Question

The height of 50 people in the Industrial Engineering department is given in the following. 180 167 159 153 162 168 161 165 156 174 181 166 171 167 168 168 166 183 177 177 172 158 160 187 179 176 170 184 169 159 150 178 171 157 181 172 161 158 176 174 167 175 174 160 174 164 179 143 188 164
a) Calculate the sample average and standard deviation.
b) Construct a histogram. (Please use 8 bins) (Hint: There is a histogram tool in MS Excel, you can use it if you wish. Or you can plot it manually. )
c) Construct a stem-and-leaf plot.
d) Find the sample median and the lower and upper quartiles.
From the previous questions, I want to test the hypothesis ??0 : ?? = 170 cm vs ??1 : ?? ? 170 ????. From Google I know that ?? = 10 ???? for the generation between 18-22 in the world.
1) What can you say about the distribution, mean and variance of ??? ?
2) Please write the appropriate hypothesis and test against, a) ?? = 0.1 b) ?? = 0.05 c) ?? = 0.01
3) Find the corresponding critical regions for a) ?? = 0.1 b) ?? = 0.05 c) ?? = 0.01
4) Can you plot the hypothesis test for ?? = 0.10 as we did in the lecture, i.e., please plot the distribution, show the values of the critical region, put the ????????? on the plot.
5) Without making any further calculations, can you say what happens to the critical region? Will it get wider or narrower?

Answer 1

Given: n=50

a)

sample mean:

standard deviation:

b)

Computational Table:

Class intervals (X)	frequency (F)
143 - 148	1
149 - 154	2
155 - 160	8
161 - 166	8
167 - 172	12
173 - 178	10
179 - 184	7
185 - 190	2
Total	50

Histogram:

c)

stem Leaf

14 0

15 0,3,6,7,8,8,9,9

16 0,0,1,1,2,4,4,5,6,6,7,7,7,8,8,8,9

17 0,1,1,2,2,4,4,4,4,5,6,6,7,7,8,9,9

18 0,1,1,3,4,7,8

d) First data arrange in ascending order

Median = size of [(n+1)/2]th item

Median = size of [(50+1)/2]th item

Median = size of 25.5^th item

Median = size of 25th item + 0.5 (size of 26th - size of 25th item)

Median = 168 + 0.5(169 - 168)

Median = 168.5

Qk = size of k [(n+1)/4]th item

For First Quartile

Q1 = size of 1(50+1)/4th item

Q1 = size of 12.75th item

Q1 = size of 12th ietm + 0.75 (size of 13th item - size of 12th item)

Q1 = 161 + 0.75(161 - 161)

Q1 = 161

Lower quartile Q1 = 161

For Third (Upper) Quartile

Q3 = size of 3(50+1)/4th item

Q3 = size of 38.25th item

Q3 = size of 38th ietm + 0.25 (size of 39th item - size of 38th item)

Q3 = 176 + 0.25 (177- 176)

Q3 = 176.25

Upper Quartile = 176.25

Q3 = 176.25