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52
64.tex
52
64.tex
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@ -28,8 +28,9 @@ P\{X<2\}=p(0)+p(1)&={10!\over0!10!}\times(0.2)^0\times(1-0.2)^{10}
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&=0.3758
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&=0.3758
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}$$
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}$$
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\bigskip
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\vfill
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\bigskip
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\eject
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2. In a large population, 80\% of the members are right-handed.
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2. In a large population, 80\% of the members are right-handed.
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\bigskip
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\bigskip
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@ -139,5 +140,52 @@ i) $P\{0<Z<c\}=0.3531$, $P\{Z<c\}=0.5+0.3531=0.8531$, $c=1.05$
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\bigskip
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\bigskip
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j) $P\{-0.54<Z<c\}=0.646$, $P\{Z<c\}=P\{Z<-0.54\}+0.646=0.9406$, $c=1.56$
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j) $P\{-0.54<Z<c\}=0.646$, $P\{Z<c\}=P\{Z<-0.54\}+0.646=0.9406$, $c=1.56$
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\vfill
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\eject
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5. Let $X$ be a normally distributed random variable with mean 550 and
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standard deviation 150.
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\bigskip
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a) $P\{X<790\}=P\{Z<(790-550)/150\}=P\{Z<1.6\}=0.9452$
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\bigskip
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b) $P\{220<X<784\}=P\{(220-550)/150<Z<(784-550)/150\}
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=P\{Z<1.56\}-P\{Z<-2.2\}=0.9406-0.0139=0.9267$
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\bigskip
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c) $P\{X>211\}=P\{Z>(211-550)/150\}=P\{Z>-2.26\}=P\{Z<2.26\}=0.9881$
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\bigskip
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d) $P\{X>c\}=0.1075, (c-550)/150=1.24, c=736$
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\bigskip
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e) $P\{X<c\}=0.2005, (c-550)/150=-0.84, c=424$
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\vfill
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\eject
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6. Let $X_i$ be independent identically distributed as $N(100,16)$
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for $i=1,2,\ldots,n$.
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\bigskip
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a) for $n=25$, what is $P\{\bar X>101.28\}$?
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\medskip
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From $N(100,16)$ we have the mean $\mu=100$ and the
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standard deviation $\sigma=\sqrt{16}=4$.
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By the Central Limit Theorem we have $\sigma_{\bar X}=\sigma/\sqrt n$.
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$$P\{\bar X>101.28\}
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=P\{Z>\sqrt{25}\,(101.28-100)/4\}
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=P\{Z>1.6\}=P\{Z<-1.6\}=0.0548$$
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\bigskip
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b) for $n=64$, what is $P\{99<\bar X<101.5\}$?
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$$\eqalign{
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P\{99<\bar X<101.5\}
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&=P\{\sqrt{64}\,(99-100)/4<Z<\sqrt{64}\,(101.5-100)/4\}\cr
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&=P\{-2<Z<3\}\cr
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&=P\{Z<3\}-P\{Z<-2\}\cr
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&=0.9987-0.0228\cr
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&=0.9759
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}$$
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\end
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\end
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