Question

the weights, in pounds, of packages on a delivery truck are shown in the stem-and-leaf plot. find the mean, the median, and the mode of the data, if possible. if any measure cannot be found or does not represent the center of the data, explain why.
0|3 8
1|0 4 5 8
2|0 0 0 7 8 8 9
3|0 0 0 2 4 5 7 8 9
4|2 3 7 8 9
5|0
key: 3|0 = 30

find the mean. select the correct choice below and, if necessary, fill in the answer box to complete your choice.
a. the mean is
(round to one decimal place as needed.)
b. the mean cannot be calculated because there is an even number of data entries.
c. the mean cannot be calculated because the sample size is too small.
d. the mean cannot be calculated because the data are at the nominal level of measurement.

Explanation:

Step1: Extract data from stem - leaf plot

From the stem - leaf plot:

• Stem 0: Leaves 3, 8. So data points: \(0\times10 + 3=3\), \(0\times10+8 = 8\)
• Stem 1: Leaves 0, 4, 5, 8. So data points: \(1\times10+0 = 10\), \(1\times10 + 4=14\), \(1\times10+5 = 15\), \(1\times10+8 = 18\)
• Stem 2: Leaves 0, 0, 0, 7, 8, 8, 9. So data points: \(2\times10+0 = 20\), \(2\times10+0 = 20\), \(2\times10+0 = 20\), \(2\times10+7 = 27\), \(2\times10+8 = 28\), \(2\times10+8 = 28\), \(2\times10+9 = 29\)
• Stem 3: Leaves 0, 0, 0, 2, 4, 5, 7, 8, 9. So data points: \(3\times10+0 = 30\), \(3\times10+0 = 30\), \(3\times10+0 = 30\), \(3\times10+2 = 32\), \(3\times10+4 = 34\), \(3\times10+5 = 35\), \(3\times10+7 = 37\), \(3\times10+8 = 38\), \(3\times10+9 = 39\)
• Stem 4: Leaves 2, 3, 7, 8, 9. So data points: \(4\times10+2 = 42\), \(4\times10+3 = 43\), \(4\times10+7 = 47\), \(4\times10+8 = 48\), \(4\times10+9 = 49\)
• Stem 5: Leaf 0. So data point: \(5\times10+0 = 50\)

Now, count the number of data points:

• Stem 0: 2 points
• Stem 1: 4 points
• Stem 2: 7 points
• Stem 3: 9 points
• Stem 4: 5 points
• Stem 5: 1 point

Total number of data points \(n=2 + 4+7 + 9+5 + 1=28\)

Step2: Calculate the sum of all data points

• Sum of stem 0 data: \(3 + 8=11\)
• Sum of stem 1 data: \(10+14 + 15+18=57\)
• Sum of stem 2 data: \(20+20 + 20+27+28+28+29 = 20\times3+27 + 28\times2+29=60 + 27+56 + 29=172\)
• Sum of stem 3 data: \(30\times3+32 + 34+35+37+38+39=90+32 + 34+35+37+38+39=90+(32 + 34)+(35 + 37)+(38 + 39)=90 + 66+72+77=305\)
• Sum of stem 4 data: \(42+43 + 47+48+49=(42 + 43)+(47 + 48)+49=85+95 + 49=229\)
• Sum of stem 5 data: \(50\)

Total sum \(S=11 + 57+172+305+229+50\)
\(S=(11 + 57)+172+305+229+50=68+172+305+229+50=(68 + 172)+305+229+50=240+305+229+50=(240 + 305)+229+50=545+229+50=(545 + 229)+50=774+50 = 824\)

Step3: Calculate the mean

The formula for the mean \(\bar{x}=\frac{S}{n}\), where \(S\) is the sum of data and \(n\) is the number of data points.

We have \(S = 824\) and \(n = 28\)

\(\bar{x}=\frac{824}{28}\approx29.4\) (rounded to one decimal place)

Answer:

A. The mean is \(29.4\)