📝 Exam Questions — Probability & Statistics 1

A farmer recorded the weights, in grams, of 50 apples harvested from a new orchard. The weights are as follows: 103, 112, 108, 115, 120, 125, 118, 105, 130, 122, 110, 117, 128, 135, 140, 123, 109, 114, 121, 129, 133, 107, 116, 124, 131, 138, 101, 111, 126, 134, 142, 119, 106, 127, 136, 104, 113, 120, 132, 139, 145, 102, 110, 125, 137, 141, 100, 118, 122, 130 (a) Construct a histogram for this data, using class intervals of 10g starting from 100g. Ensure your axes are clearly labelled and include all necessary features of a histogram. [7] (b) Comment on the distribution of the apple weights based on your histogram. [3]

A researcher is recording the ages of participants in a study. The ages are recorded as whole numbers. (a) For a continuous dataset, identify the lower and upper class boundaries for the interval 20-29. [2] (b) Calculate the frequency densities for the classes 10-19 (frequency 15) and 20-29 (frequency 25), assuming equal-width intervals based on the boundaries identified in part (a). [4]

Histograms are a common way to represent continuous data. When constructing a histogram, the choice of interval widths is important. (a) Identify the primary advantage of using equal-width intervals when constructing a histogram. [2] (b) State the formula used to calculate frequency density for a histogram with equal-width intervals. [3]

Fig 1.1 shows a cumulative frequency graph for the marks of 100 students in an exam. (a) Estimate the interquartile range from the graph. [3] (b) Calculate the number of students who scored between 50 and 70 marks. [4] (c) Explain why it is not possible to determine the exact mean from this graph. [3]

A teacher recorded the scores of 18 students in a recent mathematics test. (a) Construct a stem-and-leaf diagram for the following test scores: 65, 72, 81, 68, 75, 90, 62, 78, 85, 70, 73, 88, 60, 79, 83, 76, 92, 69. [4] (b) Calculate the mean test score from your diagram. [2] (c) Find the percentage of students who scored 80 or above. [2]

Different graphical representations are used to visualise continuous data, each with specific strengths. (a) Compare the effectiveness of a histogram versus a cumulative frequency graph for identifying the mode of a continuous dataset. [6] (b) Evaluate the strengths and weaknesses of using a histogram to represent a dataset with a very large range of values. [5]

A researcher collected data on the scores of participants in a cognitive task. The data is represented in the histogram shown in Fig 1.1. (a) Determine the frequencies for each class interval from the provided histogram. (i) For the interval 0-5. [1] (ii) For the interval 5-10. [1] (iii) For the interval 10-15. [1] (iv) State the total number of participants. [1] (b) Explain how you used the frequency density values to find the frequencies. [3]

The cumulative frequency graph in Fig 1.1 shows the distribution of 120 measurements taken during an experiment. (a) Using the graph, calculate the frequency of the class 60-70. [5] (b) Estimate the percentage of data points that fall within the range 45 to 85. [3]

The table below shows the journey times, in minutes, for 50 commuters travelling to work.

(a) Construct a cumulative frequency graph for this data on a piece of graph paper. [6] (b) Using your graph, estimate the 10th percentile of journey times. [3] (c) Comment on the skewness of the distribution based on your graph. [2]

In a study of daily commute times, data was collected for a group of employees and organised into class intervals for a histogram. (a) Calculate the frequency density for the class 15-20 minutes, given that the frequency for this class is 30 employees. [3] (b) Determine the class frequency for an interval with a class width of 10 minutes and a frequency density of 2.5 employees per minute. [4]

A retail store records the number of units sold each day over a period of time. The daily sales data is represented in the stem-and-leaf diagram shown in Fig 1.1. Fig 1.1 Daily Sales (units) Key: 5|2 means 52 units 5 | 2 4 6 8 6 | 0 1 1 3 5 7 9 7 | 0 2 4 4 5 8 8 | 1 3 (a) Discuss the strengths and limitations of using a stem-and-leaf diagram for representing large data sets compared to small data sets. [6] (b) Calculate the mean of the data set shown in Fig 1.1, ensuring you show your working. [5]

A study recorded the time taken for a new chemical reaction to complete, with the results displayed in the histogram in Fig 1.1. (a) Analyse the shape of the distribution shown in Fig 1.1, commenting on skewness and spread. [6] (b) Discuss the implications of this distribution for the variable being measured. [4]

Fig 1.2 shows a histogram representing the distribution of a continuous variable. (a) Describe the overall shape and spread of the data shown in the histogram in Fig 1.2. [4] (b) Suggest a possible real-world scenario that this distribution could represent, justifying your choice with reference to the characteristics identified in part (a). [4]

