Question

(a) Test at the 0.05 level of significance whether the sample of male BMI observations is enough to show that the mean BMI for males exceeds 25.5. Show your manual calculations (you may use Excel to summarize the sample data).

(b) Explain whether your test satisfies the underlying assumptions, with reference to a boxplot of the sample data.

bmi_male	bmi_female	OW_male	OW-female
26.9	19.4	1	0
29.9	23.1	1	0
28.2	24.8	1	0
30.5	18.4	1	0
25.6	29.9	1	1
31.3	24.5	1	0
27.6	19.8	1	0
23.3	19	0	0
25.1	22.9	0	0
29.6	17.7	1	0
22.1	25.6	0	1
24.2	25.6	0	1
26.3	22.1	1	0
31.3	23.9	1	0
22.1	27.7	0	1
23.8	22.1	0	0
26.2	28	1	1
30.3	32.3	1	1
23.4	29.1	0	1
19.7	35.2	0	1
28	22.1	1	0
27.2	19.1	1	0
27	25.2	1	0
21.7	18.9	0	0
24.9	24.3	0	0
30.5	21.9	1	0
25.6	28.1	1	1
29.3	16.3	1	0
33	18.4	1	0
27.1	23.8	1	0
30.2	22	1	0
	29		1
	24.9		0
	22		0
	25.6		1

Answer 1

(a)

Mean =

Standard Deviation =

Population Mean = 25.5 ( Given )

Sample Mean = 26.3855

Standard Deviation = 3.3080

The test statistic is given by

Ho:    v/s   H1:

t = [(26.3855-25.5)*Sqrt(31)] / 3.3080

= 1.4904

t-critical value at level of significance 0.05 ( one tail ) at 30 degrees of freedom = 1.697

As t-critical value is greater than t-statistic , we accept Ho

b)

Box-plot technique of the given data

A box-and-whisker plot, sometimes called a box plot, is a diagram that utilizes the upper and lower quartiles along with the median and the two most extreme values to depict a distribution graphically. The plot is constructed by using a box to enclose the median. This box is extended outward from the median along a continuum to the lower and upper quartiles, enclosing not only the median but also the middle 50% of the data. From the lower and upper quartiles, lines referred to as whiskers are extended out from the box toward the outermost data values.

The box-and-whisker plot is determined from five specific numbers.

1. Q1

2. Q2

3. Q3

4. Maximum value

5. Minimum value

A box is drawn around the median with the lower and upper quartiles (Q1 and Q3) as the box endpoints. These box endpoints (Q1 and Q3) are referred to as the hinges of the box.

the interquartile range (IQR) is computed by Q3 - Q1. The interquartile range includes the middle 50% of the data and should equal the length of the box. However, here the interquartile range is used outside of the box also.At a distance of 1.5 . IQR outward from the lower and upper quartiles are what are referred to as inner fences. A whisker, a line segment, is drawn from the lower hinge of the box outward to the smallest data value. A second whisker is drawn from the upper hinge of the box outward to the largest data value. The inner fences are established as follows.

Q1 - 1.5 . IQR and Q3 + 1.5 . IQR

Arranging the given data in ascending order of magnitude

Q1 = ( N+1 )th term / 4 = 8th. term = 24.2

Q2 = 2*(N+1)th term / 4 = 16th. term = 27

Q3 = 3*(N+1)th term / 4 = 23.25 th. term

                                   = 23 rd term + (0.25)*(24th term - 23rd term)

                                   = 29.6 + 0.25*( 29.9-29.6)

                                   = 29.6 + 0.25*(0.3)

                                   = 29.6 + 0.075 = 29.67

Inter Quartile Range = Q3 - Q1 = 29.67 - 24.2 = 5.47

Q1 - 1.5 . IQR = 24.2 - 1.5*5.47 = 15.995

Q3 + 1.5 . IQR = 29.67 + 1.5*5.47 = 37.875

Population size: 31
Median: 27
Minimum: 19.7
Maximum: 33
First quartile: 24.2
Third quartile: 29.67
Interquartile Range: 5.47
Outliers: none

As there are no outliers, test satisfies the underlying assumptions, with reference to a boxplot of the sample data.