Question

a survey was conducted of monthly electricity usage in kilowatt - hours for households in irvine, california and middleton, wisconsin. the results of the survey are summarized in the table at the right. use the data to sketch two box plots on the given number line.

1. describe the difference in electricity usage between the two cities. use shape and measures of center to justify your answer.

1. based on the shape of the box plot, predict the mean electricity usage for irvine, ca to be smaller, larger, or similar to the median of 510 kwh? explain your solution.

1. two additional households report their electricity usage in each city. determine if these data points would be considered outliers.

irvine: 330 kwh
middleton: 950 kwh

1. tray wants to compare the spread of the electricity usage for the two cities.

a. which measure of variability should he select to compare the spreads of the data sets? explain.

b. compare the spread of the data sets.

Explanation:

Step1: Analyze central - tendency and shape for question 3

For Irvine, median = 510, Q1 = 480, Q3 = 560. For Middleton, median = 700, Q1 = 630, Q3 = 780. Irvine has a lower median. If we assume normal - like distribution (no shape info given, but we can use central tendency), Middleton has higher usage on average.

Step2: Predict mean for question 4

In a symmetric distribution, mean = median. If right - skewed, mean>median; if left - skewed, mean<median. Since no shape info about Irvine is given, if symmetric, mean is similar to 510 kWh. If right - skewed, mean>510 kWh; if left - skewed, mean<510 kWh.

Step3: Determine outliers for question 5

For Irvine:

• IQR = Q3 - Q1=560 - 480 = 80.
• Lower fence = Q1−1.5×IQR=480−1.5×80=480 - 120 = 360.
• Upper fence = Q3 + 1.5×IQR=560+1.5×80=560 + 120 = 680. 330 kWh is an outlier as it is less than 360 kWh.

For Middleton:

• IQR = Q3 - Q1=780 - 630 = 150.
• Lower fence = Q1−1.5×IQR=630−1.5×150=630 - 225 = 405.
• Upper fence = Q3 + 1.5×IQR=780+1.5×150=780 + 225 = 1005. 950 kWh is not an outlier as it is less than 1005 kWh.

Step4: Choose measure of variability for question 6a

The inter - quartile range (IQR) should be selected. It is not affected by outliers. Since we may have outliers in the data (as seen in question 5), the IQR is a more robust measure of spread compared to the range.

Step5: Compare spread for question 6b

For Irvine, IQR = 560 - 480 = 80. For Middleton, IQR = 780 - 630 = 150. The spread of the electricity usage data in Middleton is larger than in Irvine as the IQR for Middleton (150) is greater than the IQR for Irvine (80).

Answer:

1. Middleton has a higher median electricity usage than Irvine, indicating on average, households in Middleton use more electricity.
2. Without shape information for Irvine, if symmetric, mean is similar to 510 kWh; if right - skewed, mean>510 kWh; if left - skewed, mean<510 kWh.
3. 330 kWh in Irvine is an outlier; 950 kWh in Middleton is not an outlier.
4. a. The inter - quartile range (IQR) should be selected because it is robust to outliers.

b. The spread of electricity usage in Middleton (IQR = 150) is larger than in Irvine (IQR = 80).