Contemporary Math Exam 3
Consider the hypothetical study described below. Based solely on the information given, do you have reason to question the results of the study? Explain your reasoning. A TV talk show host asks the TV audience, "Do you support new national mileage standards for automobiles?" and asks people to vote by telephone at a toll-free number.
Yes, there is reason. The TV audience might not be representative of the population. Your answer is correct. Yes, there is reason. The wording of the question might produce inaccurate or dishonest responses. This is the correct answer. Yes, there is reason. Call-in polls tend to be biased.
A report claims that the returns for the investment portfolios with a single stock have a standard deviation of 0.55, while the returns for portfolios with 33 stocks have a standard deviation of 0.323. Explain how the standard deviation measures the risk in these two types of portfolios.
A lower standard deviation means more certainty in the return and less risk. Hence, the returns for portfolios with 33 stocks have less risk than the ones with a single stock.
Figure 5.31 uses television sets to represent the number of homes with cable in 1980 and 2012. Note that the heights of the TVs represent the number of homes. Briefly explain how the graph creates a perceptual distortion that exaggerates the true change in the number of homes with cable.
A perceptual distortion is created because the heights of the TVs represent the number of homes, but the human eye tends to focus on the volumes of the TVs, exaggerating the true change to appear too large.
Consider the data displayed in the figure. Complete parts (a) and (b) below.
Consider the data displayed in the figure. Complete parts (a) and (b) below. a) Which group scored higher on the test, boys in country D or girls in country F? Estimate the test scores of each group. The boys in country D scored about 510. (Round to the nearest ten as needed.) The girls in country F scored about 530. (Round to the nearest ten as needed.) The girls in country F scored higher on the test. b) Which group scored higher on the test, boys in country B or girls in country E? Estimate the test scores of each group. The boys in country B scored about 550. (Round to the nearest ten as needed.) The girls in country E scored about 540. (Round to the nearest ten as needed.) The boys in country B scored higher on the test.
Identify the sampling method (simple random sampling, systematic sampling, convenience sampling, or stratified sampling) in the following study. A famous magazine chooses its "most-likeable presidents" by compiling responses from readers who mail in a survey printed in the magazine.
Convenience sampling
The following data give the margin of victory for a football championship over 15 years. 13 4 4 11 12 33 3 27 6 4 3 4 4 14 6 a. Find the mean and median margin of victory. b. Identify the outlier in the data set. If the outlier is eliminated, what are the new mean and median?
Find the mean and median of the weights. The mean is 9.9 The median is 6. b. Which, if any, of the margins of victory would be considered the outlier? Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The outlier is 33. (Type an integer or a decimal.) A. The mean without the outlier is 8.2 B. None of the weights would be considered an outlier. Find the median with the outlier excluded. Select the correct choice below and, if necessary, fill in the answer box to complete your choice. A. The median without the outlier is 5.
Each year, a group surveys 50,000 households to study internet usage. In one area of the study, the group is interested in finding out how many hours a day the household spends streaming video from the internet. Describe the five basic steps in a statistical study with an example of their application below.
First step State the goal of your study. In this case, it is to discover how many hours per day a household spends streaming internet video. Second Step Choose a representative sample from the population. In this case, it would be choosing a sample of 50,000 households. Third Step Collect raw data from the sample and summarize these data by finding sample statistics of interest. In this case, it would be asking the households how many hours they spend sreaming internet video and turning this data into an average. Fourth Step Use the sample statistics to infer the population parameters. In this case, based on the data gathered, the group estimates the average time per day that a household spends streaming internet video. Fifth Step Draw conclusions to determine what you learned and whether you achieved your goal. In this case, we discovered the average time per day that a household spends streaming internet video.
A researcher wants to determine the percentage given below. Describe how the researcher should apply the five basic steps in a statistical study. (Assume that all the people in the poll answered truthfully.) the percentage of businesses that have over 20 employees
First step The population is all businesses. The researcher wants to estimate the percentage in this population that have over 20 employees. Second step The researcher should gather data about having over 20 employees from the largest sample of businesses from which the researcher can gather data. Third step The sample statistic of interest is the percentage of businesses in the sample that have over 20 employees. Fourth step The researcher should use the sample statistic as an estimate for the population value of the percentage of businesses that have over 20 employees and then use the methods of statistics to determine how good that estimate is. Fifth step the researcher should use the methods of statistics to determine the quality of the estimate of the population parameter and draw conclusions based on this estimate accordingly.
Much like sound bites of news stories, statistical studies are often reduced to one- or two-sentence "stat-bites." For the stat-bite below, discuss what crucial information is missing and what more someone would want to know before acting on the study. A news station poll reveals that 77% of Americans say "Merry Christmas" rather than "Happy Holidays."
How many Americans use another greeting or say nothing? How were respondents chosen?
A recent telephone poll of 997 randomly selected adults revealed that 5 in 10 adults believe there has been progress in finding a cure for cancer in the last 30 years. Describe the population, sample, population parameters, and sample statistics.
Population All adults in the country Sample The 997 adults selected Population Parameter The percentage of adults in the country who believe there has been progress Sample statistic 5 out of 10
The following statistical study gives a sample statistic and a margin of error. Find the confidence interval and answer any additional questions. A national survey of 1522 respondents reached on land lines and cell phones found that the percentage of adults who favor legalized abortion has dropped from 53% a year ago to 45%. The study claimed that the error attributable to sampling is ±4 percentage points. Would you claim that a majority of people are not in favor of legalized abortion?