A student recorded the number of minutes they spent revising for a test each day over a two-week period. The data collected is as follows: 12, 15, 21, 23, 23, 27, 30, 31, 35, 38, 40, 42. (a) Construct a stem-and-leaf diagram to represent this data. [4] (b) Include a key for your diagram. [1]

A botanist measures the heights of 50 plants, in cm, and groups them into the following classes:

(a) Calculate the frequency density for each class interval. [4] (b) Show that the sum of the products of frequency density and class width equals the total frequency of 50 plants. [4]

A dataset contains grouped continuous data, where one class interval is 15-20. The preceding class interval is 10-15 with a frequency of 18. The total frequency up to the start of the 15-20 interval is 18. (a) Calculate the frequency density for the class interval 15-20, given its frequency is 25. [3] (b) Estimate the number of observations below 17.5 using linear interpolation. [3]

A scientist recorded the number of insect species found in 16 different samples from a nature reserve. The data is presented in the stem-and-leaf diagram shown in Fig 1.1. Fig 1.1 Key: 0|8 means 8 Stem | Leaves 0 | 8 9 1 | 0 1 3 5 7 2 | 0 2 4 6 8 9 3 | 1 3 5 (a) Determine the lower quartile and upper quartile from the stem-and-leaf diagram shown in Fig 1.1. [4] (b) State the interquartile range for this dataset. [2]

Histograms are powerful tools for visualising continuous data. However, the choice of class intervals is crucial for effective representation. (a) Explain why unequal-width intervals might be used in a histogram. [3] (b) Give an example of a scenario where unequal-width intervals would be appropriate. [3]

Data can be represented in various ways to highlight different characteristics. Cumulative frequency graphs are one such method for continuous data. (a) List three advantages of using a cumulative frequency graph. [3] (b) State one disadvantage of using a cumulative frequency graph. [2]

The histogram in Fig 1.1 shows the distribution of times, in minutes, taken by a group of students to complete a puzzle. (a) Find the frequency for each class interval represented in the histogram. [5] (b) Calculate the total number of observations represented by the histogram. [4]

Data can be represented in various ways, including cumulative frequency graphs. (a) Define what is meant by cumulative frequency. [2] (b) State three key features of a cumulative frequency graph. [3]

Fig 1.1 shows a histogram representing the distribution of values in a dataset. The first class interval is 0-5, the second is 5-10, and the third is 10-20. (a) Determine the frequencies of the last two classes of the given histogram. [4] (b) State the main reason why the heights of the bars in a histogram do not directly represent frequency when class widths are unequal. [3]

The table below shows the distribution of ages of participants in a local marathon event.

| 25 to < 30 | 35 |\ | 30 to < 40 | 50 |\ | 40 to < 55 | 30 |\ | 55 to < 70 | 15 |\ (a) Draw a histogram to represent this distribution of ages. [6] (b) Explain how the frequency density is crucial for accurate representation when class widths are unequal. [3]

A researcher collected data on the ages of a small group of participants in a study. The data is presented in the stem-and-leaf diagram in Fig 1.1. Fig 1.1 Stem | Leaves ---- | ------ 2 | 1 3 5 3 | 0 0 2 4 4 | 1 6 Key: 2|1 means 21 (a) State one advantage of using a stem-and-leaf diagram over a simple list of data. [2] (b) Identify the median and mode from the data shown in Fig 1.1. [4]

A researcher is studying the average reaction time of individuals to a specific stimulus. (a) Describe how outliers in a dataset can influence the interpretation of data and the conclusions drawn. [3] (b) Explain why it is important to check for missing data during the data collection and cleaning phase. [3]

A scientist measured the reaction times, in milliseconds, of 60 individuals to a visual stimulus. The data was grouped into the following class intervals:

(a) Calculate the frequency density for each class interval of this data, where the class width is uniform. [6] (b) Describe the steps you would take to accurately label the axes of a histogram for this data. [3]

The table below shows the distribution of waiting times, in minutes, for customers at a bank during a busy morning.

