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

The daily screen time (in hours) for a group of students was recorded. The data was organized in a box and whisker diagram as shown below:

For this data, write down

(i) the minimum daily screen time.

For this data, write down

(ii) the lower quartile.

For this data, write down

(iii) the median daily screen time.

A student, Sarah, claims that this box and whisker diagram indicates that the percentage of students who spend less than 3 hours on screen time is smaller than the percentage of students who spend more than 7.5 hours on screen time.

State whether Sarah is correct. Justify your answer.

A logistics company recorded the delivery times (in minutes) for a large batch of packages. The data is grouped in the frequency table below:

Delivery Time (minutes) | Frequency

---|---

$20 \le t < 25$ | $8$

$25 \le t < 30$ | $22$

$30 \le t < 35$ | $55$

$35 \le t < 40$ | $98$

$40 \le t < 45$ | $135$

$45 \le t < 50$ | $110$

$50 \le t < 55$ | $70$

$55 \le t < 60$ | $35$

$60 \le t < 65$ | $15$

$65 \le t < 70$ | $2$

(a) Calculate estimates of the mean and standard deviation of the delivery times.

(b) Construct a cumulative frequency table for the data, and use it to draw a cumulative frequency curve.

(c) Use your graph to estimate:

(i) the median delivery time

(ii) the lower and upper quartile of the delivery times

(iii) the interquartile range

(iv) the $85$th percentile of delivery times.

(d) Draw a box-and-whisker plot of the data.

(e) Determine, with reasons, whether any customers could be considered outliers.

A software development company tracked the completion times of 150 projects. The cumulative frequency graph shows the completion times obtained by the projects.

Find the median completion time of the projects.

The projects were assigned a performance tier from 1 to 5, depending on the completion time. The number of projects receiving each tier is shown in the following table.

Find an expression for $p$ in terms of $q$.

The mean performance tier for these projects is 3.5.

Find the number of projects that obtained a tier 5.

Find the minimum completion time needed to obtain a tier 5.

A survey was conducted to investigate the daily screen time (in minutes) of 80 students. The results are presented in the frequency table below.

Time ($t$ minutes) | Frequency

---|---

$0 \le t < 30$ | $6$

$30 \le t < 60$ | $10$

$60 \le t < 90$ | $22$

$90 \le t < 120$ | $20$

$120 \le t < 150$ | $12$

$150 \le t < 180$ | $7$

$180 \le t < 210$ | $3$

(a) State the modal class.

(b) Find the class in which the median time lies.

(c) Construct a cumulative frequency table for this data.

(d) Sketch the cumulative frequency curve.

(e) Use your curve to find estimates for the median and interquartile range.

A factory conducts quality control checks on the diameter (in mm) of components produced by Production Line A. A sample of measurements is recorded as:

18.2, 19.5, 20.1, 20.3, 20.5, 20.6, 20.8, 21.0, 21.2, 21.5, 22.0, 22.1, 22.3, 22.5, 23.0

For these data, the lower quartile is 20.3 mm and the upper quartile is 22.1 mm.

Show that a component with a diameter of 18.2 mm would not be considered an outlier.

Another production line, Line B, also produces similar components. The box and whisker diagram below displays the diameters (in mm) of a sample of components from Line B.

A quality control manager reviews the box and whisker diagrams for both lines and suggests that Production Line B generally produces components with larger diameters.

With reference to the box and whisker diagrams for Line A (from part (a) ) and Line B, state one aspect that may support the manager's opinion and one aspect that may counter it.

Dr. Anya Sharma, a sports scientist, is investigating the relationship between training habits and performance in junior athletes. She wants to collect data on the weekly training hours of junior swimmers. She decides to interview every 5th swimmer entering the training facility until she has a sample of 50 swimmers.

State the sampling method Dr. Sharma has used.

Dr. Sharma constructed the following box and whisker diagram to show the weekly training hours (in hours) of a sample of junior swimmers.

