values within a data set are. how much scores vary around the mean score
A sociologist says, "Typically, men in a certain country still earn more than women." What does this statement mean?
The center of the distribution of salaries for men in the country is greater than the center for women.
Why is it best to compare medians and interquartile ranges for these data, rather than comparing means and standard deviations?
Some of these data have outliers and/or are skewed, and the median and interquartile range are resistant to outliers.
Suppose you have a data set with the weights of all members of a high school soccer team and all members of a high school academic decathlon team (a team of students selected because they often answer quiz questionscorrectly). Which team do you think would have a larger standard deviation of weights? Explain.
The academic decathlon team would have a larger standard deviation of weights. Academics don't require a specific weight to succeed, so the distribution of weights should mirror that of the general population.
A dieter recorded the number of calories he consumed at lunch for one week. As you can see, a mistake was made on one entry. The calories are listed in increasing order below. 349, 371, 386 ,398, 412, 4190 When the error is corrected by removing the extra 0, will the mean change? Will the median? Explain without doing any calculation.
The corrected value will give a different mean but not a different median. Medians are resistant to outliers and not as affected by extreme values, but the more extreme a value is, the more the mean is affected by it
The distribution of in-state annual tuition for all colleges and universities in the United States is bimodal. What is one possible reason for this bimodality?
The distribution might be bimodal because private colleges and public colleges tend to differ in amount of tuition.
Predict the shape of the distribution of the numbers of times a group of 500 people eat breakfast in one week.
The distribution will be left-skewed. Most people will report eating breakfast every day, with a few reporting various values less than 7.
A teacher asks 90 students who drive how many speeding tickets they received in the last year. Predict the shape of the distribution and explain.
The distribution will be right-skewed. Most people will have no tickets, but there will be a few people with 1, 2, 3, or more tickets.
How could the graph be improved
The graph could be improved by making it a bar graph or a pie chart. This change would make the variable garage be seen as categories, not as numbers.
is the graph a histogram or a bar graph? How do you know?
The graph is a bar graph because the bars are separated
is the given graph a bar graph or a histogram?
The graph is a histogram because the bars touch.
The histogram shows the number of televisions in the homes of 90 community college students. Judging from the histogram, what is the approximate mean number of televisions in the homes in thiscollection? Explain.
The mean number of televisions per home is between 3 and 4. The mean is near the center, which is due to the fact that the histogram is roughly symmetric.
A study was done to see whether a smaller dose of flu vaccine could be used successfully. In this study, the usual amount of vaccine was injected into half the patients, and the other half of the patients had only a small amount of vaccine injected. The response was measured by looking at the production of antibodies. In the end, the lower dose of vaccine was just as effective as a higher dose for those under 65 years old. What more do we need to know to be able to conclude that the lower dose of vaccine was equally effective at preventing the flu for those under 65?
The patients need to be randomly assigned the full or lower dose. Without randomization there could be bias, however, with randomization we can infer causation.
Are the ranks for the rates the same as the ranks for the number of cases? If not, describe at least one difference
The ranks for the rates are different from the ranks for the number of cases. Region F had the least number of cases but had the highest rate of cases.
The histogram shows frequencies for the ages of 25 randomly selected CEOs. Approximately what is a typical age of a CEO in this sample?
The typical age of a CEO in this sample is between 56 and 60 years old.
A real estate agent claims that all things being equal, houses with swimming pools tend to sell for less than those without swimming pools. What does this statement mean?
The typical price for homes with pools is smaller than the typical price for homes without pools.
Is the variable WeightWeight numerical or categorical? Explain.
The variable is numericalnumerical because the values are numbersnumbers.
Do these men, or do these women, have greater variation in brain size? Why?
The variation in brain sizes between men and women is less than 0.3, indicating no significant difference.
Indicate whether the following study is an observational study or a controlled experiment: A researcher is interested in the effect of music on memory. She randomly divides a group of students into three groups: those who will listen to quiet music, those who will listen to loud music, and those who will not listen to music. After the appropriate music is played (or not played), she gives all the students a memory test.
This is a controlled experiment. She assigns students to the control and treatment groups at random in order to control for all relevant factors aside from the effect of music on memory, which is essential to conducting a controlled experiment.
Indicate whether the following study is an observational study or a controlled experiment: Patients with Alzheimer's disease are randomly divided into two groups. One group is given a new drug, and the other is given a placebo. After six months they are given a memory test to see whether the new drug fightsAlzheimer's better than a placebo.
This is a controlled experiment. The researchers randomly assigned patients to either a treatment or control group, and they gave the patients a test afterwards to identify the effect of the new drug. This satisfies a key criterion of controlled experiments.
Indicate whether the following study is an observational study or a controlled experiment: Records of patients who have had broken ankles are examined to see whether those who had physical therapy achieved more ankle mobility than those who did not.
This is an observational study. Since the researchers did not randomly assign subjects to the control or treatment group beforehand, they did not satisfy a key feature of controlled experiments.
Indicate whether the following study is an observational study or a controlled experiment. A local public school encourages, but does not require, students to wear uniforms. The principal of the school compares the grade point averages of students at this school who wear uniforms with the GPAs of those who do not wear uniforms to determine whether those wearing uniforms tend to have higher GPAs.
This is an observational study. The principal does not randomly assign students to either wear or not wear uniforms. Random assignment is essential to conducting a controlled experiment.
A college magazine suggested that overeating reduces brain function. Is this likely to be a conclusion from observational studies or randomized experiments? Can we conclude that overeating causes a reduction in brainfunction? Why or why not?
This is likely to be from observational studies. It would not be ethical to assign people to overeat. We cannot conclude causation from observational studies because of the possibility of confounding factors.
What percentage of men had an older sibling? ( chart that adds up)
To find the proportion, divide the number of men with an older sibling by the total number of men. Then convert the proportion to a percentage.
Some people believe that wearing copper bracelets is a good treatment for arthritis of the hand. To test this belief, suppose you recruit 100 people and supply them all with copper bracelets. After the patients wear the bracelets for a month, you ask them whether or not their pain is less than it was before they began wearing the bracelets. Explain how to improve this study.
To improve the study, the patients should be randomly divided into two groups; one group will be given the copper bracelets, and the other group will be given non-copper bracelets. After a month, the patients will be surveyed on the levels of their pain.
What type(s) of graph(s) would be more appropriate?
Two histograms or a pair of dot plots with a common horizontal axis would be more appropriate since the given data are numerical.
The circles shown to the right are similar, but not exactly the same. This is an example of _______.
Variation
The study of statistics rests on what two major concepts?
Variation And Data
A study reported on the effects of vitamin C in breast milk for breast-feeding mothers. The children whose mothers had chosen to take high doses of vitamin C had a 30% lower risk of developing allergies. Can you conclude that the use of vitamin C caused the reduction in allergies? Why or why not?
You cannot conclude that the use of vitamin C caused the reduction in allergies because the researchers did not randomly assign mothers to treatment and control groups. This step is necessary for identifying causation.
Dot Plot
a graphical display of data using dots - good for numerical data
Indicate whether the study is an observational study or a controlled experiment: A student watched picnickers with a large cooler of soft drinks to see whether teenagers were less likely than adults to choose diet soft drinks over regular soft drinks.
an observational study
The five-number summary for a distribution of final exam scores is 45,72,79,90,100. Is it possible to draw a boxplot based on this information? Why or why not?
it is possible to draw a boxplot based on this information. Both the minimum and maximum are within the bounds of the left limit and right limit, which means that all potential outliers can be displayed. This is necessary to construct the boxplot.
In 2008, a highway safety administration reported that the number of pedestrian fatalities in City A was 65 and that the number in City B was 45. Can we conclude that pedestrians are safer in City B? Why or why not?
No, in order to compare the fatalities the statistics must include the number of fatalities per pedestrian. There may be fewer pedestrians in City B causing the difference.