(a) Calculate the frequency densities for each class interval. [4] (b) Determine the total frequency of the data set from the class widths and the frequency densities. [4]

Data is collected from a survey about the heights of students in a school. (a) Identify the type of data best represented by a cumulative frequency graph. [2] (b) Explain why upper class boundaries are used when plotting cumulative frequency graphs. [2]

Fig 1.2 shows a cumulative frequency graph for the weights of 200 apples. (a) Use the graph to determine the 75th percentile of the data. [4] (b) Determine the number of items with a value greater than 80 g. [3]

In the field of medical research, different types of data are collected to study patient health and disease progression. (a) Explain the key difference between qualitative and quantitative data. [3] (b) Distinguish between discrete and continuous data, providing an example for each. [4]

The lengths, in cm, of a sample of insects are shown in the stem-and-leaf diagram in Fig 1.1. Fig 1.1 Stem | Leaves ---- | ------ 1 | 2 5 8 2 | 0 1 3 5 7 3 | 0 2 4 6 4 | 1 3 Key: 1|5 means 1.5 cm (a) Calculate the range and interquartile range of the data represented in Fig 1.1. [4] (b) Interpret what the interquartile range indicates about the spread of the data. [3]

A conservation group measured the heights of trees in a small forest area. The data collected is presented in the histogram shown in Fig 1.1. Fig 1.1 (a) Interpret the information presented in the histogram regarding the distribution of tree heights. [4] (b) Estimate the number of trees with heights between 8m and 12m. [4]

The number of goals scored by two football teams, Team A and Team B, in 10 matches each, were recorded. Team A: 1, 3, 0, 2, 1, 4, 2, 3, 1, 0 Team B: 2, 1, 3, 0, 2, 5, 1, 4, 3, 2 (a) Construct a back-to-back stem-and-leaf diagram to compare the number of goals scored by the two teams. [6] (b) Compare the median number of goals for both teams using your diagram. [2]

Histograms are powerful tools for visualising continuous data, but their construction requires specific terminology and understanding. (a) Define the term 'frequency density' as it applies to histograms. [2] (b) Define what the area of a bar in a histogram represents. [2]

Data can be represented in various ways to highlight different characteristics. Both histograms and bar charts use bars to display frequencies. (a) State one key difference between a histogram and a bar chart. [2] (b) Explain why histograms are generally preferred for representing continuous data over bar charts. [3]

A study recorded the heights of 100 students, grouped into the following frequency distribution:

(a) Draw a cumulative frequency curve for this distribution on a piece of graph paper. [5] (b) Using your curve, estimate the number of people shorter than 158 cm. [2] (c) Using your curve, find the height below which 70% of the people lie. [2]

A regional car sales manager wants to compare the performance of two dealerships, Dealership A and Dealership B, over 15 months. The number of cars sold each month is recorded as follows: Dealership A: 21, 28, 30, 32, 35, 35, 37, 40, 41, 43, 45, 48, 50, 52, 55. Dealership B: 18, 20, 22, 25, 27, 29, 30, 31, 33, 36, 38, 40, 42, 44, 46. (a) Construct a back-to-back stem-and-leaf diagram for the number of cars sold by the two dealerships. [5] (b) Find the median for each dealership. [2] (c) Comment on the central tendency of sales for both dealerships based on your medians. [2]

A market research company is conducting a survey to understand consumer preferences and habits. (a) Classify the type of data represented by 'favourite colour'. [2] (b) Give two distinct examples of quantitative data. [3]

A scientist is conducting an experiment to investigate the effect of a new fertiliser on plant growth. (a) State two reasons why data might be collected in this scientific experiment. [2] (b) State one benefit of displaying data visually. [2]

A researcher is collecting information about students in a school to better understand their demographics and living situations. (a) Identify whether 'number of siblings' is discrete or continuous data. [2] (b) Identify whether 'temperature in a room' is discrete or continuous data. [2]

A market research company collects information about consumer preferences for a new product. (a) Define the term 'data analysis' in the context of statistics. [2] (b) Give two examples of real-world scenarios where data analysis is crucial for decision-making. [3]

A large technology company is collecting user data from its new social media platform to understand user behaviour and improve its services. (a) Discuss the ethical considerations that should be taken into account when collecting personal data from individuals. [6] (b) Evaluate the potential biases that can arise during the data collection process and suggest ways to mitigate them. [4]

Data classification is a fundamental step in any statistical analysis. Incorrect classification can lead to flawed conclusions. (a) Discuss the implications of misclassifying data (e.g., treating continuous data as discrete) for statistical analysis. [6] (b) Analyse a scenario where the same variable could be considered either discrete or continuous depending on the context or measurement precision. [4]

A national health organisation is collecting various types of data related to public health, including the number of reported flu cases per week and the body temperatures of patients. (a) Compare and contrast the appropriate graphical representations for discrete data versus continuous data, explaining why certain types are more suitable. [6] (b) Evaluate a situation where data that is inherently discrete (e.g., shoe size) might be approximated as continuous for certain statistical analyses, discussing the advantages and disadvantages of such an approximation. [5]