Write down the median weekly training hours.

Calculate the interquartile range for the weekly training hours.

One swimmer in the sample reported training for 15 hours per week. Dr. Sharma believes this swimmer's training time is not an outlier.

Determine whether Dr. Sharma is correct. Support your reasoning.

Dr. Sharma also collected data on the average weekly training hours ($x$) and the competition score ($y$) for a group of athletes. These data are represented on the scatter diagram.

Describe the correlation between weekly training hours and competition score.

Dr. Sharma correctly calculates the equation of the regression line $y$ on $x$ for these athletes to be $y = -2.5x + 110$. She uses the equation to estimate the competition score for an athlete who trains 3 hours per week.

Find the competition score calculated by Dr. Sharma.

State whether it is valid to use the regression line $y$ on $x$ for Dr. Sharma's estimate in part (f). Give a reason for your answer, assuming the original data for training hours ranged from 5 to 20 hours.

Dr. Sharma investigated the relationship between an athlete's national competition rank and their average daily protein intake (in grams). She collected data for eight athletes, as shown in the table.

Dr. Sharma intends to analyse the data using Spearman's rank correlation coefficient, $r_s$.

Copy and complete the information in the following table.

Calculate the value of $r_s$.

Interpret your result.

A group of 160 participants completed a fitness challenge. The cumulative frequency graph shows the points obtained by the participants.

Find the median of the points obtained.

The participants were awarded a performance grade from 1 to 5, depending on the points obtained in the challenge.

The number of participants receiving each grade is shown in the following table.

Find an expression for $a$ in terms of $b$.

The mean grade for these participants is 3.5.

Find the number of participants who obtained a grade 5.

Find the minimum points needed to obtain a grade 5.

The lifespans, $t$, of 250 LED light bulbs are recorded in the following table.

This table is used to create a cumulative frequency graph.

Write down the mid-interval value of the class $0 \le t < 1000$.

Calculate an estimate of the mean lifespan of the 250 light bulbs.

Use the cumulative frequency curve (which would be provided in an exam) to estimate the interquartile range. Assume the lower quartile ($Q_1$) is $1300$ hours and the upper quartile ($Q_3$) is $2150$ hours.

A light bulb from the data set had a lifespan of $3400$ hours.

Use your answer to part (b) to estimate whether this light bulb's lifespan is an outlier for this data. Justify your answer.

It is believed that the lifespans of these LED light bulbs follow a normal distribution with mean $1740$ hours and standard deviation $450$ hours.

It is decided to perform a $\chi^2$ goodness of fit test on the data to determine whether this sample of 250 light bulbs could have plausibly been drawn from an underlying distribution $N(1740, 450^2)$.

Write down the null and the alternative hypotheses for the test.

As part of the test, the following table is created.

Find the value of $a$ and the value of $b$. Give your answers to one decimal place.

Hence, perform the test to a 5% significance level, clearly stating the conclusion in context.

[Maximum mark: 19]

A tech company recorded the time (in minutes) 180 customers spent completing a new online feedback survey. The data was compiled into the following cumulative frequency graph.

(a) Use the graph to find

(i) the median time;

(ii) the lower quartile;

(iii) the upper quartile;

(iv) the interquartile range.

Sarah completed the survey in 1.5 minutes.

(b) Determine whether Sarah's time is an outlier.

(a) The scores, $s$, of 200 students on a mathematics test are recorded in the following table.

(i) Write down the mid-interval value of $60 \le s < 80$.

(ii) Calculate an estimate of the mean score of the 200 students.

(b) The data from this table is used to create a cumulative frequency graph. From this graph, the first quartile ($Q_1$) is estimated to be $60$ and the third quartile ($Q_3$) is estimated to be $96$.

Use these values to estimate the interquartile range (IQR).

(c) A student, Elara, scored $155$ on the test.

