Question

the following data represent the number of grams of fat in breakfast meals offered at a local fast food res and (b) describe the shape of the distribution.
17 13 12 8 23 33
3 17 3 11 2 29
43 37 25 16 36 16
25 6 17 21
(a) construct the stem-and-leaf plot below.
0 |
1 |
2 |
3 |
4 |
legend: 5 | 1 represents 51 grams of fat.

Explanation:

Step1: List all data points

The data points are: 17, 13, 12, 8, 23, 33, 3, 17, 3, 11, 2, 29, 43, 37, 25, 16, 36, 16, 25, 6, 17, 21.

Step2: Separate stems and leaves

• Stem 0: Leaves are the units digits of numbers with tens digit 0. Numbers: 2, 3, 3, 6, 8. So leaves: 2, 3, 3, 6, 8.
• Stem 1: Leaves are units digits of numbers with tens digit 1. Numbers: 11, 12, 13, 16, 16, 17, 17, 17. So leaves: 1, 2, 3, 6, 6, 7, 7, 7.
• Stem 2: Leaves are units digits of numbers with tens digit 2. Numbers: 21, 23, 25, 25, 29. So leaves: 1, 3, 5, 5, 9.
• Stem 3: Leaves are units digits of numbers with tens digit 3. Numbers: 33, 36, 37. So leaves: 3, 6, 7.
• Stem 4: Leaves are units digits of numbers with tens digit 4. Numbers: 43. So leaves: 3.

Step3: Construct stem - leaf plot

Stem	Leaf
1	1, 2, 3, 6, 6, 7, 7, 7
2	1, 3, 5, 5, 9
3	3, 6, 7
4	3

Step4: Describe the shape (for part b)

To describe the shape, we look at the distribution of leaves. The left - most stem (stem 0) has relatively few leaves, and as we move to stem 1, the number of leaves increases. Then, as we move to higher stems (2, 3, 4), the number of leaves decreases. Also, the mean and median: let's calculate the median. There are \(n = 22\) data points. The median is the average of the 11th and 12th values when sorted. Sorted data: 2, 3, 3, 6, 8, 11, 12, 13, 16, 16, 17, 17, 17, 21, 23, 25, 25, 29, 33, 36, 37, 43. The 11th value is 17 and the 12th value is 17, so median is 17. The mean: sum of data \(\sum x=2 + 3+3 + 6+8+11+12+13+16+16+17+17+17+21+23+25+25+29+33+36+37+43\)
\(\sum x=(2)+(3 + 3)+(6 + 8)+(11)+(12)+(13)+(16+16)+(17\times3)+(21)+(23)+(25\times2)+(29)+(33)+(36)+(37)+(43)\)
\(=2 + 6+14+11+12+13+32+51+21+23+50+29+33+36+37+43\)
\(=2+6 = 8; 8 + 14=22; 22+11 = 33; 33+12 = 45; 45+13 = 58; 58+32 = 90; 90+51 = 141; 141+21 = 162; 162+23 = 185; 185+50 = 235; 235+29 = 264; 264+33 = 297; 297+36 = 333; 333+37 = 370; 370+43 = 413\)
Mean \(\bar{x}=\frac{413}{22}\approx18.77\). Since the mean (\(\approx18.77\)) is slightly greater than the median (17), and the tail on the right (higher values) is longer (more spread out for higher stems), the distribution is slightly right - skewed (positively skewed). Also, from the stem - leaf plot, we can see that the data is more concentrated in the lower stems (0 - 2) and has a longer tail towards the higher stems (3 - 4).

Answer:

(a) The stem - leaf plot is as follows:

Stem	Leaf
1	1, 2, 3, 6, 6, 7, 7, 7
2	1, 3, 5, 5, 9
3	3, 6, 7
4	3

(b) The distribution is slightly right - skewed (positively skewed). The data is more concentrated in the lower stems (0 - 2) and has a longer tail towards the higher stems (3 - 4), and the mean is slightly greater than the median.