A researcher plans to conduct a study on the average sleep duration of teenagers in a particular city. (a) Explain why it is important to carefully plan data collection methods before beginning a study. [3] (b) Outline the steps involved in the process of collecting, displaying, and analysing data to draw conclusions. [4]

A research team is planning a study to investigate the effectiveness of a new educational programme implemented across several schools. They need to gather data from students, teachers, and parents. (a) Discuss the importance of sampling methods in data collection for this study and the potential pitfalls of a poorly chosen sampling technique. [6] (b) Analyse a scenario where a company uses customer feedback data to improve a product. Propose a method for collecting and analysing this data to ensure reliable conclusions are drawn. [6]

Data can be classified as qualitative or quantitative, but sometimes the distinction is not straightforward. (a) Give an example of data that could be considered both qualitative and quantitative depending on how it is measured or recorded. [3] (b) Justify your classification in part (a). [3]

Fig 1.1 shows two cumulative frequency graphs, A and B, representing the heights of two different groups of plants. (a) Interpret the differences in the median values of the two datasets shown in Fig 1.1. [4] (b) Compare the spread of the two datasets using the interquartile range. [3]

A researcher is comparing the reaction times (in milliseconds) of two different groups of participants, Group X and Group Y, to a visual stimulus. (a) Construct a back-to-back stem-and-leaf diagram for the following two datasets: Group X: 12, 15, 18, 21, 22, 25, 27, 30, 31, 33 Group Y: 10, 14, 16, 19, 20, 23, 26, 28, 29, 32 [5] (b) Compare the two groups based on their medians and ranges from your diagram. [4]

Different types of data representation are suitable for different purposes. (a) Identify which data representation (stem-and-leaf diagram, histogram, or cumulative frequency graph) would be most suitable for each of the following scenarios, giving a brief reason: (i) showing individual data points for a small dataset. [2] (ii) illustrating the overall shape of a large continuous dataset. [2]

Fig 1.1 shows two histograms, P and Q, representing the test scores of two different classes in a recent examination. (a) Analyse the skewness of the distribution for both datasets based on the provided histograms. [5] (b) Discuss which representation (histogram or cumulative frequency graph) would be more suitable if the primary goal is to identify outliers and extreme values. [5]

Different data representations are suitable for different types of data and purposes of analysis. (a) Describe one situation where a stem-and-leaf diagram is a more appropriate representation than a histogram. [3] (b) Give one advantage of using a histogram for a large dataset compared to a stem-and-leaf diagram. [3]

Histograms and stem-and-leaf diagrams are both effective ways to represent data, but they serve different purposes and have distinct characteristics. (a) Compare the advantages and disadvantages of using a histogram versus a stem-and-leaf diagram to represent data. [5] (b) Explain when a histogram would be preferred over a stem-and-leaf diagram. [3]

Fig 1.2 shows two cumulative frequency graphs, representing the scores of Class 1 and Class 2 in a mathematics test. There are 50 students in each class. (a) Determine the percentage of students who scored above 70 marks in each test. [4] (b) Explain which class performed better overall, justifying your answer using statistical measures from the graphs. [4]

Data representation choices are crucial for effective comparison and analysis. Different types of diagrams offer unique advantages and disadvantages depending on the nature and size of the dataset. (a) Discuss the strengths and weaknesses of using a cumulative frequency graph to compare two different datasets. [7] (b) Evaluate whether a back-to-back stem-and-leaf diagram or two separate cumulative frequency graphs would be more effective for comparing the ages of students in two different schools (one with 30 students, another with 200 students). [5]

Different types of data require different methods of representation. Understanding the strengths of each representation is crucial for effective data analysis. (a) Name two types of data representation suitable for continuous data. [2] (b) Identify three pieces of information that can be easily obtained from a cumulative frequency graph but not directly from a stem-and-leaf diagram. [3]

Bar charts and histograms are often confused, but they serve different purposes in data representation. (a) Discuss the potential misinterpretations that could arise from using a bar chart instead of a histogram for continuous data. [5] (b) Suggest a scenario where a bar chart would be more appropriate than a histogram for representing numerical data. [3]

Fig 1.3 shows a histogram representing the masses of a group of children. (a) Calculate the frequency density for the class interval 50-70 kg. [3] (b) Estimate the number of children with masses less than 40 kg. [3] (c) State why a histogram is a suitable representation for this data. [2]

Fig 1.4 shows a cumulative frequency graph representing the weights of 120 puppies. (a) Estimate the median weight of the puppies. [3] (b) Calculate the percentage of puppies that weigh between 3.5 kg and 5.5 kg. [4] (c) Compare the spread of weights in the lower half of the data to the upper half of the data. [3]

