Measures of central tendency & measures of dispersion (std. Dev, var, IQR): notes and practice questions

A local library tracks the number of books borrowed by its members in one month. The data for 120 members is shown in the following table.

One of the members is chosen at random.

(a) Find the probability that this member borrowed fewer than 2 books.

(b) Calculate the mean number of books borrowed per member.

A biologist records the number of eggs in the nests of a certain species of bird. The results for 25 nests are shown in the following frequency table.

It is given that the mean number of eggs per nest is 2.6.

(a) Find the value of p and the value of q.

The biologist enters the data into a competition. The competition score is calculated using the formula $S = 10x - 5$, where $x$ is the number of eggs in a nest.

(b) Find the mean competition score.

A set of 12 spherical Matryoshka dolls, $D_1, D_2, ..., D_{12}$, are designed to fit inside one another.

The smallest doll, $D_1$, has a radius of 2 cm.

The radius of each subsequent doll $D_{n+1}$ is 20% larger than the radius of doll $D_n$, for $n \in \mathbb{Z}^+, 1 \le n \le 11$.

(a) (i) Show that the volume of doll $D_n$ is $\frac{32\pi}{3} \left(\frac{216}{125}\right)^{n-1}$ cm$^3$.

(ii) Hence, find the mean volume of the twelve dolls, giving your answer in the form $p\pi \left(\left(\frac{216}{125}\right)^{12} - 1\right)$ cm$^3$, where $p \in \mathbb{Q}^+$.

(b) Find the median volume of the twelve dolls, giving your answer in the form $q\pi \left(\frac{216}{125}\right)^5$ cm$^3$, where $q \in \mathbb{Q}^+$.

A student, Chloe, records the number of hours she spends studying each day for 11 consecutive days. Her records are as follows:

3, 5, 2, 4, 5, 6, 3, 5, 7, 2, 4.

(a) Find the mode of the number of hours Chloe studied.

(b) Find the median number of hours Chloe studied.

(c) Calculate the mean number of hours Chloe studied.

(d) Find the range of the number of hours Chloe studied.

A new coffee shop records the number of customers, $c$, per hour during its 120 opening hours in a week. The number of customers per hour is shown in the following frequency table.

(a) Find the value of $k$.

(b) Write down the modal class.

The following cumulative frequency diagram also displays these data.

(c) Use the cumulative frequency curve to estimate the median number of customers per hour.

(d) The coffee shop is considered 'busy' when there are more than 35 customers. Use the cumulative frequency curve to estimate the number of hours the coffee shop was busy.

The coffee shop manager wants to survey customers about their experience.

(e) State one disadvantage of surveying only the customers who arrive between 8 am and 9 am on a Monday.

(f) Describe how the manager could use systematic sampling to survey customers throughout a single day.

The total number of customers for the week was 2900. The following box and whisker diagram displays the amount of money, in USD, spent by customers during their visit.

(g) Estimate the number of customers who spent between $4.50 and $16.

(h) The top 25% of customers spent more than $d$ USD. Find the value of $d$.

The following week, a new promotion is introduced, which is expected to attract an additional 3 customers per hour.

(i) Calculate the new mean number of customers per hour.

(j) State, with a reason, the effect this increase would have on the range of the number of customers per hour.

The following table shows the advertising spending (in thousands of dollars) and the corresponding monthly sales (in thousands of units) for a new product over several months.

The data is also represented on the following scatter diagram.

The relationship between advertising spending (x) and monthly sales (y) can be modelled by the regression line of y on x with equation $y = ax + b$, where $a, b \in \mathbb{R}$.

Write down the value of $a$ and the value of $b$.

Use this model to predict the monthly sales (in thousands of units) when the advertising spending is 25 thousand dollars.

Write down the value of $\bar{x}$ and the value of $\bar{y}$.

Draw the line of best fit on the scatter diagram.

The box plot below shows the time, in minutes, taken by a group of students to complete a crossword puzzle.

(a) Write down the median time.

(b) Write down the minimum time.

(c) The range of the times is $25$ minutes. Find the value of $p$.

(d) The interquartile range is $9$ minutes. Find the value of $q$.

A botanist is studying the growth of a particular plant species. They record the average height of several plants (in cm) at different weeks after planting. The data collected is shown in the table below.

