Answer in Statistics and Probability for Goldie #254478


Use the traditional method in testing hypothesis.

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Developmental psychologists at a prominent California university conducted a longitudinal study investigating the effect of high levels of curiosity in early childhood on intelligence. The local population of 3-year-olds was screened via a test battery assessing curiosity. Twelve of the 3-year-olds scoring in the upper 90% of this variable were given an IQ test at age 3 and again at age 11. The following IQ scores were obtained.

Student Number: 1 2 3 4 5 6 7 8 9 10 11 12

IQ (Age of 3) : 100 105 125 140 108 122 117 112 135 128 104 98 

IQ (Age of 11) : 114 116 139 151 106 119 131 136 148 139 122 113

Using ∝= 0.01, two-tailed test, analyse the data and then interpret the result.

In this question, we have a case where our sample is dependent and we should first consider using matched/paired t-test for our analysis.

To perform this test, we begin by finding the difference between pairs of observation for each child. We shall use the notation “d,” to represent the differences and use it to make a decision.

The hypotheses to be tested are,



“H_1:\mu_d\not=0”, where “d” is the difference between each pair of observations.

To test these hypotheses, we first determine the mean of the differences“(\bar{d})” given by,

“\bar{d}=(\sum(d))/n”, where “n=12.”





Variance of the differences is given by,


“V(d)=624.666666667/11=56.78788(5\space d\space p)”.

The standard deviation “SD(d)” is given by,


The test statistic is given by,

“t^*=\bar{d}/(SD(d)/\sqrt n)”

“t^*=-11.66666667/(7.535773271/\sqrt {12})”


“t^*” is compared with the student’s t distribution with “(n-1)=12-1=11” degrees of freedom at “\alpha=0.01”.

The table value “t_{\alpha/2,11}=t_{0.01/2,11}=t_{0.005,11}=3.106” and reject the null hypothesis if “|t^*|\gt t_{0.005,11}.”

Since “|t|=5.363\gt t_{0.005,11}=3.106,” we reject the null hypothesis and conclude that sufficient evidence exists to show that high levels of curiosity in childhood affect IQ. They seem to increase the IQ.

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