Unit Activity Answer Sheet

Transcription

1 Algebra 1 Unit Activity Answer Sheet Unit: Descriptive Statistics The Unit Activity will help you meet these educational goals: Mathematical Practices You will make sense of problems and solve them, reason abstractly and quantitatively, use mathematics to model real-world situations, use appropriate tools strategically, and look for and make use of structure. Inquiry You will conduct online research in which you will make observations, analyze results, communicate your results in graphs and written form, and draw conclusions. STEM You will apply mathematical and technology tools and knowledge to analyze real-world situations; gain insight into careers in science, technology, engineering, and math; and grow in your understanding of mathematics as a creative human activity. 21st Century Skills You will employ online tools for research and analysis, use critical-thinking and problem-solving skills, communicate effectively, perform largescale data analysis, and carry out technology-assisted modeling. Directions and Analysis Task 1: The Number of Hispanics (Latinos) in the United States US Census Regions (from the Energy Information Administration) Copyright 2012 PLATO Learning, Inc. All rights reserved. 1

2 Consider the population of Hispanic (Latino) people in the United States, according to the 2010 US Census. Examine the data for the 2010 US Census in this spreadsheet. a. How do the columns titled Number and % of Total Population relate to the column titled Total? The Total column gives the total population of the state or region. The Number column gives the number of Hispanic (Latino) people in the particular state or region. The % of Total Population column gives the percentage of Hispanic (Latino) people in the total population of the state or region. b. Make a histogram of the state data in the column titled % of Total Population. (If you need help, follow these instructions for using the online probability tools. Note that you can copy a column of data from the spreadsheet and paste it into the Histogram tool.) Set useful limits and intervals, and label the histogram appropriately. Export an image of the histogram, and paste it below. The use of an interval of 5 gives a good distribution of the data and helps in identifying the outliers. It also shows skewness of the data. c. Generate a box plot of the state data in the column titled % of Total Population. (You can copy a column of data from the spreadsheet and paste it into the Box Plot tool.) Be sure to add appropriate labels to your box plot. Export an image of your box plot, and paste it below. 2

3 d. Describe the spread, shape, and skewness, if any, of the graph. The range of the data (the difference between the smallest and largest values) is The graph is skewed strongly to the right. e. What information about central tendencies can you determine from the histogram and the box plot? The histogram itself doesn t display the mean and the median. However, the Histogram tool does display values for both the mean (12.28) and the median (8.3). The box plot itself shows the value of the median (8.3). f. Return to the Box Plot tool, click the Exclude Outliers checkbox, and then click Update. The plot boundaries contract, with outlier data points showing beyond the whiskers of the plot. Outliers are generally considered to be points that are more than 1.5 times (interquartile range) below Q1 or above Q3. What are the minimum and maximum values for the box plot once you exclude outliers? Based on your box plot, how many outliers do you have? Minimum (excluding outliers): 2.28% ( %) = -4.31% (impossible) Maximum (excluding outliers): 8.88% + ( %) = 15.47% It looks like there are five outliers. All are above the maximum of 15.47% 3

4 g. Which states are represented by the outlier data? What do these states have in common that might contribute to making them outliers? The outliers represent Puerto Rico (98.7%) and the states of New Mexico (42.1%), California (32.4%), Texas (32.0%), Arizona (25.3%), and Nevada (19.7%). Puerto Rico was a Spanish colony at one time and has been Hispanic for centuries. The rest are Southwestern states that are close to Mexico and Latin America. Because of geographical proximity, Hispanics may have immigrated to these states over a great many years. h. According to the US Census data, the Hispanic (Latino) population of the United States as a whole is 16.3% of the total 2010 US population (as shown in cell G5). Where would this percentage fit into the list of the distribution of the individual states on your latest box plot? Does it seem surprising that it would fit there? How might you explain this situation? The maximum value of the box plot (once outliers are excluded) is 15.47%. At 16.3%, the US as a whole is an outlier compared with the individual states. This does seem surprising. Looking at the states with a high Hispanic population, though, it s clear that some of the biggest (such as California, with 33.9 million people) have high Hispanic populations. Vermont, on the other hand, has a very low Hispanic population, but it has a very small population (about 1/50 the size of California.) As a result, the United States as a whole has a much higher Hispanic population percentage than most of its individual states. Task 2: Diamonds If you were in the diamond business, you would have to know how to price diamonds accurately. Otherwise you would either be losing money (selling too cheaply) or losing customers (selling too expensively.) Many people buy diamond engagement rings, and it is often a significant personal purchase. In fact, buying a diamond and buying a house are similar in two ways: Both a house and a diamond can be significant investments of your hard-earned money. No two are exactly alike (unlike two new cars), so you can t just shop around and clearly see who has the best price. So, having some sort of price ruler can be very useful for a lot of people. In this lesson, you ll practice building that ruler, based on data. Read about the 4 Cs of diamonds (cut, clarity, color, and carat weight) for some background information. With this data, you will have a pretty good idea of what size diamond you can afford, and you will know whether a diamond you see at the jeweler s is worth the cost. a. Affordable Diamond Pricing Imagine you have set out to purchase a diamond, and you want all the information you can possibly get before the purchase. You run an Internet search on diamonds to learn 4

