Questions: Loyd and Abby surveyed people they met at the theater about how many times they ate out in an average month. The results are given below. Average times dining out per month 5, 8, 12, 4, 11, 6 7, 2, 3, 5, 4, 9 9, 13, 12, 1, 1, 3 8, 8, 13, 4, 5, 6 2, 10, 9, 21, 3, 3 5, 1, 14, 7, 12, 0 Are there any outliers in this data? Explain your answer and give any applicable outliers.

Solution Steps

The data collected from the survey is as follows:

\[ \text{Data} = [5, 8, 12, 4, 11, 6, 7, 2, 3, 5, 4, 9, 9, 13, 12, 1, 1, 3, 8, 8, 13, 4, 5, 6, 2, 10, 9, 21, 3, 3, 5, 1, 14, 7, 12, 0] \]

After sorting, the data becomes:

\[ \text{Sorted Data} = [0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 4, 4, 4, 5, 5, 5, 5, 6, 6, 7, 7, 8, 8, 8, 9, 9, 9, 10, 11, 12, 12, 12, 13, 13, 14, 21] \]

To find \( Q_1 \), we use the formula:

\[ \text{Rank} = Q \times (N + 1) = 0.25 \times (36 + 1) = 9.25 \]

Since the rank is not an integer, we average the values at positions 9 and 10:

\[ Q_1 = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{3 + 3}{2} = 3.0 \]

Thus,

\[ Q_1 = 3.0 \]

To find \( Q_3 \), we use the formula:

\[ \text{Rank} = Q \times (N + 1) = 0.75 \times (36 + 1) = 27.75 \]

Again, since the rank is not an integer, we average the values at positions 27 and 28:

\[ Q_3 = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{9 + 10}{2} = 9.5 \]

Thus,

\[ Q_3 = 9.5 \]

The IQR is calculated as:

\[ \text{IQR} = Q_3 - Q_1 = 9.5 - 3.0 = 6.5 \]

The lower and upper bounds for identifying outliers are calculated as follows:

\[ \text{Lower Bound} = Q_1 - 1.5 \times \text{IQR} = 3.0 - 1.5 \times 6.5 = -6.75 \]

\[ \text{Upper Bound} = Q_3 + 1.5 \times \text{IQR} = 9.5 + 1.5 \times 6.5 = 19.25 \]

Any data point below the lower bound or above the upper bound is considered an outlier. In this case, the outlier is:

\[ \text{Outliers} = [21] \]

Final Answer