Question

archie is fed up with waiting in line at his local post office and writes his congresswoman with his complaint. she stubbornly responds that, unless he can demonstrate that the mean waiting time at his branch is greater than 13 minutes (which is the threshold set for her district), there is nothing she can do to help him. over the course of the next few months, archie records the waiting times for each of a random selection of 22 post - office visits made by him and other customers. these waiting times (in minutes) are summarized in the following histogram. based on the histogram, using the midpoint of each data class, estimate the mean waiting time for archies sample. carry your intermediate computations to at least four decimal places, and round your answer to one decimal place.

Explanation:

Step1: Calculate mid - points of each class

For the class 0 - 6, mid - point $x_1=\frac{0 + 6}{2}=3$; for 7 - 13, $x_2=\frac{7+13}{2}=10$; for 14 - 20, $x_3=\frac{14 + 20}{2}=17$; for 21 - 27, $x_4=\frac{21+27}{2}=24$; for 28 - 34, $x_5=\frac{28 + 34}{2}=31$.

Step2: Calculate the product of mid - point and frequency for each class

The frequencies are $f_1 = 2$, $f_2=9$, $f_3 = 5$, $f_4=5$, $f_5 = 1$.
The products are $f_1x_1=2\times3 = 6$, $f_2x_2=9\times10 = 90$, $f_3x_3=5\times17 = 85$, $f_4x_4=5\times24 = 120$, $f_5x_5=1\times31 = 31$.

Step3: Calculate the sum of frequencies and the sum of products

The sum of frequencies $\sum f_i=2 + 9+5 + 5+1=22$.
The sum of products $\sum f_ix_i=6 + 90+85+120+31=332$.

Step4: Calculate the mean

The mean $\bar{x}=\frac{\sum f_ix_i}{\sum f_i}=\frac{332}{22}\approx15.1$.

Answer:

15.1