The relationship between the week number (x) and the plant height (y) can be modelled by the regression line of y on x with equation $y = ax + b$, where $a, b \in \mathbb{R}$.

Write down the value of $a$ and the value of $b$.

Use this model to predict the height of a plant after 15 weeks.

Write down the mean week number, $\bar{x}$, and the mean plant height, $\bar{y}$.

Draw the line of best fit on a scatter diagram for this data.

The daily commute time to work (in minutes) for a group of employees is shown in the following table.

The median commute time is 17.5 minutes.

Find the value of x.

Using the value of x found in part (a), find the standard deviation of the commute times.

A survey was conducted among a group of students to determine the number of video games they played per week. The results are shown in the following frequency table.

The median number of games played is 3.5.

(a) Find the value of x.

(b) Find the standard deviation of the number of games played per week.

A botanist conducted an experiment to compare the growth of a particular plant species under two different lighting conditions: natural sunlight and artificial grow lights. A random sample of 10 plants was grown under each condition for a month, and their increase in height (in cm) was recorded.

The box and whisker diagrams for the height increase are shown below.

Consider the box and whisker diagram representing the height increase for plants grown in natural sunlight.

(a) State the median height increase for plants grown in natural sunlight.

(b) Verify that the measurement of 24.5 cm is not an outlier for plants grown in natural sunlight.

(c) For plants grown in natural sunlight, state why it appears that the mean height increase is greater than the median height increase.

(d) Now consider the two box and whisker diagrams. Comment on whether these box and whisker diagrams provide any evidence that might suggest that artificial grow lights cause an increase in plant height.

A customer service center recorded the number of calls received per hour over several days. The data is presented in the following cumulative frequency table.

Find the values of $m$ and $n$.

Write down the value of the mean number of calls received per hour.

Find the variance of the number of calls received per hour.

A student records the number of hours they study for an exam over a period of 6 consecutive days. The data is shown below:

$12, 17, 9, 15, x, y$

where $x$ and $y$ are integers representing the hours on the last two days, and $x > y$.

The mode of the data is 15 hours. The median of the data is 14 hours.

(a) Find the value of $x$ and the value of $y$.

(b) Find the mean number of hours the student studied over the 6 days.

State the mathematical condition used to identify outliers in a set of data.

A botanist measures the heights, in cm, of 11 seedlings. The results, ordered from smallest to largest, are shown below.

$5.2, 6.1, 6.5, 6.8, 7.2, 7.5, 7.9, 8.3, 8.5, 9.1, 12.5$

Find the median, the lower quartile, the upper quartile, and the interquartile range for these heights.

Using the condition from part (a), identify, with a reason, any outliers for this set of data.

(a) A small artisanal bakery recorded its daily revenue for $8$ days, finding a mean revenue of $450$ USD. On the $9$th day, a special promotion was run, and the mean revenue for all $9$ days increased to $465$ USD. Find the revenue generated on the $9$th day.

(b) For the $9$ days, the lower quartile ($Q_1$) of the daily revenue was $420$ USD and the upper quartile ($Q_3$) was $500$ USD. Determine, with justification, whether the revenue on the $9$th day (found in part (a) ) is considered an outlier.

A group of students participated in a puzzle-solving competition. The time, $t$ minutes, taken by each student to complete the puzzle was recorded and grouped into the following frequency table.

State the total number of students who participated in the competition.

Find the midpoint of the modal class.

Estimate the mean time taken to complete the puzzle.

Estimate the standard deviation of the times.

A quick calculation suggests the median is $27.5$ minutes. Find a more precise estimate for the median time by considering its position within the interval it belongs to. Give your answer to the nearest integer.

(a) Data on the number of goals scored by a football team in each of their $50$ matches during a season is represented in the table below.

State whether this data is discrete or continuous.

(b) Find the mode.

(c) (i) Find the mean.

(c) (ii) Find the standard deviation.

(d) (i) Find the median.

(d) (ii) Find the lower quartile ($Q_1$).

(d) (iii) Find the upper quartile ($Q_3$).

(e) Hence draw a box-and-whisker plot for this data using a scale of $2$ cm for $1$ goal.

(f) Identify with justification any outliers.