Answer to Question #88387 in Statistics and Probability for Randi Carpenter

Question #88387
suppose that 1 out of 10 plasma televisions shipped with a defective speaker. out of a shipment of n=400 plasma televisions. find the probability that there are
a) at most 40 with defective speakers (Hint: Use the dishonest- coin principle with P= 1/10=0.1 to the find the mean and standard deviation.
B) Most than 52 with defective speakers
1
Expert's answer
2019-04-24T09:59:18-0400

Dishonest-Coin Principle

Let X denote the number of heads in n tosses of a coin (assume n30)n\geq30). Let p denote the probability of heads on each toss.Then, X has an approximately normal distribution with mean μ=np\mu=n\cdot p and standard deviation σ=np(1p)\sigma =\sqrt{n\cdot p\cdot (1-p)}.

We have that p=0.1,n=400p=0.1, n=400. Then μ=4000.1=40,σ=4000.1(10.1)=6.\mu=400\cdot 0.1=40, \sigma =\sqrt{400\cdot 0.1\cdot (1-0.1)}=6.

a) at most 40 with defective speakers


z=xμσ=40.5406=0.083333z={x-\mu \over \sigma}={40.5-40 \over 6}=0.083333

P(X40)=P(X<40.5)=P(Z<0.083333)=0.533207P(X\leq40)=P(X<40.5)=P(Z<0.083333)=0.533207

b) Most than 52 with defective speakers


z=xμσ=52.5406=2.083333z={x-\mu \over \sigma}={52.5-40 \over 6}=2.083333

P(X>52)=P(X>52.5)=0.018610P(X>52)=P(X>52.5)=0.018610

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