Question

Consider the accompanyi g data en fuxural strength (MPaforcencrete bams fcertain ty 07 8.1867.0 7 78 6.3 7 7.4 11.6 9.7 63 S3 7.4 72 6.3 78 7,7 70 7 6.8 113 7.3 9.0 18 8.7 Stat& "hich ?'timator ,ou used nterpret thia point eztimate

Answer 1

we are given the following data:

10.7	8.1	8.6	7.0	5.7	7.8	6.3
7.7	7.4	11.6	9.7	6.3	6.8	7.4
7.2	6.5	7.8	7.7	7.0	7.9	6.8
9.7	11.3	7.3	9.0	11.8	8.7

ANSWER TO :

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

total number of observations, n = 27

We use the sample mean, $\bar{x}$ to estimate the population mean $\mu$

the point estimate of the mean value of strength for the conceptual population of all beams manufactured in this fashion will be given by:

$\bar{x}$ = sum of all the observations / total number of observations

$\bar{x}$ = $\Sigma$x_i / n

given, $\Sigma$x_i = 219.8

therefore, $\bar{x}$ = 219.8 / 27

$\bar{x}$ = 8.141

The estimator we used is the sample mean, i.e., $\bar{x}$

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

we use the sample median.

Recall that, a median is the middle value in a sequence which is sorted in ascending order, i.e., from lowest to highest.

hence, the median will be the value which separates the weakest 50% of beams from the strongest 50%.

to calculate the median first we arrange the data in ascending order

1 - 5.7

2 - 6.3

3 - 6.3

4 - 6.5

5 - 6.8

6 - 6.8

7 - 7.0

8 - 7.0

9 - 7.2

10 - 7.3

11 - 7.4

12 - 7.4

13 - 7.7

14 - 7.7

15 - 7.8

16 - 7.8

17 - 7.9

18 - 8.1

19 - 8.6

20 - 8.7

21 - 9.0

22 - 9.7

23 - 9.7

24 - 10.7

25 - 11.3

26 - 11.6

27 -11.8

here the total no of observations = 27 i.e., odd

the median for odd number of observation is

median, $\tilde{x}$ = [(n+1)/2]^{th term} from the sorted sequence of data

hence,

$\tilde{x}$ = [(27+1)/2]^{th term} from the sorted sequence of data

$\tilde{x}$ = 28/2^{th term} from the sorted sequence of data

$\tilde{x}$ = 14^{th term}   from the sorted sequence of data

1 - 5.7

2 - 6.3

3 - 6.3

4 - 6.5

5 - 6.8

6 - 6.8

7 - 7.0

8 - 7.0

9 - 7.2

10 - 7.3

11 - 7.4

12 - 7.4

13 - 7.7

14 - 7.7 - median

15 - 7.8

16 - 7.8

17 - 7.9

18 - 8.1

19 - 8.6

20 - 8.7

21 - 9.0

22 - 9.7

23 - 9.7

24 - 10.7

25 - 11.3

26 - 11.6

27 -11.8

Hence, the median is 7.7 and the estimator we used is the median of the data i.e., $\tilde{x}$

PART (c) Calculate and interpret a point estimate of the population standard deviation

teh point estimate of the population standard deviation is the sample standard deviation

the sample standard deviation, s is given by:

s = $\sqrt{}$[ ($\Sigma$x_i² -($\Sigma$x_i)²/n)/ (n-1)]

given that,

$\Sigma$x_i² = 1862.34

$\Sigma$x_i = 219.8

n = 27

therefore,

s = $\sqrt{}$[(1862.34 - (219.8)²/27)/(27-1)]

s = $\sqrt{}$(1862.34 - 1789.34)/26]

s = $\sqrt{}$73/26

s = $\sqrt{}$2.80769 = 1.6756 = 1.676 (rounded off)

hence, the estimate for population standard deviation is 1.676 and the estimator we used is sample

standard deviation, s

we know the standard deviation describes the spread or dispersion of the data.

therefore, interpreting the point estimate, s

The estimate describes the spread of the data.

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

let the "success" be = all the observations greater than 10 MPa

therefore, the successes are:

10.7

11.3

11.6

11.8

total number of successes = 4

therefore the proportion of all such beams = total number of successes / total number of observations

we know,

total number of successes = 4

total number of observations = 27

therefore, required proportion, $\hat{p}$ = 4/27

$\hat{p}$ = 0.1481 = 0.148 (rounded off)

point estimate of the proportion of all such beams whose flexural strength exceeds 10 MPa is 0.148

PART (e) Calculate a point estimate of the population coefficient of variation $\sigma$/$\mu$

the point estimate of population coefficient of variation will be

sample standard deviation / sample mean = s / $\bar{x}$

s / $\bar{x}$ = 1.676 / 8.141

s / $\bar{x}$ = 0.20587

s / $\bar{x}$ = 0.2059 (rounded off)