Fig 1.4 shows a histogram representing the time spent by customers in a supermarket. (a) Calculate the frequency of the class interval 100-120 minutes. [4] (b) Estimate the number of customers who spent less than 70 minutes in the supermarket. [4] (c) Comment on the distribution of customer spending times. [2]

Fig 1.3 shows a histogram representing the duration of phone calls made from an office. (a) Calculate the frequency of the class 20-30 minutes. [3] (b) Estimate the proportion of calls that lasted longer than 40 minutes. [3] (c) Justify why there are no gaps between the bars in the histogram. [2]

Fig 1.4 shows a stem-and-leaf diagram representing the exam scores of a group of students. (a) Find the number of students who scored exactly 75 marks. [2] (b) State the highest score achieved by a student. [2]

Fig 1.3 shows a cumulative frequency graph illustrating the weights of 100 apples, represented by both a cumulative frequency polygon and a smooth cumulative frequency curve. (a) Estimate the median weight of the apples using the smooth curve. [3] (b) Calculate the interquartile range of the apple weights using the smooth curve. [4] (c) Compare the distribution of apple weights represented by the polygon and the curve, discussing their suitability. [5]

Fig 1.2 shows a histogram representing the mathematics test scores of a group of students. (a) Calculate the total number of students represented in the histogram. [3] (b) Estimate the number of students who scored between 50 and 70 marks. [3] (c) Comment on the skewness of the distribution of test scores. [3]

Fig 1.2 shows a histogram representing the race times of a group of athletes. (a) Calculate the frequency density for the class interval 15.5-18.5 min. [3] (b) Determine the total number of athletes represented in the histogram. [3]

Fig 1.3 shows a histogram titled 'Production Output per Hour' representing the number of units produced per hour by a factory. (a) Calculate the frequency density for the class interval 100-150 units. (b) Estimate the total number of items produced that are represented in the histogram. (c) Explain why the class 0-50 units has the highest frequency density but not necessarily the highest frequency.

Fig 1.3 shows a cumulative frequency graph of student exam scores. (a) Estimate the number of students who scored less than 60 marks. [2] (b) Determine the 75th percentile score from the graph. [3]

Fig 1.2 shows a back-to-back stem-and-leaf diagram representing the daily rainfall (mm) in a certain town for the months of November and December. (a) Find the number of days in November with rainfall less than 15 mm. (b) State the median daily rainfall for December.

Fig 1.4 shows a cumulative frequency graph titled 'Student Exam Scores' representing the scores of 150 students in an exam. (a) Estimate the number of students who scored between 50 and 70 marks. (b) Calculate the percentage of students who scored above 80 marks.

Fig 1.1 shows a cumulative frequency graph illustrating the heights of 100 plants. (a) Estimate the 20th percentile for the heights of the plants. [3] (b) Calculate the number of plants that have a height greater than 40 cm. [3] (c) Estimate the value of the lower quartile. [2]

Fig 1.1 shows a cumulative frequency graph illustrating the lifetimes of 200 electronic components. (a) Estimate the number of components that lasted between 600 and 900 hours. [3] (b) Calculate the lifetime that is exceeded by 10% of the components. [4] (c) Interpret the steepness of the curve between 400 and 800 hours compared to between 800 and 1200 hours in terms of component lifetimes. [3]

Fig 1.2 shows a histogram representing the speeds of cars on a highway. (a) Calculate the total number of cars whose speeds are recorded. [3] (b) Determine the frequency of cars travelling between 40 km/h and 60 km/h. [3] (c) Explain why the height of the bar for 60-80 km/h is not directly proportional to the frequency for that class. [3]

Fig 1.1 shows a histogram illustrating the race completion times for a group of athletes. (a) Determine the frequency of the class interval 10-15 seconds. [4] (b) Calculate the percentage of athletes who completed the race in less than 20 seconds. [4] (c) Comment on the shape of the distribution shown by the histogram. [2]

Fig 1.2 shows a cumulative frequency graph illustrating the recovery times of 150 patients. (a) Estimate the number of patients whose recovery time was between 5 and 10 days. [3] (b) Find the recovery time that was achieved by 80% of the patients. [3]

Fig 1.1 shows a histogram illustrating the volume of liquid distributed to a sample of containers. (a) Calculate the frequency of the class interval 10-20 litres. [3] (b) Determine the total volume of liquid distributed, assuming the mid-point of each class for calculation. [3] (c) Discuss why frequency density is used on the y-axis instead of frequency. [2]