Studdy AI Logo
Math  /  Data & Statistics

QuestionThe table gives information about the times taken by 80 people to run a race.
Time taken ( tt minutes) Cumulative Frequency 50<t≤601550<t≤703150<t≤805250<t≤906650<t≤1007450<t≤11080\begin{array}{ll} 50<t \leq 60 & 15 \\ 50<t \leq 70 & 31 \\ 50<t \leq 80 & 52 \\ 50<t \leq 90 & 66 \\ 50<t \leq 100 & 74 \\ 50<t \leq 110 & 80 \end{array}
This information is shown on the cumulative frequency graph below.
Use this graph to find an estimate for the interquartile range of the times taken.

Studdy Solution

STEP 1

What is this asking? Find the spread of the middle 50% of race times using the cumulative frequency graph. Watch out! Don't confuse cumulative frequency with regular frequency; they accumulate!

STEP 2

1. Understand the cumulative frequency
2. Find the quartiles
3. Calculate the interquartile range

STEP 3

Alright, let's dive into this!
Cumulative frequency is all about **adding up** the frequencies as you go.
It's like a running total.
In this problem, the cumulative frequency tells us how many runners finished the race in a certain amount of time or less.
For example, **52** people finished in **80 minutes** or less.

STEP 4

To find the quartiles, we need to identify the **lower quartile (Q1)** and the **upper quartile (Q3)**.
These are the times at which 25% and 75% of the runners finished, respectively.
Since there are **80** runners, let's calculate:
- **Q1**: 14×80=20 \frac{1}{4} \times 80 = 20 - **Q3**: 34×80=60 \frac{3}{4} \times 80 = 60
These calculations tell us that **Q1** is the time at which the **20th** runner finished, and **Q3** is the time at which the **60th** runner finished.

STEP 5

Now, let's use the graph to find these times.
Locate the **20th** and **60th** runners on the cumulative frequency axis (y-axis) and read across to the time taken (x-axis).
- For **Q1** (20th runner), find the corresponding time on the x-axis. - For **Q3** (60th runner), do the same.

STEP 6

The **interquartile range (IQR)** is the difference between the upper quartile and the lower quartile.
It's a measure of the spread of the middle 50% of the data.
So, let's calculate it:
IQR=Q3−Q1\text{IQR} = Q3 - Q1
Plug in the values you found from the graph for **Q3** and **Q1** to get the **IQR**.

STEP 7

The interquartile range of the times taken is the difference between the times for the **60th** and **20th** runners on the graph.
This gives us the spread of the middle 50% of the race times.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord