a) To Explain: the shape, centre and variability of the distribution of the random variable X-Y. b) To calculate: the probability that a randomly selected DVD will fit inside randomly selected case. c) To find: the probability that all DVDs fit in their cases.

Question

a) To Explain: the shape, centre and variability of the distribution of the random variable X-Y.  b) To calculate: the probability that a randomly selected DVD will fit inside randomly selected case.  c) To find: the probability that all DVDs fit in their cases.

Mayme · Accepted Answer

a) Given: μx=5.3 σx=0.01 μx=5.26 σy=0.02 Assume X= diameter of randomly selected case Y= diameter of randomly selected DVD The mean of the difference of 2 random variables is μx−y=μx−μy =5.3−5.26 =0.04 The variance of the difference of 2 random variables is σ2x−y=σ2x+σ2y =0.012+0.022 =0.0001+0.0004 =0.0005 The standard deviation is σx−y=σ2x−y =0.0005 =0.02236 X-Y is important to the DVD manufacture, because if the difference X-Y is positive then the DVD will fit in the case but if the difference X-Y negative, and then the DVD will not fit in the case. b) Given: μ=0.04 σ=0.0005 x=0 Formula used: z=x−μσ Calculation: The DVD fits in the case, when the difference X-Y is positive. The z-score is z=x−μσ =0−0.040.02236 =−1.7 Find the corresponding probability using the normal probability table P(z&amp;lt;−1.79) is given in the row starting with -1.7 and in the column starting with .09 of the standard normal probability table. P(X−Y&amp;gt;0)=P(Z≻1.79) =1−P(Z&amp;lt;−1.79) =1−0.0367 =0.9633 =96.33% c) Given: n=100 p=0.9633 Formula used: P(X=k)=nCk⋅pk⋅(1−p)n−k Calculation: Binomial probability is P(X=k)=nCk⋅pk⋅(1−p)n−k Find the definition of binomial probability at k=100: P(X=100)=100C100⋅0.9633100⋅(1−0.9633)100−100 =0.02378 =2.378% There is a 2.378% possibility that all of the 100 DVDs fit in their cases.

a) To Explain: the shape, centre and variability of the

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