Question

Consider the accompanying data on flexural strength (MPa) for concrete beams of a certain type.

7.4	6.3	10.7	7.0	7.5	9.7	7.2
9.0	7.3	11.3	6.8	7.9	5.5	7.7
8.6	8.1	6.5	11.6	7.8	9.7	11.8
6.3	6.8	8.7	7.7	7.0	7.9

(a) Calculate a point estimate of the mean value of strength for the conceptual population of all beams manufactured in this fashion. [Hint: Σx_i = 219.8.] (Round your answer to three decimal places.)
MPa


(b) Calculate a point estimate of the strength value that separates the weakest 50% of all such beams from the strongest 50%.
MPa


(c) Calculate a point estimate of the population standard deviation σ. [Hint: Σx_i² = 1863.16.] (Round your answer to three decimal places.)
MPa

(d) Calculate a point estimate of the proportion of all such beams whose flexural strength exceeds 10 MPa. [Hint: Think of an observation as a "success" if it exceeds 10.] (Round your answer to three decimal places.)


(e) Calculate a point estimate of the population coefficient of variation σ/μ. (Round your answer to four decimal places.)

Answer 1

Solution:

We are given data on flexural strength (MPa) for concrete beams of a certain type.

7.4	6.3	10.7	7.0	7.5	9.7	7.2
9.0	7.3	11.3	6.8	7.9	5.5	7.7
8.6	8.1	6.5	11.6	7.8	9.7	11.8
6.3	6.8	8.7	7.7	7.0	7.9

Part a) Calculate a point estimate of the mean value of strength for the conceptual population of all beams manufactured in this fashion.

Given:  

and n = Number of observations = 27

Since sample mean is an unbiased estimator of population mean.

Thus a point estimate of the mean value of strength in of all beams manufactured in this fashion is given by:

MPa

Part b) Calculate a point estimate of the strength value that separates the weakest 50% of all such beams from the strongest 50%.

That is we have to find median.

Thus we need to arrange data in ascending order.

Sr No
1	5.5
2	6.3
3	6.3
4	6.5
5	6.8
6	6.8
7	7
8	7
9	7.2
10	7.3
11	7.4
12	7.5
13	7.7
14	7.7
15	7.8
16	7.9
17	7.9
18	8.1
19	8.6
20	8.7
21	9
22	9.7
23	9.7
24	10.7
25	11.3
26	11.6
27	11.8

when n is odd

Thus a point estimate of the strength value that separates the weakest 50% of all such beams from the strongest 50% is 7.7 MPa.

Part c) Calculate a point estimate of the population standard deviation σ.

Sample standard deviation is an unbiased estimator of population standard deviation and it is given by:

That is:

Part d) Calculate a point estimate of the proportion of all such beams whose flexural strength exceeds 10 MPa.

Success = beams whose flexural strength exceeds 10 MPa.

From given data we have number of observations > 10 MPa = 4

Thus

Part e)  Calculate a point estimate of the population coefficient of variation σ/μ.

Coefficient of Variation ( CV ) is given by: