Questions: Create a Dataset Give a positive integer less than 100 as the last data value in each of the following datasets so that the resulting dataset satisfies the given condition. Part 1 (a) The mean of the numbers is substantially less than the median. 51,52,53,54,

Solution Steps

We start with the dataset \( D = \{51, 52, 53, 54, x\} \), where \( x \) is a positive integer less than 100 that we need to determine.

The mean \( \mu \) of the dataset is calculated as follows:

\[ \mu = \frac{51 + 52 + 53 + 54 + x}{5} = \frac{210 + x}{5} \]

To find the median, we first need to sort the dataset. The median \( m \) is the middle value when the dataset is ordered. For the dataset \( D \), if \( x = 1 \), the ordered dataset becomes \( \{1, 51, 52, 53, 54\} \). The median is the third value:

\[ m = 52 \]

We found that when \( x = 1 \):

• The mean is calculated as:

\[ \mu = \frac{210 + 1}{5} = \frac{211}{5} = 42.2 \]

• The median is:

\[ m = 52 \]

To satisfy the condition that the mean is substantially less than the median, we check:

\[ \mu < m - 1 \]

Substituting the values:

\[ 42.2 < 52 - 1 \implies 42.2 < 51 \]

This inequality holds true.

Final Answer