Exploring Data

Which of the following is least affected if an extreme high outlier is added to your data?

(a) Median

(b) Mean  

(c) Standard deviation  

(d) Range  

(e) Maximum 

What are all the values that a standard deviation $s_x$ can possibly take?

(a)$s_x \ge 0$

(b)$s_x > 0$

(c) $0 \le s_x \le 1$

(d) $-1 \le s_x \le 1$

(e) Any number 

We used Census At School’s “Random Data Selector” to choose a sample of $50$ Canadian students who completed a survey from $2007-2008$

The bar graph displays data on students’ responses to the question “Which of these methods do you most often use to communicate with your friends?” 

(a) Would it be appropriate to make a pie chart for these data? Why or why not? 

(b) Jerry says that he would describe this bar graph as skewed to the right. Explain why Jerry is wrong. 

Here are the heights (in centimeters) of 50 randomly selected Canadian students who participated in Census At School in 2007–2008: 

Make an appropriate graph to display these data. Describe the shape, center, and spread of the distribution. Are there any outliers? 

How tall are you? (1.2) Here are the heights (in centimetres) of $50$ randomly selected Canadian students who participated in CensusAtSchool in :

Make an appropriate graph to display these data. Describe the shape, centre, and spread of the distribution. Are there any outliers?

The $2007$ Freshman Survey asked first-year college students about their “habits of mind”—specific behaviors that college faculty have identified as being important for student success. One question asked students, “How often in the past year did you revise your papers to improve your writing?” Another asked, “How often in the past year did you seek feedback on your academic work?” The figure is a bar graph comparing male and female responses to these two questions.

What does the graph tell us about the habits of mind of male and female college freshmen?

According to the Internet Movie Database, Avatar is tops based on box office sales worldwide. The following table displays data on several popular movies.

(a) What individuals does this data set describe?

 (b) Clearly identify each of the variables. Which are quantitative? In what units do you think they are measured? 

(c) Describe the individual in the highlighted row

Hit movies According to the Internet Movie Database, Avatar is tops based on box office sales worldwide. The following table displays data on several popular movies.

(a) What individuals does this data set describe?

(b) Clearly identify each of the variables. Which are quantitative? In what units do you think they are measured?

(c) Describe the individual in the highlighted row.

•  Make a bar graph of the distribution of a categorical variable

or, in general, compare related quantities.

•  Recognize when a pie chart can and cannot be used.

Movie ratings The movie rating system we use today was first established on November $1$, $1968$. Back then, the possible ratings were G, PG, R, and X. In 1984, the $PG-13$ rating was created. And in $1990, NC-17$ replaced the X rating. Here is a summary of the ratings assigned to movies between 1968 and $2000: 8\%$ rated G, $24\%$ rated PG, 10% rated $PG-13, 55\%$ rated R, and $3\%$ rated $NC-17.48$ Make an appropriate graph for displaying these data.

• Identify what makes some graphs deceptive.

The movie rating system we use today was first established on November 1, 1968. Back then, the possible ratings were G, PG, R, and X. In 1984, the PG-13 rating was created. And in 1990, NC-17 replaced the X rating. Here is a summary of the ratings assigned to movies between 1968 and 2000: 8% rated G, 24% rated PG, 10% rated PG-13, 55% rated R, and 3% rated NC-17.48 Make an appropriate graph for displaying this data  

Forty students took a statistics examination having a maximum of 50 points. The score distribution is given in the following stem-and-leaf plot:

The third quartile of the score distribution is equal to (a) 45. (b) 44. (c) 43. (d) 32. (e) 23.

You are interested in how much time students spend on the Internet each day. Here are data on the time spent on the Internet (in minutes) for a particular day reported by a random sample of 30 students at a large high school

(a) Construct a histogram of these data. (b) Are there any outliers? Justify your answer. (c) Would it be better to use the mean and standard deviation or the median and )12 to describe the center and spread of this distribution? Why? 

In a July 2008 survey of over 2000 U.S. teenagers by Harris Interactive, 47% said that “their social life would end or be worsened without their cell phone.”49 One survey question asked the teens how important it is for their phone to have certain features. The figure below displays data on the percent who indicated that a particular feature is vital. 

(a) Explain how the graph gives a misleading impression.

 (b) Would it be appropriate to make a pie chart to display these data? Why or why not? 

 (c) Make a graph of the data that isn’t misleading 

Is there a relationship between Facebook use and age among college students? The following two-way table displays data for the $219$ students who responded to the survey.

(a) What percent of the students who responded were Facebook users? Is this percent part of a marginal distribution or a conditional distribution? Explain.

(b) What percent of the younger students in the sample were Facebook users? What percent of the Facebook users in the sample were younger students? 

The density of the earth In $1798$, the English scientist Henry Cavendish measured the density of the earth several times by careful work with a torsion balance. The variable recorded was the density of the earth as a multiple of the density of water. Here are Cavendish’s $29$ measurements:

(a) Present these measurements graphically in a stem plot.

(b) Discuss the shape, center, and spread of the distribution. Are there any outliers?

