TRY IT TOURSELF prices \( (\$) \) of 8 jerseys are listed below. The modal price is \( \$ 35 \) and the median price is \( \$ 31 \). It is given that \( x

To solve the problem, we need to analyze the given information about the prices of the jerseys. We know: - The modal price is \( \$ 35 \), which means that \( 35 \) is the most frequently occurring price in the dataset. - The median price is \( \$ 31 \), which means that when the prices are arranged in ascending order, the average of the 4th and 5th prices is \( 31 \). Let's denote the prices of the jerseys as follows: \[ a_1, a_2, a_3, a_4, a_5, a_6, a_7, a_8 \] where \( a_1 \leq a_2 \leq a_3 \leq a_4 \leq a_5 \leq a_6 \leq a_7 \leq a_8 \). ### Step 1: Finding \( x \) and \( y \) Since the median is \( 31 \), we have: \[ \frac{a_4 + a_5}{2} = 31 \implies a_4 + a_5 = 62 \] Given that the mode is \( 35 \), it means that \( 35 \) appears more frequently than any other price. To satisfy this condition, we can assume that at least three of the prices are \( 35 \). Let's assume the following arrangement for the prices: - \( a_1, a_2, a_3 \) are less than or equal to \( 31 \) - \( a_4 = 31 \) - \( a_5 = 31 \) - \( a_6, a_7, a_8 \) are \( 35 \) or higher To satisfy the condition \( a_4 + a_5 = 62 \), we can set: - \( a_4 = 31 \) - \( a_5 = 31 \) Now, we need to determine \( x \) and \( y \) such that \( x < y \) and they fit into the arrangement. Since \( a_1, a_2, a_3 \) must be less than or equal to \( 31 \), we can set: - \( a_1 = x \) - \( a_2 = y \) - \( a_3 = 31 \) (or another value less than \( 31 \)) To ensure that \( 35 \) is the mode, we can set: - \( a_6 = 35 \) - \( a_7 = 35 \) - \( a_8 = 35 \) Now we have: - \( a_1 = x \) - \( a_2 = y \) - \( a_3 = 31 \) - \( a_4 = 31 \) - \( a_5 = 31 \) - \( a_6 = 35 \) - \( a_7 = 35 \) - \( a_8 = 35 \) Since \( x < y \) and both must be less than or equal to \( 31 \), we can choose: - \( x = 30 \) - \( y = 31 \) However, since \( y \) must be strictly less than \( 31 \) to maintain \( x < y \), we can set: - \( x = 30 \) - \( y = 31 \) Thus, we can conclude: - \( x = 30 \) - \( y = 31 \) ### Step 2: Finding the Mean Now we can calculate the mean of the dataset. The prices are: - \( 30, 30, 31, 31, 31, 35, 35, 35 \) Calculating the sum: \[ 30 + 30 + 31 + 31 + 31 + 35 + 35 + 35 = 30 + 30 + 93 + 105 = 393 \] Now, we divide by the number of jerseys (8): \[ \text{Mean} = \frac{393}{8} = 49.125 \] ### Final Answers (a) The values of \( x \) and \( y \) are: - \( x = 30 \) - \( y = 31 \) (b) The mean of the set of data is: - \( 49.125 \)