L2 Probability
Question Bank
A class's test scores were summarised in a box plot shown below. The minimum score was 55, and the maximum score was 95.
(a) What is the interquartile range (IQR) for the test scores?
(b) What percentage of students scored between 65 and 95?
A survey asked 50 students how many books they read in the last month. The results are shown in the frequency table below.
| Number of Books | Frequency |
|---|---|
| 0 | 5 |
| 1 | 8 |
| 2 | 12 |
| 3 | 15 |
| 4 | 7 |
| 5 | 3 |
| Total | 50 |
(a) Using a graphics calculator, calculate the measures of centre: mean, median, and mode.
(b) Using a graphics calculator, calculate the measures of spread: range, interquartile range (IQR), and standard deviation.
A set of daily sales figures for a small business is summarised in the box plot below. The sales data is noticeably skewed.
Based on the plot:
(a) Calculate the range of the central $50\%$ of the data. What does the position of the median line tell you about the skew of the sales data?
(b) What is the longest range of scores that still contains $25\%$ of the data, and what does this imply about the sales figures in that range?
The time (in minutes) that 50 customers spent queuing at a supermarket self-checkout is summarised in the frequency table below.
| Time in Minutes | Frequency |
|---|---|
| $0 \le t < 2$ | 15 |
| $2 \le t < 4$ | 20 |
| $4 \le t < 6$ | 10 |
| $6 \le t < 8$ | 5 |
| Total | 50 |
(a) Calculate the estimated mean queuing time for the 50 customers.
(b) State the modal class and explain why the mean might be a better measure of centre than the median for this data set.
The weekly study hours of 80 students preparing for exams are shown in the histogram below. The vertical axis represents the frequency ($f$).
(a) Using a graphics calculator, estimate the measures of centre: mean, median, and modal class.
(b) Using a graphics calculator, estimate the measures of spread: range, interquartile range (IQR), and standard deviation.
The daily temperature (in degrees Celsius) recorded over 60 days in a coastal town is shown in the histogram below. The vertical axis represents the frequency ($f$).
(a) Using a graphics calculator, estimate the measures of centre: mean, median, and modal class.
(b) Using a graphics calculator, estimate the measures of spread: range, interquartile range (IQR), and standard deviation.
A small health blog publishes a study that claims, "Drinking a certain herbal tea daily increases a person's resting heart rate by an average of 10 beats per minute." The study involved 8 participants over a single week.
A major news outlet reports this finding as a scientific fact. A student comments: "The finding may be true for the group tested, but it is not reliable for the general population."
Comment on the student's statement by identifying two major flaws in the original study that affect the reliability of the claim.
The number of strokes taken by golfers in a local tournament was recorded. The data from the final round is displayed in the box plot below. All scores are within 40 to 90 strokes.
(a) What is the range of scores that contains the middle $50\%$ of all golfers? What is the median score, and what does its position relative to the box indicate?
(b) Compare the spread of the best $25\%$ of golfers (Minimum to $Q_1$) to the spread of the worst $25\%$ of golfers ($Q_3$ to Maximum).
A fast-food restaurant conducted a survey of customer satisfaction by handing out questionnaires to all customers who dined in between 12:30 pm and 1:30 pm on a Tuesday.
The survey results showed that $95\%$ of the customers who responded rated their satisfaction as "Excellent". The restaurant manager concludes that "over $90\%$ of all our customers across the entire week are highly satisfied."
(a) Identify two reasons why the manager's conclusion may not be justified.
(b) Suggest one change to the sampling method that would improve the generalisability of the results to all customers of the restaurant.
The ages (in years) of 55 children attending a holiday program are shown in the histogram below. The vertical axis represents the frequency ($f$).
Age (a) in Years
(a) State the modal age group. What is the estimated mean age of the children?
(b) Estimate the median age and explain why the mean is likely an over-estimate for the true mean in this case.
A supermarket wants to estimate the proportion of shoppers who use their self-checkout service. A staff member collects data by standing near the service on a busy Saturday morning between 9:00 am and 11:00 am. They survey every 10th customer who walks past them in the main aisle, asking if they have used the self-checkout today.
Out of the $N=120$ customers surveyed, $80$ confirmed they had used the service.
(a) Calculate the proportion of people in the sample which had used the self checkout.
(b) The supermarket concludes, "Around $67\%$ of all our weekly customers use the self-checkout." Discuss the reliability of the supermarkets conclusion.