(c) What is your estimate of the density of the earth based on these measurements? Explain.

Facebook and age Use the data in the previous exercise to determine whether there is an association between Facebook use and age. Give appropriate graphical and numerical evidence to support your answer.

• Make a dot-plot or stem plot to display small sets of data.
• Describe the overall pattern (shape, center, spread) of distribution, and identify any major departures from the pattern (like outliers).

Guinea pig survival times Here are the survival times in days of $72$guinea pigs after they were injected with infectious bacteria in a medical experiment.$52$ Survival times, whether of machines under stress or cancer patients after treatment, usually have distributions that are skewed to the right.

(a) Make a histogram of the data and describe its main features. Does it show the expected right-skew?

(b) Now make a boxplot of the data. Be sure to check for outliers. 

Use the data in the previous exercise to determine whether there is an association between Facebook use and age. Give appropriate graphical and numerical evidence to support your answer 

Household incomes Rich and poor households differ in ways that go beyond income. Following are histograms that compare the distributions of household size (number of people) for low-income and high-income households.$53$ Low-income households had annual incomes less than $\backslash (15,000$ and high-income households had annual incomes of at least $\backslash )100,000$

(a) About what percent of each group of households consisted of two people?

(b) What are the important differences between these two distributions? What do you think explains these differences?

R1.9. Mercury in tuna A histogram and some computer output provide information about the mercury concentration in the sampled cans (in parts per million, ppm).

(a) Interpret the standard deviation in context.

(b) Determine whether there are any outliers.

(c) Describe the shape, center, and spread of the distribution.

Household incomes Rich and poor households differ in ways that go beyond income. Following are histograms that compare the distributions of household size (number of people) for low-income and high-income households.53 Low-income households had annual incomes less than \(15,000, and high-income households had annual incomes of at least \)100,000.

(a) About what percent of each group of households consisted of two people?

(b) What are the important differences between these two distributions? What do you think explains these differences?

A histogram and some computer output provide information about the mercury concentration in the sampled cans (in parts per million, ppm).

(a) Interpret the standard deviation in context.

(b) Determine whether there are any outliers.

(c) Describe the shape, center, and spread of the distribution.

Mercury in tuna Is there a difference in the mercury of light tuna and albacore tuna? Use the side by side boxplots and the computer output to write a few sentences comparing the two distributions. 

Which of the following bar graphs is equivalent to the pie chart in Question T1.2?

You record the age, marital status, and earned income of a sample of 1463 women. The number and type of variables you have recorded is 

(a) 3 quantitative, 0 categorical 

(b) 4 quantitative, 0 categorical 

(c) 3 quantitative, 1 categorical 

(d) 2 quantitative, 1 categorical 

(e) 2 quantitative, 2 categorical

Consumers Union measured the gas mileage in miles per gallon of 38 vehicles from the same model year on a special test track. The pie chart provides information about the country of manufacture of the model cars tested by Consumers Union. Based on the pie chart, we conclude

(a) Japanese cars get significantly lower gas mileage than cars from other countries. 

(b) U.S. cars get significantly higher gas mileage than cars from other countries. (c) Swedish cars get gas mileages that are between those of Japanese and U.S. cars. 

(d) Mercedes, Audi, Porsche, and BMW represent approximately a quarter of the cars tested. 

(e) More than half of the cars in the study were from the United States.

Movie ratings The movie rating system we use today was first established on November 1, 1968. Back then, the possible ratings were G, PG, R, and X. In 1984, the PG-13 rating was created. And in 1990, NC-17 replaced the X rating. Here is a summary of the ratings assigned to movies between 1968 and 2000: 8% rated G, 24% rated PG, 10% rated PG-13, 55% rated R, and 3% rated NC-17.48 Make an appropriate graph for displaying these data. 

Mercury in Tuna

A study among the Pima Indians of Arizona investigated the relationship between a mother’s diabetic status and the appearance of birth defects in her children. The results appear in the two-way table below. 

(a) Fill in the row and column totals in the margins of the table. (b) Compute (in percents) the conditional distributions of birth defects for each diabetic status. (c) Display the conditional distributions in a graph. Don’t forget to label your graph completely. (d) Comment on any clear associations you see. 

The percentage of students with less than $10 in their possession is closest to 

(a) 30%. (b) 35%. (c) 50%. (d) 60%. (e) 70% 

Which of the following statements about this distribution is not correct? 

(a) The histogram is right-skewed. 

(b) The median is less than $\backslash (20$

(c) The IQR is $\backslash )35$

(d) The mean is greater than the median. 

(e) The histogram is unimodal.

Earthquake intensities are measured using a device called a seismograph, which is designed to be most sensitive to earthquakes with intensities between $4.0$ and $9.0$ on the Richter scale. Measurements of nine earthquakes gave the following readings:

where L indicates that the earthquake had an intensity below $4.0$ and an H indicates that the earthquake had an intensity above $9.0$ The median earthquake intensity of the sample is

(a) $5.75$ (c) $6.47$ (e) Cannot be determined.