Use your answer to part (b) to estimate whether Elara's score is an outlier for this data. Justify your answer.

(d) It is believed that the scores of students on this mathematics test follow a normal distribution with mean $77.5$ and standard deviation $20$.

It is decided to perform a $\chi^2$ goodness of fit test on the data to determine whether this sample of 200 students could have plausibly been drawn from an underlying distribution $N(77.5, 20^2)$.

Write down the null and the alternative hypotheses for the test.

(e) As part of the test, the following table is created, where some categories have been combined to ensure expected frequencies are not too low.

(i) Find the value of $a$ and the value of $b$.

(ii) Hence, perform the test to a 5% significance level, clearly stating the conclusion in context.

[Maximum mark: 4]

A salesperson records the number of items sold per day over several weeks. The data is presented in the following frequency distribution table.

| Frequency (f)

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

1 | 3

2 | 5

3 | 8

4 | p

5 | 6

6 | 4

)

For this distribution, the mean number of items sold per day is 3.8.

(a) Write down the total number of days the salesperson recorded data in terms of p.

(b) Calculate the value of p.

A tech company launched two new smartphone models, "Voyager" and "Explorer". They collected customer satisfaction scores (out of 100) from a large sample of users for both models. The results are summarized in the following box and whisker diagram.

Identify which two of the following statements must be true according to the box and whisker diagram. Indicate your choices by placing tick marks in the second column of the following table.

Statement | True (✓)

---|---

The satisfaction scores for Model Voyager are normally distributed. |

A higher percentage of customers gave a score less than 70 for Model Voyager than for Model Explorer. |

A higher percentage of customers gave a score greater than 90 for Model Explorer than for Model Voyager. |

The interquartile range for Model Explorer is less than the interquartile range for Model Voyager. |

A product manager believes there is no significant difference in the average customer satisfaction scores between the two models. She plans to conduct a t-test at the 10% significance level. Write down the null and alternative hypotheses for her test.

The t-test yielded a p-value of 0.0783. Find the p-value for her test.

Write down the conclusion to the test. Give a reason for your answer.

A fitness enthusiast, Alex, recorded the number of steps (in hundreds) he took each day over a period of $28$ days. The data is given below in ascending order:

$55, 60, 62, 65, 68, 70, 72, 75, 78, 80, 82, 85, 88, 90, 92, 95, 98, 100, 102, 105, 110, 115, 120, 125, 130, 140, 160, 250$

(a) Find the median number of steps.

(b) Find the lower quartile (Q1).

(c) Find the upper quartile (Q3).

(d) Find the range of the data.

(e) Determine whether there are any outliers in the data.

(f) Draw a box-and-whisker diagram for the above data, marking any outliers as required.

P2: A farmer, Mr. Jensen, is analyzing the harvest from a new variety of apple trees. He recorded the weights (in grams) of 25 randomly selected apples.

$101.4$, $102.7$, $104.2$, $106.1$, $107.5$, $120.3$, $121.5$, $122.9$, $123.6$, $124.8$, $125.1$, $126.4$, $140.1$, $141.3$, $142.5$, $143.7$, $144.9$, $146.1$, $147.3$, $148.5$, $160.2$, $161.4$, $162.6$, $180.1$, $181.3$

Copy and complete the following grouped frequency table.

Weight, ($w$ g) | Frequency

---|---

$100 \le w < 120$ |

$120 \le w < 140$ |

$140 \le w < 160$ |

$160 \le w < 180$ |

$180 \le w < 200$ |

Find an estimate for the mean weight, using the frequency table. Give your answer to one decimal place.

Find an estimate for the variance, using the frequency table. Give your answer to three significant figures.

Find an estimate for the standard deviation, using the frequency table. Give your answer to three significant figures.

Mr. Jensen also harvested apples from an older variety of trees. For these apples, the mean weight was $145.0$ g and the standard deviation was $20.0$ g. Compare the weights of the apples from the new variety and the older variety, drawing specific conclusions.

