Questions: Six movies based on Marvel comic book characters for the U.S. box office as of fall 2017 are shown in the accompanying domestic gross rounded to the nearest million. Complete parts (a) through (c) below. Movie Domestic Gross (millions) 520 471 434 423 408 384 b. Using the sorted data, find Q1 and Q3. Then find the interquartile range and interpret it in context. Find Q1. Q1 = million dollars (Type an integer or a decimal. Do not round.) Find Q3. Q3 = million dollars (Type an integer or a decimal. Do not round.) Find the interquartile range (IQR). IQR= million dollars (Simplify your answer. Type an integer or a decimal. Do not round.) Interpret the interquartile range in context. Choose the correct answer below.

Solution Steps

The domestic gross of the six movies is sorted in ascending order: \[ \text{Sorted data} = [384, 408, 423, 434, 471, 520] \]

To find the first quartile \( Q_1 \), we use the formula for the rank: \[ \text{Rank} = Q \times (N + 1) = 0.25 \times (6 + 1) = 1.75 \] Since the rank is not an integer, we take the average of the values at ranks 1 and 2: \[ Q_1 = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{384 + 408}{2} = 396.0 \] Thus, \[ Q_1 = 396.0 \text{ million dollars} \]

To find the third quartile \( Q_3 \), we use the formula for the rank: \[ \text{Rank} = Q \times (N + 1) = 0.75 \times (6 + 1) = 5.25 \] Again, since the rank is not an integer, we take the average of the values at ranks 5 and 6: \[ Q_3 = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{471 + 520}{2} = 495.5 \] Thus, \[ Q_3 = 495.5 \text{ million dollars} \]

The interquartile range is calculated as: \[ \text{IQR} = Q_3 - Q_1 = 495.5 - 396.0 = 99.5 \] Thus, \[ \text{IQR} = 99.5 \text{ million dollars} \]

Final Answer