Presentation of data (frequency distribution tables, histograms, box & whisker, cumulative frequency graphs + finding median quartiles, percentiles, range, iqr): notes and practice questions

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 group of students were timed in minutes on how long it took them to solve a logic puzzle. The results are summarized in the following box and whisker diagram, where $L$ and $U$ represent the lower and upper quartiles respectively.

The interquartile range is 8 minutes, and the minimum time recorded was 5 minutes. There are no outliers in the data.

(a) Find the maximum possible value of $L$.

(b) Hence, find the maximum possible value of $U$.

In a large university, 200 students were surveyed. Of those, 120 were undergraduates (U) and the rest postgraduates (P).

Each student in the survey was asked whether they preferred quiet zones (Q) or collaborative areas (C) for studying. It was found that 75 of the undergraduates preferred quiet zones. The total number of students who preferred collaborative areas was 100. This information is shown in the following table.

Find the value of

$p$;

$q$.

Three students are chosen at random from those surveyed. Find the probability that all three are postgraduates.

Given that $P(P|Q) = \frac{1}{4}$, find the value of $x$.

A student is chosen at random from those surveyed. Write down the probability that they are a postgraduate who prefers quiet zones.

Determine if the events P (Postgraduate) and Q (prefers Quiet Zones) are independent. Justify your answer.

It can be assumed that the survey results are representative of the university population. Ten students from the university are chosen at random. Find the probability that at least five of them prefer quiet zones.

A biologist is studying a species of fish. The length, $L$ cm, and weight, $W$ g, of each fish in a sample are recorded.

The lengths of the fish are summarized in the following box and whisker diagram.

Find the largest value of $L$ that would not be considered an outlier.

The regression line of $W$ on $L$ is $W = 25L - 150$. The regression line of $L$ on $W$ is $L = 0.03W + 10.5$.

One of the fish in the sample weighs 200 g. Estimate the length of this fish.

Find the mean weight of the fish in the sample.

A sports scientist records the time, in seconds, for a group of athletes to run 400 metres. The results are summarized in the following box and whisker diagram, where $L$ and $U$ are the lower and upper quartiles respectively.

The interquartile range is 6 seconds and there are no outliers in the data.

(a) Find the maximum possible value of $L$.

(b) Hence, find the maximum possible value of $U$.

A tech company is testing the battery life of its new smartphone. A sample of 120 phones are tested to see how long their batteries last under continuous video playback. The results are shown in the cumulative frequency graph below.

(a) Find the median battery life.

(b) The lowest 25% of battery lives in the sample are less than $k$ hours. Find the value of $k$.

(c) The same data is represented by the following frequency table.

Find the value of $p$ and the value of $q$.

(d) The company manufactures a batch of 10,000 of these smartphones. Estimate the number of phones in the batch that will have a battery life of more than 10 hours.

(e) The company wishes to advertise the 'typical' battery life of the phone based on this test.

(i) Explain why this testing method might not provide an accurate representation of the battery life for a typical user.

(ii) Suggest a more appropriate method for testing the battery life to represent a typical user.

A study was conducted to investigate the relationship between the number of hours, $h$, a student spends studying for an exam and their score, $s$ (%), in that exam.

The number of hours spent studying is summarized in the following box and whisker diagram.

(a) Find the largest value of $h$ that would not be considered an outlier.

The regression line of $s$ on $h$ is $s = 5h + 25$. The regression line of $h$ on $s$ is $h = 0.1s + 2.5$.

(b) (i) One of the students scored 90% on the exam. Estimate the number of hours they studied.

(ii) Find the mean score of all the students in the study.

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 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 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.

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 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.

A local bakery recorded the number of chocolate croissants sold each day over a 30-day period. The results are shown in the following table.

(a) Write down the modal number of croissants sold.

(b) Find the median number of croissants sold.

(c) Calculate the mean number of croissants sold.

A local food delivery service recorded the delivery times for all orders received during a busy weekend. The results are presented in the cumulative frequency diagram below.

The horizontal axis represents the delivery time in minutes, from $0$ to $60$. The vertical axis represents the cumulative frequency (number of deliveries), from $0$ to $200$. The curve starts at $(0, 0)$ and ends at $(60, 200)$. Key points on the curve are approximately:

• $(22, 50)$

• $(33, 100)$

• $(38, 130)$

• $(42, 150)$

• $(48, 180)$

• $(49, 185)$

(a) State the total number of deliveries recorded.

(b) Determine the median delivery time.

(c) Show that the interquartile range for the delivery times is $20$ minutes.

(d) Find the number of deliveries that took more than $48$ minutes.

(e) A customer complains that their delivery took $49$ minutes. Would this delivery be in the $90$th percentile or higher? Justify your answer.

(f) The delivery service aims to complete $130$ deliveries within a certain time limit, $k$ minutes. Use the diagram to find the value of $k$.

A group of students recorded the number of hours they spent studying for a mathematics test ($x$) and their corresponding test score ($y$). The results for eight students are shown in the table below.

On graph paper, draw a scatter diagram to represent this data. Use a scale of $1$ cm for $1$ unit on the $x$-axis (starting from $x=0$) and $1$ cm for $5$ units on the $y$-axis (starting from $y=50$).

By considering the scatter diagram, state the type of linear correlation that is shown in this example.

i Find the mean of the $x$-values.

ii Find the mean of the $y$-values.

iii Mark the point $(\bar{x},\bar{y})$ on the scatter diagram using the symbol $\otimes$.

A group of students participated in a puzzle-solving challenge. The time, $t$, in 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 puzzle challenge.

Identify the modal class interval and state its midpoint.

Calculate an estimate for the mean time taken to complete the puzzle.

Calculate an estimate for the standard deviation of the time taken to complete the puzzle.

The competition organizer initially estimated the median time to be $12$ 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 tech company launched a new mobile application and collected customer satisfaction ratings from $100$ users. The ratings were on a scale of $1$ (very dissatisfied) to $5$ (very satisfied). The discrete data showing the ratings is given in the table below.

Rating | $1$ | $2$ | $3$ | $4$ | $5$

---|---|---|---|---

Frequency | $10$ | $25$ | $30$ | $25$ | $10$

Sketch a bar chart to represent this data.

For this data find:

i the mode

ii the median

iii the mean.

Explain any similarities between the answers to parts b(ii) and b(iii) by referring to a geometrical property of the bar chart drawn in part (a).

A local library organized a 'Summer Reading Challenge' for high school students. The number of books read by 60 participating students over the summer is recorded below:

$0110210011$

$2203111020$

$1100213110$

$0221101021$

$0112031102$

$1001121011$

(a) Tabulate the results in a frequency table and describe the frequency distribution.

(b) Calculate the mean number of books read and the standard deviation.