The data provided comes from a university that wanted to see if there was a rela

Question

The data provided comes from a university that wanted to see if there was a relationship between IQ scores and the
GPA’s of students after their freshman year. Use the data for questions 1 through 6.
1) Using a 95% level of confidence, test the claim that IQ’s have a mean of 100.

IQ GPA
126 3.68
105 3.10
104 2.72
118 3.61
126 4.0
154 3.89
118 3.51
98 2.64
108 3.33
108 3.32
95 2.91
113 3.81
128 3.53
134 3.95
116 3.57
103 3.22
119 4.0
104 2.32
100 2.65
116 3.42
123 3.55
95 2.94
105 2.75
107 2.6
133 3.56

Explanation / Answer

TRADITIONAL METHOD
given that,
sample mean, x =114.24
standard deviation, s =14.0957
sample size, n =25
I.
stanadard error = sd/ sqrt(n)
where,
sd = standard deviation
n = sample size
standard error = ( 14.0957/ sqrt ( 25) )
= 2.819
II.
margin of error = t /2 * (stanadard error)
where,
ta/2 = t-table value
level of significance, = 0.05
from standard normal table, two tailed value of |t /2| with n-1 = 24 d.f is 2.064
margin of error = 2.064 * 2.819
= 5.819
III.
CI = x ± margin of error
confidence interval = [ 114.24 ± 5.819 ]
= [ 108.421 , 120.059 ]
-----------------------------------------------------------------------------------------------
DIRECT METHOD
given that,
sample mean, x =114.24
standard deviation, s =14.0957
sample size, n =25
level of significance, = 0.05
from standard normal table, two tailed value of |t /2| with n-1 = 24 d.f is 2.064
we use CI = x ± t a/2 * (sd/ Sqrt(n))
where,
x = mean
sd = standard deviation
a = 1 - (confidence level/100)
ta/2 = t-table value
CI = confidence interval
confidence interval = [ 114.24 ± t a/2 ( 14.0957/ Sqrt ( 25) ]
= [ 114.24-(2.064 * 2.819) , 114.24+(2.064 * 2.819) ]
= [ 108.421 , 120.059 ]
-----------------------------------------------------------------------------------------------
interpretations:
1) we are 95% sure that the interval [ 108.421 , 120.059 ] contains the true population mean
2) If a large number of samples are collected, and a confidence interval is created
for each sample, 95% of these intervals will contains the true population mean

we don't have evidence that that IQ’s have a mean of 100, it is exceeded the value of 100

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