Question

archie is fed up with waiting in line at his local post office, and plans to contact his congresswoman. archie knows that his complaint will not be well received without data to back it up. so to improve his case, over the next few months archie records the waiting time for a selection of 26 post office visits made by him and other customers. these waiting times (in minutes) are summarized in the following frequency distribution.

waiting time (minutes)	0 to 6	7 to 13	14 to 20	21 to 27	28 to 34

based on the frequency distribution, using the midpoint of each data class, estimate the mean waiting time for the data archie collected. for your intermediate computations, use four or more decimal places, and round your answer to one decimal place.

Explanation:

Step1: Find midpoints of each class

For class \(0\) to \(6\): midpoint \(x_1=\frac{0 + 6}{2}=3\)
For class \(7\) to \(13\): midpoint \(x_2=\frac{7+13}{2}=10\)
For class \(14\) to \(20\): midpoint \(x_3=\frac{14 + 20}{2}=17\)
For class \(21\) to \(27\): midpoint \(x_4=\frac{21+27}{2}=24\)
For class \(28\) to \(34\): midpoint \(x_5=\frac{28 + 34}{2}=31\)

Step2: Calculate \(f_i\times x_i\) for each class

• For \(x_1 = 3\), \(f_1 = 4\): \(f_1x_1=4\times3 = 12\)
• For \(x_2 = 10\), \(f_2 = 9\): \(f_2x_2=9\times10 = 90\)
• For \(x_3 = 17\), \(f_3 = 7\): \(f_3x_3=7\times17 = 119\)
• For \(x_4 = 24\), \(f_4 = 4\): \(f_4x_4=4\times24 = 96\)
• For \(x_5 = 31\), \(f_5 = 2\): \(f_5x_5=2\times31 = 62\)

Step3: Sum of \(f_i\times x_i\) and sum of \(f_i\)

Sum of \(f_i\times x_i\): \(12+90 + 119+96+62=379\)
Sum of \(f_i\): \(4 + 9+7+4+2=26\)

Step4: Calculate the mean

Mean \(\bar{x}=\frac{\sum f_ix_i}{\sum f_i}=\frac{379}{26}\approx14.6\) (rounded to one decimal place)

Answer:

\(14.6\)