Mat2044 Introduction To Statistics Assessment Answer

Question 1This question is designed to discover the Central Limit Theorem(CLT) and learn to describe the predictable pattern of the generating empirical distributions of sample means.Consider a random sample 12,,...,nX XXfrom the assigned distribution based onyourStudent ID’snumber(refer to Appendix).(a)Describe the parameters and the probability density function (pdf) of the random variable. Plot the density graph of thedistirbution. (b)Generate random samples ofsizen= 5, n= 50 and n= 100 from the assigned distribution.(c)Calculate the sample mean, sample standard deviationfrom the samplesgenerated in part (b).(d)Perfom500repetitions forpart (b) and (c). Plot the relative frequency histogram for the values of the sample means for 500 samples, each with n= 5, n= 50and n= 100. (e)Compare graphically the histogram in part (d) to the density of a suitable Normal distribution. You might find the following R command useful: Curve(dnorm(x,mean=, sd= , add=TRUE, lwd=2, col=”red”))(f)Commenton your findings in the context of the CLT.aQuestion 2A new study indicates that babies maychoose not to learn from someone they donottrust. A group of 60 babies, aged 13 to 16 months, were randomly divided into two groups. Each baby watched an adult express great excitement while looking into a box. The babies were then checkedthe box and it either had a toy in it (the adult could be trusted) or it was empty (the adult was not reliable). The same adult then turned ona push-on light with her/his forehead, and the number of babies who imitated the adult’s behaviour by doing the same thing was counted. The results are in the table below.ImitatedDid not imitateReliableUnreliable18101220Test whether there is evidence thatbabies are more likely to imitate those they consider reliable at(a)1% significance level?(b)5% significance level?Justify your findings. Notes for Question 2:(I)You may consider using function prop.testto perform the above hypothesis test.(II)Present complete procedures of hypothesis testing for the above problem such as null hypothesis, alternative hypothesis, significance level, test statistics value, p-value etc..in your findings.(III)State your conclusion clearly