3. Suppose that Z N(0,1) and X N(3,2). Answer the following questions by stating

Question

3. Suppose that Z N(0,1) and X N(3,2). Answer the following questions by stating the correct R function and answer. (a) P(Z0.7) (c) P(Z=0) (d) P(0.35 1.28) (h) Height of the probability density function for X at x = 3. (i) Height of the probability density function for Z at z = 0. (j) If the random variable X (and the X axis) had units of inches, what would the units of dnorm(x) for any value of x on the x-axis be? (k) Find the value of a such that P(Z < a) = 0.648 (l) Find the value of b such that P(|Z| < b) = 0.95

Explanation / Answer

a)P(Z<-0.7)=0.241964

c) P(Z=0)=0 as the probability at a certain point for a continuous distribution is zero

d)P(0.35<Z<1.28)=P(Z<1.28)-P(Z<0.35)=0.262896

h) At x=3 height is 0.1994711

R code

dnorm(3, mean = 3, sd =2, log = FALSE)

i) Height of Z distrubution when z=0 is 0.3989423

R code

dnorm(0, mean = 0, sd = 1, log = FALSE)

j) Unitless as it gives the height or density at that point

k)P(Z<0.379926)=0.648

#R code

qnorm(0.648, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)

l) P(|Z|<1.959964)=0.95

#R code

qnorm(0.025, mean = 0, sd = 1, lower.tail = TRUE, log.p = FALSE)

As P(-b<Z<+b)=0.95 then P(Z<-b)=0.025 using this we get the above result

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