Question

Online restaurant ratings are shown for the last nine visitors to Joe's Pizza Kitchen.

Visitor # 1 2 3 4 5 6 7 8 9
Number of Stars 3 3 3 1 5 3 3 2 2
Use the 1.5 IQR rule to determine whether each number of stars is an outlier.

Outlier Not an outlier
1 star
2 stars
3 stars
5 stars

Answer 1

To determine whether any of the ratings are outliers using the 1.5 IQR (Interquartile Range) rule, follow these steps:

1. **Sort the Ratings**:
The ratings in order are:
```
1, 2, 2, 3, 3, 3, 3, 3, 5
```

2. **Find the Quartiles**:
- The median (Q2) is the 5th value in the sorted list: **3**.
- The lower quartile (Q1) is the median of the first half (1, 2, 2, 3) which is the average of the 2nd and 3rd values:
\( Q1 = \frac{2 + 2}{2} = 2 \).
- The upper quartile (Q3) is the median of the second half (3, 3, 3, 5) which is the average of the 7th and 8th values:
\( Q3 = \frac{3 + 3}{2} = 3 \).

3. **Calculate the IQR**:
\[
IQR = Q3 - Q1 = 3 - 2 = 1
\]

4. **Determine the Outlier Boundaries**:
- Lower Bound: \( Q1 - 1.5 \times IQR = 2 - 1.5 \times 1 = 2 - 1.5 = 0.5 \)
- Upper Bound: \( Q3 + 1.5 \times IQR = 3 + 1.5 \times 1 = 3 + 1.5 = 4.5 \)

5. **Identify Outliers**:
Any rating less than 0.5 or greater than 4.5 is considered an outlier.

Now, evaluate each star rating:

- **1 star**: Less than 0.5 (Outlier)
- **2 stars**: Between 0.5 and 4.5 (Not an outlier)
- **3 stars**: Between 0.5 and 4.5 (Not an outlier)
- **5 stars**: Greater than 4.5 (Outlier)

### Summary of Outliers:
- **1 star**: Outlier
- **2 stars**: Not an outlier
- **3 stars**: Not an outlier
- **5 stars**: Outlier

So, the final results are:

- Outlier: 1 star, 5 stars
- Not an outlier: 2 stars, 3 stars