(b) $6.00$ (d) $8.70$

Forty students took a statistics examination having a maximum of $50$ points. The score distribution is given in the following stem-and-leaf plot: 

The third quartile of the score distribution is equal to

$\left(a\right) 45. \left(b\right) 44. \left(c\right) 43. \left(d\right) 32. \left(e\right) 23.$

Earthquake intensities are measured using a device called a seismograph, which is designed to be most sensitive to earthquakes with intensities between 4.0 and 9.0 the Richter scale. Measurements of nine earthquakes gave the following readings: 

where L indicates that the earthquake had an intensity below 4.0 and an H indicates that the earthquake had an intensity above 9.0. The median earthquake intensity of the sample is (a) 5.75. (c) 6.47. (e) Cannot be determined. (b) 6.00. (d) 8.70. 

The mean salary of all female workers is $\backslash (35,000$ The mean salary of all male workers is $\backslash )41,000$ What must be true about the mean salary of all workers?

 (a) It must be $\backslash (38,000$

 (b) It must be larger than the median salary.

 (c) It could be any number between $\backslash )35,000$ and $\backslash (41,000$

 (d) It must be larger than $\backslash )38,000$

 (e) It cannot be larger than $\$40,000$

A survey was designed to study how business operations vary according to their size. Companies were classified as small, medium, or large. Questionnaires were sent to $200$ randomly selected businesses of each size. Since not all questionnaires in a survey of this type are returned, researchers decided to investigate the relationship between the response rate and the size of the business. The data are given in the following two-way table.

What percent of all small companies receiving questionnaires responded? $\left(a\right) 12.5\% \left(c\right) 33.3\% \left(e\right) 62.5\% \left(b\right) 20.8\%$

An experiment was conducted to investigate the effect of a new weed killer to prevent weed growth in onion crops. Two chemicals were used: the standard weed killer (C) and the new chemical (W). Both chemicals were tested at high and low concentrations on a total of 50 test plots. The percent of weeds that grew in each plot was recorded. Here are some boxplots of the results. Which of the following is not a correct statement about the results of this experiment? 

 (a) At both high and low concentrations, the new chemical (W) gives better weed control than the standard weed killer (C). 

(b) Fewer weeds grew at higher concentrations of both chemicals. 

(c) The results for the standard weed killer are less variable than those for the new chemical. 

(d) High and low concentrations of either chemical have approximately the same effects on weed growth. 

(e) Some of the results for the low concentration of weed killer W show fewer weeds growing than some of the results for the high concentrate 

A survey was designed to study how business operations vary according to their size. Companies were classified as small, medium, or large. Questionnaires were sent to 200 randomly selected businesses of each size. Since not all questionnaires in a survey of this type are returned, researchers

decided to investigate the relationship between the response rate and the size of the business. The data are given in the following two-way table: 

Which of the following conclusions seems to be supported by the data?

(a) There are more small companies than large companies in the survey.

(b) Small companies appear to have a higher response rate than medium or big companies.

(c) Exactly the same number of companies responded as didn’t respond.

(d) Small companies dislike larger companies.

(e) If we combined the medium and large companies, then their response rate would be equal to that of the small companies.

A survey was designed to study how business operations vary according to their size. Companies were classified as small, medium, or large. Questionnaires were sent to 200 randomly selected businesses of each size. Since not all questionnaires in a survey of this type are returned, researchers decided to investigate the relationship between the response rate and the size of the business. The data are given in the following two-ways

Which of the following conclusions seems to be supported by the data? 

(a) There are more small companies than large companies in the survey. 

(b) Small companies appear to have a higher response rate than medium or big companies. 

(c) Exactly the same number of companies responded as didn’t respond. 

(d) Small companies dislike larger companies. 

(e) If we combined the medium and large companies, then their response rate would be equal to that of the small company

A study among the Pima Indians of Arizona investigated the relationship between a mother’s diabetic status and the appearance of birth defects in her children. The results appear in the two-way table below. 

(a) Fill in the row and column totals in the margins of the table.

(b) Compute (in percentages) the conditional distributions of birth defects for each diabetic status.

(c) Display the conditional distributions in a graph. Don’t forget to label your graph completely.

(d) Comment on any clear associations you see.

During the early part of the $1994$ baseball season, many fans and players noticed that the number of home runs being hit seemed unusually large. Here are the data on the number of home runs hit by American League and National League teams in the early part of the $1994$ season: 

Compare the distributions of home runs for the two leagues graphically and numerically. Write a few sentences summarizing your finding

The back-to-back stem plot shows the lifetimes of several Brand X and Brand Y batteries.

(a) What is the longest that any battery lasted? 

(b) Give a reason someone might prefer a Brand X battery.

(c) Give a reason someone might prefer a Brand Y battery

You are interested in how much time students spend on the Internet each day. Here are data on the time spent on the Internet (in minutes) for a particular day reported by a random sample of $30$ students at a large high school:

(a) Construct a histogram of these data.

(b) Are there any outliers? Justify your answer.

(c) Would it be better to use the mean and standard deviation or the median and IQR to describe the center and spread of this distribution? Why?