Brosnahan et al. (2004) investigated the relationship between mental health and

Question

Brosnahan et al. (2004) investigated the relationship between mental health and physical exercise in youths. Suppose as part of the study a sample of 155 non-hispanic white boys aged fifteen was taken, and the body mass index was recorded on each (in kg/m^2). In the population of non-hispanic fifteen-year-old white boys, assume that body mass index follows a distribution with mean and standard deviation 3.86 Brosnahan, J., Steffen, L.M., Lytle, L., Patterson, J., and Boostrom, A. (2004): The relation between physical activity and mental health among hispanic and non-hispanic white adolescents. Arch. Pediatr Adolesc. Med. 158, 818-823 Giving your answer to two decimal places, what is the standard deviation of the sample mean? Use Chebyshev's inequality to give an upper bound on the probability that the sample mean differs from mu by more than 1.96 times 3.86/Squareroot 155. Give your answer to two decimal places. Provide an approximation of the probability that the sample mean differs from u by more than 1.96 times 3.86/155. Give your answer to two decimal places. Suppose now we assume that body mass indices are Normally distributed within the population. What is the probability that the sample mean differs from Au by more than 1.96 times 3.86/V155? Give your answer to two decimal places. In order to find the probability in (d), which of the following best describes your reasoning? We can use the Central Limit Theorem since the sample size is large, so the distribution of the sample mean is approximately Normal with mean u and standard deviation 3.86. The distribution of the sample mean is Normal with mean u and standard deviation 3.86/Squareroot 155 since each observation is from a Normal distribution with mean u and standard deviation 3.86 We can use the Central Limit Theorem since the sample size is large, so the distribution of the sample mean is approximately Normal with mean u and standard deviation 3.86/155 The distribution of the sample mean is Normal with mean mu and standard deviation 3.86 since each observation is from a Normal distribution with mean mu and standard deviation 3.86 The bound from Chebyshev's inequality is a good approximation here since the sample size is large

Explanation / Answer

1) standard deviation of mean =std deviation of population/.(n)1/2 =0.31

b)as it lies with in 1.96 standard deviation from mean

hence from chebychev's probabilty =(1-1/k2) where k=1.96

=0.74

c) probabilty that it differs more then 1.96 =1-P(it differs at most 1.96) =1-0.74 =0.26

d)from normal value table for value more then 1.96 ; probability =0.05

e)option C is correct

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