Question: User: a) Work out an estimate for the mean delivery time to 1 d.p.
b) Does the data in the table support the takeaway’s claim?
c) The takeaway gives a refund if a delivery takes longer than 30 minutes.
Work out the percentage of deliveries that received a refund to the nearest 1%
Delivery time, t (minutes) Frequency
10 < t 80 20 19
20 < t 80 30 20
30 < t 80 40 6
40 < t 80 50 1
1. **State the problem:**
We have delivery times grouped in intervals with frequencies. We need to estimate the mean delivery time, check if the data supports the takeaway's claim, and find the percentage of deliveries that took longer than 30 minutes (which get a refund).
2. **Estimate the mean delivery time:**
Use the midpoint of each interval as an estimate for delivery time in that interval.
- Midpoints:
- $10 < t \leq 20$: midpoint = $\frac{10 + 20}{2} = 15$
- $20 < t \leq 30$: midpoint = $\frac{20 + 30}{2} = 25$
- $30 < t \leq 40$: midpoint = $\frac{30 + 40}{2} = 35$
- $40 < t \leq 50$: midpoint = $\frac{40 + 50}{2} = 45$
- Frequencies: $19, 20, 6, 1$
Calculate total frequency:
$$19 + 20 + 6 + 1 = 46$$
Calculate sum of midpoint times multiplied by frequencies:
$$15 \times 19 + 25 \times 20 + 35 \times 6 + 45 \times 1 = 285 + 500 + 210 + 45 = 1040$$
Estimate mean delivery time:
$$\text{mean} = \frac{1040}{46} \approx 22.6$$
3. **Check if data supports the takeaway's claim:**
Assuming the takeaway claims delivery times are generally under 30 minutes.
The mean delivery time is approximately $22.6$ minutes, which is under 30 minutes.
4. **Calculate percentage of deliveries that received a refund:**
Refunds are given if delivery time $> 30$ minutes.
Number of deliveries over 30 minutes:
$$6 + 1 = 7$$
Percentage of deliveries refunded:
$$\frac{7}{46} \times 100 \approx 15.2\%$$
Rounded to nearest 1%:
$$15\%$$
**Final answers:**
- a) Mean delivery time $\approx 22.6$ minutes (to 1 d.p.)
- b) Yes, the data supports the claim since mean is under 30 minutes.
- c) Percentage refunded $= 15\%$