Question

Consider the accompanying data on flexural strength (MPa) for concrete beams of a certain type.
7.3 6.8 8.6 9.7 11.6 9.7 6.8 7.7 11.8 7.4 8.1 8.7 6.3 7.9 7.0 7.9 7.8 6.5 6.3 7.0 7.5 7.7 7.2 11.3 9.0 10.7 5.1
(a) Calculate a point estimate of the mean value of strength for the conceptual population of all beams manufactured in this fashion. [Hint: Σxi = 219.4.] (Round your answer to three decimal places.) State which estimator you used.

a) s b) x c) p? d) x tilde e) s / x

(b) Calculate a point estimate of the strength value that separates the weakest 50% of all such beams from the strongest 50%. State which estimator you used.

a) s b) x c) p? d) x tilde e) s / x

(c) Calculate a point estimate of the population standard deviation σ. [Hint: Σxi2 = 1858.92.] (Round your answer to three decimal places.)
Interpret this point estimate.

a. This estimate describes the center of the data.
b. This estimate describes the bias of the data.
c. This estimate describes the linearity of the data.
d. This estimate describes the spread of the data.

Answer 1

Answer:

Step-by-step explanation:

Hello!

The variable of interest is X: flexural strength of concrete beams.

Data:

7.3 6.8 8.6 9.7 11.6 9.7 6.8 7.7 11.8 7.4 8.1 8.7 6.3 7.9 7.0 7.9 7.8 6.5 6.3 7.0 7.5 7.7 7.2 11.3 9.0 10.7 5.1

a.

The point estimate of the mean value is the sample mean. You calculate it using the following formula:

X[bar]= ∑X/n= 219.40/27= 8.1259≅ 8.13

b)x

b.

The point estimate that divides the sample in 50%-50% is the measure of central tendency called Median(Me).

To reach the median you have to calculate it's position (PosMe)

For uneven samples PosMe= (n+1)/2= (27+1)/2= 14 ⇒ This means that the Median is the 14th observation of the data set.

So next is to order the data from lower to heighest and identify the value in the 14th position:

5,1 ; 6,3; 6,3; 6,5 ; 6,8; 6,8; 7; 7; 7,2; 7,3; 7,4 ; 7,5 ; 7,7 ; 7,7 ; 7,8 ; 7,9 ; 7,9 ; 8,1 ; 8,6 ; 8,7 ; 9 ; 9,7 ; 9,7 ; 10,7 ; 11,3 ; 11,6 ; 11,8

Me= 7.7

d) x tilde

c.

The point estimate of the population standard deviation is the sample standard deviation. The standard deviation is the square root of the variance, so the first step is to calculate the sample variance:

$S^2= \frac{1}{n-1}*[sumX^2-\frac{(sumX)^2}{n} ]$

∑X²= 1858.92

$S^2= \frac{1}{26}*[1858.92-\frac{219.40^2}{27} ] = 2.926$

S= √2.926= 1.71

The standard deviation is a measurement of variability, it shows how to disperse the data is concerning the sample mean.

d. This estimate describes the spread of the data.

I hope it helps!