Review: Statistical Problem Solving

A-Level Year 1 review: Data collection, processing, and visual representation.

Topics include sampling, sample standard deviation, bivariate data, and statistical diagrams.

Answers Worked Solutions
1)

A botanist wants to study the plants in a large greenhouse containing exactly 800 plants. She creates a numbered list of all the plants and uses a random number generator to select 40 plants for her study.

For each selected plant, she records the weight in grams and the number of fully formed leaves.

  1. State the name of the sampling method the botanist has used.
  2. Give one advantage and one disadvantage of using this sampling method compared to taking a census.
  3. State, with a reason, whether the variable "weight in grams" is discrete or continuous.
  4. State, with a reason, whether the variable "number of fully formed leaves" is discrete or continuous.
2)

The marks out of 60 for 15 students in a recent statistics test are recorded below:

14, 21, 38, 26, 45, 33, 21, 51, 34, 42, 28, 39, 44, 21, 35
  1. Draw an ordered stem and leaf diagram to represent this data. Include a key.
  2. Find the median mark ($Q_2$).
  3. Find the lower quartile ($Q_1$) and upper quartile ($Q_3$).
  4. Calculate the interquartile range (IQR).
3)

A second class, Class B, takes the same test from Question 2. The summary statistics for their marks are as follows:

  • Lowest mark: 8 (the next lowest is 22)
  • Lower quartile: 34
  • Median: 38
  • Upper quartile: 44
  • Highest mark: 58

An outlier is defined as any value that is more than $1.5 \times \text{IQR}$ below the lower quartile, or more than $1.5 \times \text{IQR}$ above the upper quartile.

  1. Show mathematically that the lowest mark (8) is an outlier, but the highest mark (58) is not an outlier.
  2. Using the blank grid provided below, draw a box plot to represent the marks of Class B. Ensure you clearly mark the outlier.
0 10 20 30 40 50 60
4)

The distances, in metres, of 6 consecutive throws by a javelin athlete are recorded as a sample:

45.2,  48.1,  42.9,  49.5,  46.0,  47.3

  1. Calculate the mean, $\bar{x}$.
  2. Calculate the sample standard deviation. Give your answer to 3 significant figures.
    Recall the formula: $s = \sqrt{\frac{\sum x^2 - \frac{(\sum x)^2}{n}}{n-1}}$
5)

The time taken, $t$ minutes, for 100 people to complete a logic puzzle is summarised in the table below. Note that the data is continuous.

Time, $t$ (mins) Frequency
$0 < t \le 10$ 15
$10 < t \le 15$ 20
$15 < t \le 20$ 25
$20 < t \le 30$ 30
$30 < t \le 50$ 10
  1. Explain why a histogram is an appropriate diagram to represent this data.
  2. A student draws a histogram to represent this data. The bar representing the $10 < t \le 15$ class is drawn with a width of 4cm and a height of 8cm.

    Using the principle that the area of the bar is proportional to the frequency, find the width and height of the bar representing the $20 < t \le 30$ class.
  3. Estimate the number of people who took between 18 and 25 minutes to complete the puzzle.
6)

The waiting times, $w$ minutes, for 80 patients at a medical clinic were recorded. The cumulative frequency diagram below represents this data.

0 10 20 30 40 50 60 Wait Time (minutes) 0 20 40 60 80 Cumulative Freq.
  1. Use the graph to estimate the median wait time.
  2. Use the graph to estimate the interquartile range (IQR).
  3. Patients who wait longer than 45 minutes are given a formal apology. Estimate the number of patients who received an apology.
7)

Two groups of plants, Group A and Group B, are grown under different lighting conditions. Their heights, in cm, are measured after one month.

For Group A, there are 15 plants. The data is summarised as follows:

$\sum x = 540, \quad \sum x^2 = 20350$

For Group B, there are 25 plants. The mean height is $38\text{ cm}$ and the sample standard deviation ($s$) is $5.2\text{ cm}$.

  1. Calculate the mean and sample standard deviation for the plants in Group A.
  2. Calculate the overall mean height of all 40 plants combined.
  3. By first calculating $\sum y^2$ for Group B, determine the sample standard deviation of the combined group of 40 plants.
8)

A researcher investigates the relationship between the number of hours spent studying per week ($x$) and the score achieved on a rigorous test ($y$, out of 100) for a sample of 8 students. The data is bivariate.

The summary statistics for the 8 students are:

$\sum x = 120, \quad \sum y = 560$

  1. Calculate the mean number of study hours ($\bar{x}$) and the mean test score ($\bar{y}$).
  2. A scatter diagram of the data shows a strong positive correlation. A journalist writes an article claiming, "Studying more hours directly causes higher test scores." Explain why the journalist's claim might be flawed.
  3. The equation of the regression line of $y$ on $x$ is calculated to be $y = 28 + 2.8x$. Interpret the meaning of the value $2.8$ in the context of this study.
  4. Use the regression equation to estimate the score of a student who studied for 30 hours. Comment on the reliability of this estimate if the maximum hours studied by any student in the original sample was 22 hours.