5 more about the different cuts, carat ranges, color values, and so on. After having researched a bit, you ve decided on the basic cut, quality, color, and weight range that you can probably afford. This diamond pricing spreadsheet gives three snapshots of a large set of data on diamond pricing. The carats tab gives the 2012 data on diamonds ranging from 0.30 to 0.40 carats. 1. Go to the Subset tab, which gives a subset of randomly chosen data from the first tab. Plot the data of weight versus price in this tab using the Scatter Plot tool. (If you need help, follow these instructions for using the online probability tools.) Export an image of the plot, and paste it in the space below. Find the equation of the regression line and the value of the correlation coefficient (r). The equation of the regression line is y = 5,405.71x The value of the correlation coefficient is What can you say about the relationship between the price and the weight? The value of the correlation coefficient is Since it s a positive value, the correlation between price and weight of diamonds is positive. Also, the value is close to +1, which shows that the linear relation is strong. 5

6 3. What can you say about the slope of the regression line? The slope of the line is 5, Since the slope is positive, it shows that the line is sloping upward the right, indicating a positive correlation. This means that price increases with weight, as one would expect. Another way to interpret the slope is to say that for every increase of one carat, the price goes up $5, Does the y-intercept make sense? The y-intercept of the line is This value means that for a diamond weighing zero carats, the price is $ The weight cannot be zero, and the price cannot be negative, so the y-intercept does not make sense. 5. The VVS carats tab is also a subset of the data in the first tab; it contains data on diamonds with VVS1 clarity. There are about 200 diamonds in this subset. Plot the data of weight versus price in this tab. Export an image of the plot, and paste it in the space below. Find the equation of the regression line and the value of the correlation coefficient (r). The equation of the regression line is y = 4,760.19x The value of the correlation coefficient is

7 6. What can you say about the relationship between the price and the weight? The value of the correlation coefficient is Since it s a positive value, the correlation between the price and the weight of diamonds is positive. Also, the value is close to +1, which shows that the correlation is very strong. 7. How does this graph differ from the one you did before, in which the diamonds clarity was not specified? In this graph, the correlation between price and weight is stronger than in the previous graph. In the earlier graph, the data was random and the variable clarity could take any value. The price of the diamond varied based on weight as well as clarity. In this graph, clarity is a constant, so weight has a stronger influence on price. If factors other than weight are allowed to vary, there will be greater fluctuations in price. So, the correlation coefficient is higher because narrowing the choices (restricting the clarity) lessened the variation. b. More Extravagant Diamonds Open the VVS carats tab, which gives data on diamonds with VVS1 clarity that weigh more than 1.00 carat. 1. Plot the data of weight versus price in this tab. Export an image of the plot, and paste it in the space below. Also record the equation of the regression line and the value of the correlation coefficient. 7

