Questions: Bids were placed in a silent auction for a sword reputed to have been used at the Battle of Hastings, worth a reported 20,000. The respective bids (in thousands of dollars) placed by the 7 bidders were as follows. 25,23,16,24,13,15,12 (a) What is the median of this data set? If your answer is not an integer, round your answer to one decimal place. (b) What is the mean of this data set? If your answer is not an integer, round your answer to one decimal place. (c) How many modes does the data set have, and what are their values? Indicate the number of modes by clicking in the appropriate circle, and then indicate the value(s) of the mode(s), if applicable.

Bids were placed in a silent auction for a sword reputed to have been used at the Battle of Hastings, worth a reported 20,000. The respective bids (in thousands of dollars) placed by the 7 bidders were as follows.
25,23,16,24,13,15,12
(a) What is the median of this data set? If your answer is not an integer, round your answer to one decimal place.
(b) What is the mean of this data set? If your answer is not an integer, round your answer to one decimal place.
(c) How many modes does the data set have, and what are their values? Indicate the number of modes by clicking in the appropriate circle, and then indicate the value(s) of the mode(s), if applicable.

Transcript text: Bids were placed in a silent auction for a sword reputed to have been used at the Battle of Hastings, worth a reported $20,000. The respective bids (in thousands of dollars) placed by the 7 bidders were as follows. \[ 25,23,16,24,13,15,12 \] (a) What is the median of this data set? If your answer is not an integer, round your answer to one decimal place. (b) What is the mean of this data set? If your answer is not an integer, round your answer to one decimal place. (c) How many modes does the data set have, and what are their values? Indicate the number of modes by clicking in the appropriate circle, and then indicate the value(s) of the mode(s), if applicable.

Solution

Solution Steps

Step 1: Calculate the Median

To find the median of the bids, we first sort the data set: \[ \text{Sorted data} = [12, 13, 15, 16, 23, 24, 25] \] The formula for the rank of the median $ Q $ is given by: \[ \text{Rank} = Q \times (N + 1) = 0.5 \times (7 + 1) = 4.0 \] The quantile is at position 4, which corresponds to the value: \[ \text{Median} = 16 \]

Step 2: Calculate the Mean

The mean $ \mu $ of the data set is calculated using the formula: \[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{128}{7} \approx 18.3 \]

Step 3: Determine the Modes

To find the modes, we count the frequency of each bid. The maximum frequency is 7, indicating that all values are modes: \[ \text{Modes} = [25, 23, 16, 24, 13, 15, 12] \] Thus, the number of modes is: \[ \text{Number of modes} = 7 \]