Exercises: Statistics with technology and critical reading

The figure shows a spreadsheet with the ages of the members of a book club. In cells B11 and B12 the sheet shows the results of the formulas $=\text{AVERAGE}(B2:B9)$ and $=\text{MEDIAN}(B2:B9)$, respectively. The top bar shows the formula of cell B11, which is the selected cell.

1. a) What does the formula in cell B11, $=\text{AVERAGE}(B2:B9)$, calculate?

1. b) And the one in cell B12, $=\text{MEDIAN}(B2:B9)$?

1. c) Calculate both values by hand from the ages in B2 to B9 and check that they match the ones the sheet shows in B11 and B12.

A television channel showed this bar chart about the audience of two programs, as shown in the figure.

1. a) At first glance, how many times more audience does program B seem to have than A?

1. b) Read the actual values from the bars and calculate the true proportion between both audiences.

1. c) What trick does the graph use to exaggerate the difference? How should the vertical axis be designed so as not to be misleading?

You can do it on paper and send a photo with your answer.

A statistical calculator indicates that the mean of the $10$ marks of a group is $6.4$ and the median is $6.5$.

1. a) If the teacher decides to give $0.5$ extra marks to all students, what will the new mean and the new median be? Reason your answer without knowing the individual marks.

1. b) If instead they multiply all the marks by $1.1$, what happens to the mean and the median?

1. c) The sum of the $10$ marks was $64$. After checking, a mark of $4$ is corrected to $7$. What is the new mean?

For each of these data sets, choose the most suitable chart (bar chart, circle graph, line graph, frequency polygon or histogram) and justify your choice:

1. a) The percentage of time a teenager spends sleeping, studying, on leisure and on screens.

1. b) The month-by-month evolution of the average temperature of a city over a year.

1. c) The math grades of a class of $28$ students.

1. d) The number of pets that the students in a grade have at home.

The figure shows a spreadsheet with the points per match scored by a basketball player, as shown below.

1. a) What value will cell B10 show if it contains the formula $=\text{PROMEDIO}(B2:B8)$? Calculate it.

1. b) Cell B11 contains $=\text{MAX}(B2:B8) - \text{MIN}(B2:B8)$. Which statistical parameter does it calculate and what is its value?

1. c) Write in your own words the formula that should be entered to obtain the number of matches in which she scored more than $15$ points.

A soft drinks brand published this circle graph about consumer preferences, as shown in the figure.

1. a) Add up the percentages of all the sectors. What result do you get? Is this possible in a circle graph?

1. b) Explain what might have happened in the survey to obtain these data and why a circle graph is not suitable to present them.

1. c) What type of chart would you choose to represent these percentages correctly?

The student government of a secundaria with $850$ students wants to decide which new extracurricular activity to offer. A student surveys $40$ students outside the gym and concludes: "$70\ \%$ of students want more sport, so we need to set up a climbing club".

1. a) What is the sample size? What is the population?

1. b) What flaw is there in the way the sample was chosen? How does it affect the conclusion?

1. c) Propose a better way to select the $40$ survey respondents and explain why the conclusion would be more reliable.

1. d) What percentage of the population does the sample represent? Express it to two decimal places.

A coach uses an app to compare the minutes played by two basketball teams across $10$ matches. The app shows these results:

• North Team: mean $= 18.4$ min, standard deviation $= 7.9$ min.

• South Team: mean $= 17.8$ min, standard deviation $= 2.1$ min.

1. a) Which team distributes the minutes more evenly among its players? Which parameter tells you this?

1. b) If the North coach wants a specific player to play a lot, is it predictable that they will manage it? What about in the South?

1. c) The North team's mean is slightly higher. Does that mean that all of its players play more than those in the South? Reason your answer.

Decision-making task. An online newspaper publishes the headline "Teenagers' spending on video games soars!" alongside the graph in the figure, as shown below.

1. a) Read the actual values for annual spending from the graph and calculate the true percentage increase between the first and last year.

1. b) Why does the graph give a much more alarming impression than the data? Point out at least two design decisions that cause this.

1. c) Rewrite the headline honestly using the data.

1. d) What questions would you ask about the survey sample before accepting even that moderate increase?

To calculate the standard deviation of the data $4$, $6$, $7$, $7$, $9$, $11$ using a spreadsheet, a table is constructed with auxiliary columns:

Data $x_i$	$x_i - \bar{x}$	$(x_i - \bar{x})^2$
$4$
$6$
$7$
$7$
$9$
$11$

1. a) Calculate the mean $\bar{x}$ and complete the table in your exercise book.

1. b) What spreadsheet formula would give the value of the cell in the second column for data $4$? And for the third column?

1. c) Calculate the mean of the squares in the third column (the variance) and, using a calculator, find the square root to obtain the standard deviation.

1. d) The sum of the second column is $0$. Is this a coincidence or would it happen with any data set? Justify your answer.

You can do it on paper and send a photo with your answer.

An athletics club needs to sign a relay runner and has the times of two candidates in their last $8$ $100\ \text{m}$ races:

• Candidate 1: mean $= 12.40\ \text{s}$, standard deviation $= 0.45\ \text{s}$.

• Candidate 2: mean $= 12.55\ \text{s}$, standard deviation $= 0.10\ \text{s}$.

1. a) The coach needs a runner who will most reliably beat $12.9\ \text{s}$ in the championship, because a bad time will eliminate the team. Who should they sign? Reason your answer using the mean and standard deviation.

1. b) If instead they were looking for the runner capable of achieving the best time of the day, would your choice change? Why?

1. c) Calculate the coefficient of variation for each candidate, $CV = \frac{\sigma}{\bar{x}}$, and explain what information it adds compared to the standard deviation.

State whether each statement about the use of technology in statistics is true or false and justify your answer:

1. a) "The mean calculated using a spreadsheet from data grouped into intervals (using class midpoints) is exactly the same as the one calculated using the original data."

1. b) "If a statistical chart is produced with computer software, it cannot be misleading."

1. c) "A scientific calculator can provide the standard deviation even if we do not understand the formula, but interpreting it remains our job."

1. d) "To compare the dispersion of two data sets with very different means, it is better to use the standard deviation than the coefficient of variation."