Question

-- g a wversity mo school a. Calcualte a point estimate of the mean value of IQ for the population of first graders at this school What estimator did you use? b. Calculate a point estimate of the IW value that separates the lowert 50% of all such students from the highest 50% of students. c. Calculate a 99% confidence interval for the population mean. What sampling distribution did you use? Why? d. Test the hypothesis that this population of first graders is not smarter nor less smart than the wider population (which has average IQ equal to 100)

82
96
99
102
103
103
106
107
108
108
108
108
109
110
110
111
113
113
113
113
115
115
118
118
119
121
122
122
127
132
136
140
146

Answer 1

a) Sample mean is the point  estimator of population mean

We find sample mean

$\bar{x} = \frac{\sum x}{n}= 113.73$

b) Median divides the data into two equal parts

Point estimate of population median is the sample median

For median we arrange the data in ascending order

n= 33 , median = (33+1)/ 2= 17th observation

Median = 113

c) For 99% confidence interval we use t distribution

though sample size is large (>30) , we use t distribution as population standard deviation is not known

99% confidence interval is

$\bar{X}\pm t_{c}*s /\sqrt{n}$

where

$\bar{X}= 113.73, s= 12.74 , n= 33$

For 99% confidence with df = 33-1 =32

tc = 2.74 ( from t table)

Therefore , 99% confidence interval is

$113.73\pm 2.74*12.74 /\sqrt{33}$

$=113.73\pm 6.08$

= (107.65 , 119.81)

d) The null and alternative hypothesis

$H0:\mu =100$

$Ha:\mu \neq 100$

Test statistic

X s/Vn

$=\frac{113.73-100}{12.74/\sqrt{33}}$

= 6.19

df = 32

P value is less than 0.0001

We reject H0

There is sufficient evidence to conclude that IQ is not equal to average IQ of 100

Calculation

	x	(x-113.73)^2		data in ascending order
	82	1006.7929		82
	96	314.3529		96
	99	216.9729		99
	102	137.5929		102
	103	115.1329		103
	103	115.1329		103
	106	59.7529		106
	107	45.2929		107
	108	32.8329		108
	108	32.8329		108
	108	32.8329		108
	108	32.8329		108
	109	22.3729		109
	110	13.9129		110
	110	13.9129		110
	111	7.4529		111
	113	0.5329		113	median
	113	0.5329		113
	113	0.5329		113
	113	0.5329		113
	115	1.6129		115
	115	1.6129		115
	118	18.2329		118
	118	18.2329		118
	119	27.7729		119
	121	52.8529		121
	122	68.3929		122
	122	68.3929		122
	127	176.0929		127
	132	333.7929		132
	136	495.9529		136
	140	690.1129		140
	146	1041.3529		146
sum/ss	3753	5196.5457
mean =sum/33	113.7273
s^2=ss/32		162.3920531
s		12.74331406