8 The equation of the regression line is y = 71,762x 67,255. The value of the correlation coefficient is Looking at the graph, is this line a good fit for predicting the price of a diamond? No, the line graph consistently overestimates the price for the smaller diamonds and underestimates the price for the larger diamonds. This consistent pattern indicates that a better model is needed. 3. Complete the equation for the best-fit line. Seeing that relationship, estimate the price of diamonds weighing more than 3.5 carats and enter the values in the table. Comment on how well these estimates match the actual sale prices for these three diamonds. The equation of the regression line is y = 71,762x 67,255. Estimated Price: Weight Actual Price Linear Relationship 3.64 carats $254,392 $193, carats $301,671 $256, carats $374,480 $279,355 The estimated prices are significantly lower than the corresponding actual prices. 4. To obtain the regression line, you clicked Line of Best Fit. But the actual data values do not seem to be quite linear. There s an upward curve. Let s investigate another option. Click Custom Fit. You will see a default quadratic equation. That might be a good choice for this kind of curve. Click Update to see how this curve fits your data. (It will graph a curve in green.) Export an image of the plot, and paste it in the space below. Comment on the data fit as compared with the best-fit line, and record the correlation coefficient value for this quadratic curve. 8

9 The graph looks like a much closer fit than the linear equation. The correlation value for the quadratic equation is That s even closer than the 0.94 correlation calculated for the linear relationship. 5. In the Notes section, the tool displays the coefficients for your best-fit quadratic equation. In the space below: Record the equation for the quadratic equation (using the coefficients from the Notes section). Using this equation, estimate the price of diamonds weighing more than 3.5 carats and record them in the table. Comment on which model provides the best estimate for those three diamonds, the quadratic estimate or the linear estimate (from question 3, above). The quadratic equation is y = 18,128x 2 9,348x + 4,874. Estimated Price: Weight Actual Price Quadratic Relationship 3.64 carats $254,392 $211, carats $301,671 $331, carats $374,480 $382,644 Using the quadratic estimate, the estimated price in each case is closer to the actual price than the estimated price from the linear relationship. For making predictions like this, the quadratic estimate seems to be a better choice. 9

10 Task 3: Worldwide Health and Wealth In one of the lessons of this unit, you watched the video Joy of Stats. In this video, Professor Hans Rosling uses a special tool to see 200 years of world history statistics on the relationship between health (life expectancy) and wealth (income per person). Now it s your turn to use this tool. Open this interactive graph. Click the How to use button to watch a video tutorial on how to change the axes, find the raw data, interpret the circles, and so on. You ll use the tool to analyze the health and wealth example demonstrated in the video clip. For this example, note that wealth is on a log scale, meaning that powers of 10 are equal distances apart on the horizontal scale. a. Mathematical Relationships 1. In general, what can you say about the relationship between health (life expectancy at birth) and wealth? Mention possible reasons for this relationship. One would expect life expectancy to increase as wealth increases. Some reasons for this positive linear relationship could be better medical care and higher education levels (which empower people to make wise choices about diet, medication, and so on) for wealthier people. Such people might also find more time to work out and keep fit. 2. Set the graph to present time. Switch the Income per person scale from log to lin(ear), using the tab beside the x-axis label. What shape do the spots now seem to make? The spots make a nonlinear exponential growth curve; after a point, the curve becomes parallel to the x-axis. 3. Switch back to the log scale. Describe the mathematical relationship you observe. The log scale allows us to get a straight-line graph. 4. Does a nonlinear relationship make sense for this data? The nonlinear relationship makes sense, as there is an upper limit to age at a bit over 100 years. Making anyone twice as rich will not enable them to live to 200 years. There are limits beyond which a relationship will not apply. This is true for all regression equations. A 140-pound athlete can be expected to hurl a discus to 10 meters, and a 150-pound athlete can be expected to hurl it to 11 meters, but every 10- pound increase in the discus thrower s weight does not mean the throw will go 1 meter farther. For example, we cannot extend this relationship to a person who weighs 400 pounds or 3 pounds. So, such linear relationships do not hold beyond a particular interval in most of cases. 10

