Question

Consider the strength data for beams given in Example

1.2.

a. Construct a stem-and-leaf display of the data. What appears to be a representative strength value? Do the observations appear to be highly

concentrated about the representative value or rather spread out?

b. Does the display appear to be reasonably symmetric about a representative value, or would you describe its shape in some other way?

c. Do there appear to be any outlying strength values?

d. What proportion of strength observations in this sample exceeds 10 MPa?

Short answer

a. The stem and leaf display for the provided scenario is,

Unit: 6|3=6.3 MPa.

b. The distribution is positively skewed.

c. There is no outlying strength value.

d. The proportion of strength observations in this sample that exceeds 10 MPa is 0.148.

Step-by-step solution

Given information

The strength data of the beams is provided as,

5.9	7.2	7.3	6.3	8.1	6.8	7.0	7.6	6.8	6.5	7.0	6.3	7.9	9.0
8.2	8.7	7.8	9.7	7.4	7.7	9.7	7.8	7.7	11.6	11.3	11.8	10.7

Construct a stem and leaf diagram and comment on the spread.

a.

A stem-and-leaf display provides a visual representation of the dataset.

The steps to construct a stem-and-leaf display are as follows,

1) Select the leading digit for the stem and trailing digits for the leaves.

2) Represent the stem digits vertically and similarly the trailing digits corresponding to the stem digits.

3) Mention the units for the display.

The stem and leaf display for the provided scenario is,

Unit: 6|3=6.3 MPa.

From the above display, it can be observed that the representative strength value; that is the middle value is 7.7.

The data does not appear to be highly concentrated about the representative value or rather spread out.

Describe the shape

b.

From the stem-and-leaf display, it can be interpreted that the observations are concentrated towards the right of the graph.

Therefore, thedistribution is positively skewed.

State the outliers

c.

From the above-represented display, it can be observed that there is no outlying strength value.

Compute the proportion of strength values that exceed 10 MPa

The number of strength values that exceed 10 MPa is 4.

The total number of observations is27.

The proportion of strength observations in this sample that exceeds 10 MPa is computed as,

\(\frac{4}{{27}} = 0.148\)

Thus, the proportion of strength observations in this sample that exceeds 10 MPa is 0.148.