In a lot of 10 components 2 are sampled at random for inspec

In a lot of 10 components, 2 are sampled at random for inspection. Assume that in fact exactly 2 of the 10 components in the lot are defective. Let X be the number of sampled components that are defective. Find P(X = 0). Find P(X = 2). Find the probability mass function of X. Find the mean of X. Find the standard deviation of X. A new concrete mix is being designed to provide adequate compressive strength for concrete blocks. The specification for a particular application calls for the blocks to have a mean compressive strength mu greater than 1350 kPa. A sample of 100 blocks is produced and tested. Their mean compressive strength is 1356 kPa and their standard deviation is 70 kPa. Do you believe it is plausible that the blocks do not meet the specification, or are you convinced that they do? Students in grades 4-6 were asked whether good grades, athletic ability, or popularity was most important to them. A two-way table separating the students by grade and by choice of most important factor is shown below. Do these data provide evidence to suggest that goals vary by grade?

Solution

X the no of defectives is binomial with n =2

since i) the trials are independent

ii)there are exactly two outcomes

p = Prob for defective = 2/10 = 0.2

q = Prob for non defective = 8/10 = 0.8

a) P(x=0) = (0.8)² = 0.64

b) P(X=2) = 2C2 (0.2)² = 0.04

c) Prob mass funciton of x is binomial

x 0 1 2

P(x) 0.04 0.32 0.64

d) Mean = np = 2(0.2) = 0.4

e) Std of deviation of X = rt of variance

variance = npq = 0.4(0.8) = 0.32

Std dev = 0.5657