Question

4) a) Suppose \( X_{n} \xrightarrow{P} X \) and \( a \) is a constant. Prove that \( a X_{n} \xrightarrow{P} a X \). b) If \( T_{n}=\sum_{i=1}^{n} X_{i} \) is the sum of a random sample of size \( n \) with mean \( \mu \) and variance \( \sigma^{2} \), show that \( T_{n} \approx A N\left(n \mu, n \sigma^{2}\right) \). c) Let \( X_{i}, i=1,2, \ldots, n \) be independent Bemoulli trials with a success probability \( p \), and let \( Y_{n}=\frac{1}{n} \sum_{i=1}^{n} X_{i} \) (i) Show that \( \sqrt{n}\left(Y_{n}-p\right) \xrightarrow{D} N(0, p(1-p)) \). Clearly state theorem(s) used in the proof. (ii) A college knows from past experience that only \( 30 \% \) of students admitted to first year courses will actually attend classes. If 450 were admitted in a certain year, approximate the probability that more than 150 students actually attend the class. [State the assumptions and conditions checked when applying the approximation].

          4) a) Suppose \( X_{n} \xrightarrow{P} X \) and \( a \) is a constant. Prove that \( a X_{n} \xrightarrow{P} a X \).
b) If \( T_{n}=\sum_{i=1}^{n} X_{i} \) is the sum of a random sample of size \( n \) with mean \( \mu \) and variance \( \sigma^{2} \), show that \( T_{n} \approx A N\left(n \mu, n \sigma^{2}\right) \).
c) Let \( X_{i}, i=1,2, \ldots, n \) be independent Bemoulli trials with a success probability \( p \), and let \( Y_{n}=\frac{1}{n} \sum_{i=1}^{n} X_{i} \)
(i) Show that \( \sqrt{n}\left(Y_{n}-p\right) \xrightarrow{D} N(0, p(1-p)) \). Clearly state theorem(s) used in the proof.
(ii) A college knows from past experience that only \( 30 \% \) of students admitted to first year courses will actually attend classes. If 450 were admitted in a certain year, approximate the probability that more than 150 students actually attend the class.
[State the assumptions and conditions checked when applying the approximation].

4) a) Suppose XnP X and a is a constant. Prove that a XnP a X.
b) If Tn=∑i=1^n Xi is the sum of a random sample of size n with mean μ and variance σ^2, show that Tn≈ A N(n μ, n σ^2).
c) Let Xi, i=1,2, …, n be independent Bemoulli trials with a success probability p, and let Yn=(1)/(n)∑i=1^n Xi
(i) Show that √(n)(Yn-p) D N(0, p(1-p)). Clearly state theorem(s) used in the proof.
(ii) A college knows from past experience that only 30 % of students admitted to first year courses will actually attend classes. If 450 were admitted in a certain year, approximate the probability that more than 150 students actually attend the class.
[State the assumptions and conditions checked when applying the approximation].

Added by Christopher W.

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