10. Two neighbors work at the same place. One travels to work by bus, the other cycles to work. The times taken (in minutes) by each transport to get to work on a sample of 8 days were given as follows: ■ Bus $$ 33,28,40,32,41,32,38,42 $$ $$ \begin{array}{l} ext { Cyclist } \ 26,33,27,31,31,30,28,24 \ ext { Required: Calculate the mean and } \ ext { standard deviation for each set of times } \ ext { and use them to compare the travel } \ ext { times for the two commuters }\end{array} $$

Question

10. Two neighbors work at the same place. One travels to work by bus, the other cycles to work. The times taken (in minutes) by each transport to get to work on a sample of 8 days were given as follows: ■ Bus $$ 33,28,40,32,41,32,38,42 $$ $$ \begin{array}{l}	ext { Cyclist } \ 26,33,27,31,31,30,28,24 \ 	ext { Required: Calculate the mean and } \ 	ext { standard deviation for each set of times } \ 	ext { and use them to compare the travel } \ 	ext { times for the two commuters }\end{array} $$

UpStudy AI · Accepted Answer

To compare the travel times of the two neighbors who commute by bus and cycling, we'll calculate the **mean** and **standard deviation** for each set of times. These statistics will help us understand the average travel time and the variability in their commuting times.

### **1. Calculating the Mean**

The **mean** (average) is calculated by summing all the observations and dividing by the number of observations.

**Bus Times:** $$ 33, 28, 40, 32, 41, 32, 38, 42 $$

$$
\text{Mean}_{\text{Bus}} = \frac{33 + 28 + 40 + 32 + 41 + 32 + 38 + 42}{8} = \frac{286}{8} = 35.75 \text{ minutes}
$$

**Cyclist Times:** $$ 26, 33, 27, 31, 31, 30, 28, 24 $$

$$
\text{Mean}_{\text{Cyclist}} = \frac{26 + 33 + 27 + 31 + 31 + 30 + 28 + 24}{8} = \frac{230}{8} = 28.75 \text{ minutes}
$$

### **2. Calculating the Standard Deviation**

The **standard deviation** measures the amount of variation or dispersion in a set of values. We'll use the **sample standard deviation** formula:

$$
s = \sqrt{\frac{\sum (x_i - \bar{x})^2}{n - 1}}
$$

#### **a. Bus Standard Deviation**

1. **Calculate Squared Deviations:**

$$
\begin{align*}
(33 - 35.75)^2 &= (-2.75)^2 = 7.5625 \
(28 - 35.75)^2 &= (-7.75)^2 = 60.0625 \
(40 - 35.75)^2 &= 4.25^2 = 18.0625 \
(32 - 35.75)^2 &= (-3.75)^2 = 14.0625 \
(41 - 35.75)^2 &= 5.25^2 = 27.5625 \
(32 - 35.75)^2 &= (-3.75)^2 = 14.0625 \
(38 - 35.75)^2 &= 2.25^2 = 5.0625 \
(42 - 35.75)^2 &= 6.25^2 = 39.0625 \
\end{align*}
$$

2. **Sum of Squared Deviations:**

$$
7.5625 + 60.0625 + 18.0625 + 14.0625 + 27.5625 + 14.0625 + 5.0625 + 39.0625 = 185.5
$$

3. **Calculate Standard Deviation:**

$$
s_{\text{Bus}} = \sqrt{\frac{185.5}{7}} \approx \sqrt{26.5} \approx 5.15 \text{ minutes}
$$

#### **b. Cyclist Standard Deviation**

1. **Calculate Squared Deviations:**

$$
\begin{align*}
(26 - 28.75)^2 &= (-2.75)^2 = 7.5625 \
(33 - 28.75)^2 &= 4.25^2 = 18.0625 \
(27 - 28.75)^2 &= (-1.75)^2 = 3.0625 \
(31 - 28.75)^2 &= 2.25^2 = 5.0625 \
(31 - 28.75)^2 &= 2.25^2 = 5.0625 \
(30 - 28.75)^2 &= 1.25^2 = 1.5625 \
(28 - 28.75)^2 &= (-0.75)^2 = 0.5625 \
(24 - 28.75)^2 &= (-4.75)^2 = 22.5625 \
\end{align*}
$$

2. **Sum of Squared Deviations:**

$$
7.5625 + 18.0625 + 3.0625 + 5.0625 + 5.0625 + 1.5625 + 0.5625 + 22.5625 = 63.5
$$

3. **Calculate Standard Deviation:**

$$
s_{\text{Cyclist}} = \sqrt{\frac{63.5}{7}} \approx \sqrt{9.0714} \approx 3.01 \text{ minutes}
$$

### **3. Comparing the Travel Times**

- **Average Travel Time:**
  - **Bus:** 35.75 minutes
  - **Cyclist:** 28.75 minutes

The cyclist has a **7-minute shorter** average commute compared to the bus.

- **Variability in Travel Time:**
  - **Bus Standard Deviation:** 5.15 minutes
  - **Cyclist Standard Deviation:** 3.01 minutes

The cyclist's travel times are **less variable**, indicating more consistency in commuting times compared to the bus.

### **Conclusion**

Based on the calculated mean and standard deviation:

- **Cyclist Commute:**
  - **Faster on Average:** By 7 minutes per trip.
  - **More Consistent:** Lower variability in travel times.

- **Bus Commute:**
  - **Slower on Average:** Longer commute time.
  - **Less Consistent:** Higher variability in travel times.

Therefore, **cycling** is both a **faster** and **more reliable** mode of commuting for the neighbor in this sample.
The cyclist has an average commute of 28.75 minutes, which is 7 minutes faster than the bus's average of 35.75 minutes. Additionally, the cyclist's commute times are more consistent, with a standard deviation of 3.01 minutes compared to the bus's 5.15 minutes.

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