(Q1) (8 points) A computer supply house receives a large shipment of computer disks each week. Past experience has shown that the number of flaws per disk can be described by the following probability distribution, where the random variable X represents the number of flaws per computer disk.
number of flaws per computer disk, $x$ probability $p(x)$
0 0.65
1 0.20
2 0.10
3 0.05
(a) (2pt) Calculate the mean and variance of the number of flaws per disk.
(b) (3pt) Suppose that we randomly select a sample of 100 disks. Find the mean and variance of the sample mean $\bar{X}$. What distribution does $\bar{X}$ follow (approximately)? Justify your answer.
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(c) (2pt) Find the probability that the sample mean, $\bar{X}$, of the sample of 100 disks is less than 0.4.
(d) (1pt) Find the probability that, for a single disk, the number of flaws is less than 0.4.