(a) State whether the following set of data is discrete or continuous, and, in each case, construct a frequency table.

The number of cars passing a specific checkpoint in 10-minute intervals during rush hour:

$5 \quad 5 \quad 5 \quad 6 \quad 6 \quad 7 \quad 7 \quad 9 \quad 9 \quad 9 \quad 10 \quad 11 \quad 11 \quad 12 \quad 12 \quad 12 \quad 12 \quad 12$

(b) State whether the following set of data is discrete or continuous, and, in each case, construct a frequency table using appropriate class intervals.

The heights of saplings (in cm) in a plant nursery:

$15.7 \quad 17.3 \quad 18.4 \quad 19.1 \quad 20.2 \quad 21.3 \quad 22.6 \quad 24.4 \quad 25.2 \quad 26.0 \quad 26.5 \quad 27.5 \quad 28.5 \quad 29.0 \quad 29.7$

(c) State whether the following set of data is discrete or continuous, and, in each case, construct a frequency table.

The number of correct answers on a 10-question multiple-choice quiz for a group of students:

$4 \quad 4 \quad 4 \quad 4 \quad 5 \quad 5 \quad 5 \quad 5 \quad 5 \quad 5 \quad 6 \quad 6 \quad 7 \quad 8 \quad 8 \quad 9 \quad 9 \quad 9 \quad 9 \quad 10$

The following table shows the student enrollment figures for various faculties at a major university.

(a) Calculate the total number of students enrolled at the university.

(b) Determine the percentage of students enrolled in the Faculty of Engineering, giving your answer to one decimal place.

A city's energy department records the total annual electricity consumption (in GWh) for the years $2018$ to $2022$. The results are shown in the table.

Year | $2018$ | $2019$ | $2020$ | $2021$ | $2022$

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

Consumption [GWh] | $15500$ | $16200$ | $17000$ | $16800$ | $18100$

(a) Calculate the mean annual electricity consumption for these five years.

(b) Calculate the standard deviation of the annual electricity consumption over these five years.

(c) Calculate the percentage increase in annual electricity consumption from $2018$ to $2019$.

(d) The city council publishes an infographic to highlight the trend in electricity consumption using a bar chart where the vertical axis starts at $15000$ GWh.

(i) Explain why this diagram may give a misleading picture.

(ii) State reasons why the bar chart might be drawn in this way.

The grouped frequency table shows the daily commute times (in minutes) for the $180$ employees at a technology company.

Commute time, $t$ (minutes) | Frequency

---|---

$0\le t<15$ | $15$

$15\le t<30$ | $30$

$30\le t<45$ | $55$

$45\le t<60$ | $40$

$60\le t<75$ | $25$

$75\le t<90$ | $10$

$90\le t<105$ | $5$

Construct a cumulative frequency table for this data.

Plot the points and draw the cumulative frequency curve for the data.

Use your curve or calculations to find an approximate value for:

(i) the median commute time;

(ii) the interquartile range.

The lowest commute time recorded was $5$ minutes and the greatest was $100$ minutes.

Draw a box-and-whisker plot to represent the data.

A small online retail company, 'GadgetHub', tracks its daily advertising expenditure and the corresponding number of units sold for a new product over a period of ten days.

Draw a scatter graph to represent this information. Label the axes clearly.

Describe the correlation between daily advertising spend and units sold.

State whether you think the daily advertising spend has an effect on the number of units sold. Give a reason for your answer.

P1: The box-and-whisker diagram below illustrates the daily screen time (in hours) for a sample of teenagers.

(a) Find the range of daily screen times.

(b) Find the interquartile range (IQR) of daily screen times.

(c) Find the percentage of teenagers who spend between $3.5$ and $7$ hours on screen time daily.

(d) A new survey participant reported spending $12$ hours on screen time daily.

Determine whether this time would be counted as an outlier.