Question

Write only the numerical answer for the questions below. 6 points for each question. 1. Suppose that the probability of a day being rainy in Istanbul is 0.4 independent of the other days. What is the probability that more than 2 days are rainy in a week. 2. Find P(X < x + 4) where X is the mean of a sample with size 4. 3. Suppose, in a statistics class, there are 10 female students with a mean GPA of 2.70, and 30 malo students with a mean GPA of 2.90. What would be the mean GPA of all 40 students? 4. If you flip a fair coin 64 times, what would be the standard deviation of sample proportion of heads? 5. Suppose that the regression model between Final (Y) and Midterm (X) scores is Y - 20+0.7.X. If the difference between midterm scores of Ali and Ayse is 30 points, what is the expected value of the difference between their final scores?

Answer 1

1. Let х be more the no. of rainy days in a week. Then, X ~ Binomial (1, 0.4) Probability that than two days in a week P(X> 2) 1 - P(X2) P(x+2) P (X=0) + P(x-1) + P(X=2) (0) (0.4)º(1-0.4)'+ ()(0.4)(0.6)6 (1) (0.4)(0.675 0.028 + 0.131+ 0.261 are rainy + 0.42 "Required probability 1 - P(X2) 10.42 0.58 (Am). 2 t P(X<M+ /2) i-PC > H+0/2) PCX-M>0/2) P[C-M)2 > 0°/4) According to Markov's inequality P(X> +) = E(X) :: P[(-u > 014) < E[C#-4°] var (W) 02/4. 02/4 Now, Var (X) = 0 var () 0?/4. PCCB-M'>0/2] ? 1. Ans PC <+0/2) >0. o²/n

C 3. n = 10 Ž,- 2.70 ^2 = 30 z = 2.90 Grand mean (10* 2.7) + (30 x 2.9) não + M₂ X2 nit 12 10 + 30 27 + 87 2.85 40 . Mean GPA of all students = 2.85 4 at x be the number of heads in 64 tosses. Binomial ( 64, 1/2) var (X) Betty np (1-P) = 64 x 0.5 0-5 = 16. Sample proportion - Vour (p) (1-P) 0.5 * 0.5 n Standard deviat? Ivar (p) 0.5 0.0625 8 Standard deviation of sample yeroportion of heads = 0.0625 5. Y. 20 + 0.7% Xi - Xi 30 Expected difference Yi - Yj (20 + 0.7X;) - (20 + 0-7X;) 0.7 (Xi-;? - 21.