Question

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

11.3	9.7	6.3	5.5	6.8	9.7	7.0
8.2	7.4	7.4	7.9	6.8	7.2	8.9
10.7	11.8	7.3	6.5	7.7	9.0	7.8
8.7	7.0	6.3	7.7	11.6	8.1

(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 = 220.3.] (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 = 1871.75.] (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

Part a

Point estimate for mean = Xbar = ∑Xi/n

We are given

∑Xi = 220.3, n = 27

Point estimate for mean = ∑Xi/n = 220.3/27 = 8.159259

Point estimate for mean = 8.159

Part b

Here, we have to calculate a point estimate of the strength value that separates the weakest 50% of all such beams from the strongest 50%.

This means we have to calculate the point estimate for median.

Point estimate for population median = sample median

From given data, we have

Sample median = 7.7

Required point estimate = 7.7

Part c

Point estimate for population standard deviation σ = Sample standard deviation

Sample standard deviation = sqrt[(∑x^2 – (∑x)^2/n)/(n – 1)]

∑Xi = 220.3, n = 27

∑Xi^2 = 1871.75

Sample standard deviation =sqrt[(1871.75 – (220.3)^2/27)/(27 – 1)]

Sample standard deviation = sqrt(2.856353)

Sample standard deviation = 1.690075

Point estimate for population standard deviation σ = 1.690

Part d

Here, we have to calculate a point estimate of the proportion of all such beams whose flexural strength exceeds 10 MPa.

Number of observations whose flexural strength exceeds 10 MPa = x = 4

Sample size = n = 27

Point estimate of proportion = x/n = 4/27 = 0.148148

Point estimate of proportion = 0.148

Part e

Point estimate for population coefficient of variation = sample coefficient of variation = SD/Xbar

Point estimate for population coefficient of variation = 1.690075/8.159259 = 0.207136

Point estimate for population coefficient of variation = 0.2071 or 20.71%