The confidence interval for the study is 41% to 49%. Yes, the majority of people are not in favor of legalized abortion because the confidence interval for those in favor of legalized abortion is below 50%.
The figure below is a time-series graph showing overall death rates (from all causes) in a country for 1900 to 2015. The spike in 1919 was due to a worldwide epidemic of influenza. Write a few sentences summarizing the overall trend, describing how much the death rate changed over this period of time and putting the 1919 spike into context in terms of its impact on the population.
The death rates have decreased by about 30% since the 1920s. The death rate during the influenza epidemic was over twice its current rate.
Consider the headline and brief summary below. Discuss whether the headline accurately represents the study. Headline: "Sex more important than jobs" Story summary: A survey found that 82% of 500 people interviewed by phone ranked a satisfying sex life as important or very important, while 79% ranked job satisfaction as important or very important.
The headline does not accurately represent the study because respondents are not asked to compare sex life and job satisfaction directly.
Blood alcohol concentrations of drivers involved in fatal crashes and then given jail sentences are shown below. Find the mean, median, and mode of the listed numbers. 0.26 0.17 0.17 0.16 0.13 0.23 0.29 0.23 0.14 0.16 0.11 0.16
The mean is .184 The median is .165 The mode is (are) .16
Describe how the range of a distribution is defined and calculated.
The range of a data set is the difference between its highest and lowest data values.
Decide whether the statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. The survey of religious beliefs suffered from selection bias because the questionnaires were handed out only at Catholic churches.
The statement makes sense because no religions other than Catholicism will be represented in the responses.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. My bar chart contains more information than yours, because I made my bars three-dimensional.
The statement does not make sense because making the bars three-dimenional does not necessarily mean that information was added. It could mean that only the appearance of the graph was changed.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. The cumulative frequency of C grades in our class of 33 students was 45.
The statement does not make sense because the frequency cannot be greater than the total.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. I wanted to test the effects of vitamin C on colds, so I gave the treatment group vitamin C and gave the control group vitamin D.
The statement does not make sense. The control group should only receive a placebo, not another treatment.
Decide whether each of the following statements makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. I made a distribution of 15 apartment rents in my neighborhood. One apartment had a much higher rent than all of the others, and this outlier caused the mean rent to be higher than the median rent.
The statement makes sense because an outlier with a large value increases the mean, but does not affect the median.
Decide whether the following statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. For the 30 students who took the test, the high score was 80, the median was 75, and the low score was 40.
The statement makes sense because it is possible that when sorting the 30 scores from low to high, the first value was 40, the highest value was 80, and 75 was halfway between the 15th and the 16th score
Determine whether the statement makes sense (or is clearly true) or does not make sense (or is clearly false). Explain your reasoning. There is a strong negative correlation between the price of tickets and the number of tickets sold. This suggests that in order to sell a lot of tickets, the price should be lowered.
The statement makes sense. To increase the ticket sales, the price should be decreased since there is a strong negative correlation.
Determine whether the following study described is observational or an experiment. If the study is an experiment, identify the control and treatment groups, and discuss whether single- or double-blinding is necessary. If the study is observational, state whether it is a retrospective study, and if so, identify the cases and controls. A study of 2482 runners looked for relationships between specific types of running injuries and variables such as height, weight, body mass index, age, and running history.
The study is observational -study is not an experiment -study is not an experiment -the study is observational and it is a retrospective study. -the cases would be the runners with at least one of the running injuries. The controls would be the runners without any of the running injuries.
The following table gives the average hours of traditional (network and cable) television watched per week in five age categories. For the age categories, use the data points 6, 15, 21, 40, 65. Complete parts (a) through (c).
The two variables appear to have strong positive correlation. As people get older, they have more free time and can watch more TV.
Determine whether the following variable is qualitative or quantitative, and explain why. The responses of customers on a satisfaction survey that used a scale from 0=terrible to 5=fantastic
The variable is qualitative because satisfaction is subjective.
Consider the following statement about a correlation. State the correlation clearly. Then state whether the correlation is most likely due to a coincidence, a common underlying cause, or a direct cause. Explain your answer. Over the past three decades, the number of miles of freeways in Los Angeles has grown, and traffic congestion has worsened.
There is a positive correlation between the number of miles of freeways and the traffic congestion in Los Angeles. The correlation is most likely due to the common underlying cause of Los Angeles having a very large population.
For the following pair of variables, state the units that might be used to measure each variable. Then state whether you believe that they are correlated. If you believe they are correlated, state whether the correlation is positive or negative. Explain your reasoning. Weight and how frequently a person attends rock concerts
To measure weight, the unit pounds might be used. To measure the frequency a person attends rock concerts, the unit number of attendances might be used. The variables are not correlated.
Consider the scatterplot to the right. a. State whether the diagram shows a positive correlation, a negative correlation, or no correlation. If there is a positive or negative correlation, is it strong or weak? b. Summarize any conclusions that can be drawn from the diagram.
a. There is a strong negative correlation. b. Heavier cars generally get lower gas mileage.