11 For the given data, the income per person and the life expectancy at birth would exhibit a nonlinear relationship. 5. Run the graph over time from 1800 to present. You can move the graph manually or press the Play button below the x-axis. There s a lot of change in the data over time. Describe the basic trend in the data. In the overall picture from 1800 to the present, both income per person and life expectancy at birth have increased for almost all countries. Although the rate of growth is different for different nations, the basic trend is that increased wealth has increased life expectancy. 6. Comment on whether the mathematical relationship is the same over the past 200 years (same basic best-fit line) or whether the relationship changes. Support your statement with reasons. Over the past 200 years, the mathematical relationship has been considerably linear. However, the best-fit line would change over time. This is because as wealth has increased over the years, so has life expectancy. So, there is a shift in both the x and the y values. One would expect the scatter graph to shift continually upward and to the right, thereby giving a new best-fit line. The positive correlation between the two variables is maintained over the entire period. b. Association and Causation Possibilities In this section, you will try to explain specific observations. In some cases, you will have to do research to do so. Please record any resources you use to answer these questions in the Resources section near the end of this document. 1. How can you explain health as a result of wealth (using third variables)? What are some reasons that a wealthy individual might live longer? What are some reasons that citizens of a wealthy society might live longer? Wealthy people can get better medical care and more education, which empowers them to make wise choices about diet, medication, and so on. They can also find more time for exercise because their skilled labor typically earns more per hour than unskilled labor, so they have more time for nonwork activities. A wealthy society also usually provides a cleaner environment and better civic facilities to keep people healthy. 2. How can you explain wealth as a result of health (using third variables)? What are some reasons that a healthy individual might accumulate more wealth? What are some reasons that a healthy society might generate more wealth per person? 11

12 A healthy individual will be more active and productive. More work will result in a higher income. A healthy society provides a comfortable and disease-free lifestyle that can make people more focused and productive at work, leading to higher personal incomes. 3. In the video, it was pointed out that there was a worldwide dip in life expectancy from 1917 through 1919 because of the worldwide influenza epidemic. Slowly run through the years from 1900 to the present. (Use the slowest setting on the controller.) Click on any country that appears to go through a notable dip in longevity. Research one such country s history and explain why this dip might have occurred at that time (for example, Russia in the 1930s or China from 1958 through 1960). Answers will vary according to the country selected, but might resemble the following. The life expectancy of the Russian population shows a dip during the years 1914 to 1920 and a significant dip in The period between 1914 and 1921 saw a number of wars and revolutions in Russia. Russia lost two entire armies in the war in By the middle of 1915, many parts of Russia were overrun by the German army. The year 1917 was characterized by food riots, demonstrations, and mutinies. The years saw civil war between the Reds and Whites. Russia also faced a severe famine in What are the current outliers today? Identify a country that is rich but not so healthy. Interpret the results in terms of average wealth and average health. Research on the country online to find out what you can about the wealth and health of its people. Answers will vary. Two possible candidates for choice of country are Botswana and Equatorial Guinea. Countries like Botswana and Equatorial Guinea have a higher than average income per person, but life expectancy is lower than average. There is a high prevalence of AIDS in Botswana. More than one in three individuals is either infected with HIV or has developed the disease, so life expectancy has been dramatically reduced in this country. Equatorial Guinea is an oil-producing country. However, only a small percentage of the population has benefited from the oil riches. More than half of the population does not have access to drinking water, and one-fifth of children die before the age of five. Some countries have a few very rich people and a lot of poor people. Those very rich people bring up the average wealth, but although they live a little longer, they cannot significantly increase the average longevity for the country. Such countries do not show a linear relationship between health and wealth and so are considered outliers. 12

13 5. Looking at these country dips and current outliers, what factors besides wealth (or more specific than wealth) seem to strongly affect longevity? Factors other than wealth that affect longevity include diseases prevalent in the region, natural pandemics such as famines, and socio-economic disparity. Resources Document any references you used for this project below. At minimum, include a title and URL for any Internet resource. Evaluation This project will be evaluated on a rubric that is based on the completeness, clarity, and thinking you exhibit in the Directions and Analysis section above. Total Points: 10 Task 1: The Number of Hispanics (Latinos) in the United States a c. Relating and displaying data d h. Interpreting data regarding central tendency and distribution Task 2: Diamonds a. Affordable Diamond Pricing 1 4: Analyzing data in subset of the carats tab using scatter plots Task points: 3 1 point 2 points Task points: 3 1 point 5 7: Analyzing data in the VVS carats tab using scatter plots b. More Extravagant Diamonds 1 5: Analyzing data in the VVS carats tab using scatter plots 1 point 1 point Task 3: Worldwide Health and Wealth a. 1 6: Finding relationships in data sets Task points: 4 2 points b. 1 5: Determining association and causation possibilities 2 points 13