Questions: In the past month, Ryan has had five pizzas delivered. He kept a record of how long it took to get the pizzas: 20 minutes, 28 minutes, 35 minutes, 40 minutes, and 25 minutes. If the sixth time he orders pizza, it takes 95 minutes, how will this affect his data? a.) It will have little to no effect on both the mean and the median. b.) It will have little to no effect on the median, but it will make the mean a less useful measure of center. c.) It will make both the mean and the median less useful measures of center. d.) It will have little to no effect on the mean, but it will make the median a less useful measure of center.

In the past month, Ryan has had five pizzas delivered. He kept a record of how long it took to get the pizzas: 20 minutes, 28 minutes, 35 minutes, 40 minutes, and 25 minutes.

If the sixth time he orders pizza, it takes 95 minutes, how will this affect his data?
a.) It will have little to no effect on both the mean and the median.

b.) It will have little to no effect on the median, but it will make the mean a less useful measure of center.
c.) It will make both the mean and the median less useful measures of center.
d.) It will have little to no effect on the mean, but it will make the median a less useful measure of
center.

Transcript text: In the past month, Ryan has had five pizzas delivered. He kept a record of how long it took to get the pizzas: 20 minutes, 28 minutes, 35 minutes, 40 minutes, and 25 minutes. If the sixth time he orders pizza, it takes 95 minutes, how will this affect his data? a.) It will have little to no effect on both the mean and the median. b.) It will have little to no effect on the median, but it will make the mean a less useful measure of center. c.) It will make both the mean and the median less useful measures of center. d.) It will have little to no effect on the mean, but it will make the median a less useful measure of center.

Solution

Solution Steps

Step 1: Calculate Initial Mean

The initial delivery times for the pizzas are \(20\), \(28\), \(35\), \(40\), and \(25\) minutes. The mean \( \mu \) is calculated as follows:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{20 + 28 + 35 + 40 + 25}{5} = \frac{148}{5} = 29.6 \]

Thus, the initial mean is \(29.6\).

Step 2: Calculate Initial Median

To find the median, we first sort the delivery times: \([20, 25, 28, 35, 40]\). The formula for the rank of the median \(Q\) is given by:

\[ \text{Rank} = Q \times (N + 1) = 0.5 \times (5 + 1) = 3.0 \]

The median corresponds to the value at position \(3\), which is \(28\). Therefore, the initial median is \(28\).

Step 3: Calculate Updated Mean

When Ryan orders a sixth pizza that takes \(95\) minutes, the updated delivery times become \(20\), \(28\), \(35\), \(40\), \(25\), and \(95\). The updated mean \( \mu \) is calculated as follows:

\[ \mu = \frac{\sum_{i=1}^N x_i}{N} = \frac{20 + 28 + 35 + 40 + 25 + 95}{6} = \frac{243}{6} = 40.5 \]

Thus, the updated mean is \(40.5\).

Step 4: Calculate Updated Median

The sorted updated delivery times are \([20, 25, 28, 35, 40, 95]\). The formula for the rank of the median \(Q\) is:

\[ \text{Rank} = Q \times (N + 1) = 0.5 \times (6 + 1) = 3.5 \]

Since \(3.5\) is not an integer, we average the values at positions \(3\) and \(4\):

\[ Q = \frac{X_{\text{lower}} + X_{\text{upper}}}{2} = \frac{28 + 35}{2} = 31.5 \]

Thus, the updated median is \(31.5\).

Step 5: Analyze the Effect on Mean and Median

Initial Mean: \(29.6\), Updated Mean: \(40.5\)
Initial Median: \(28\), Updated Median: \(31.5\)

Both the mean and the median have changed significantly. The mean increased from \(29.6\) to \(40.5\), while the median increased from \(28\) to \(31.5\). This indicates that the new data point has affected both measures of center.