Respuesta :
Answer:
.0287
Step-by-step explanation:
For the binomial distribution, μ = np = 100(.5) = 50
σ = √(np(1-p)) = √(100(.5)(1-.5)) = √(100(.5)(.5)) = √25 = 5
Then, P(X >= 60) = using the continuity correction, P(X >= 59.5)
= P(Z >= (59.5-50)/5)
=P(Z >= 1.9)
=1 - P(Z <= 1.9)
(use the table in your book; I will use normsdist and report the answer to 4 decimal places, as is typical)
1 - normsdist(1.9) = .0287
(Note: the exact solution may be found in Excel using 1-binomdist (59,100,.5,TRUE) = .0284; note how the above result is close)
The probability that no more than [tex]25[/tex] were victims of e-mail fraud is [tex]\fbox{0.0278}[/tex].
Further explanation:
Given:
The probability [tex]p[/tex] that an appliance is currently repaired is [tex]0.5[/tex].
The number of complex [tex]n[/tex] are [tex]100[/tex].
Calculation:
The [tex]\bar{X}[/tex] follow the Binomial distribution can be expressed as,
[tex]\bar{X}\sim \text{Binomial}(n,p)[/tex]
Use the normal approximation for [tex]\bar{X}[/tex] as
[tex]\bar{X}\sim \text{Normal}(np,np(1-p))[/tex]
The mean [tex]\mu[/tex] is [tex]\fbox{np}[/tex]
The standard deviation [tex]\sigma\text{ } \text{is} \text{ } \fbox{\begin{minispace}\\ \sqrt{np(1-p)}\end{minispace}}[/tex]
The value of [tex]\mu[/tex] can be calculated as,
[tex]\mu=np\\ \mu= 100 \times0.5\\ \mu=50[/tex]
The value of [tex]\sigma[/tex] can be calculated as,
[tex]\sigma=\sqrt{100\times0.5\times(1-0.5)} \\\sigma=\sqrt{100\times0.5\times0.5}\\\sigma=\sqrt{25}\\\sigma={5}[/tex]
By Normal approximation \bar{X} also follow Normal distribution as,
[tex]\bar{X}\sim \text{Normal}(\mu,\sigma^{2} )[/tex]
Substitute [tex]50[/tex] for [tex]\mu[/tex] and [tex]25[/tex] for [tex]\sigma^{2}[/tex]
[tex]\bar{X}\sim\text {Normal}(50,25)[/tex]
The probability that at least [tex]60[/tex] are currently being repaired can be calculated as,
[tex]\text{Probability}=P\left(\bar{X}>60)\right}\\\text{Probability}=P\left(\frac{{\bar{X}-\mu}}{\sigma}>\frac{{(60-0.5)-50}}{\sqrt{25} }\right)\\\text{Probability}=P\left(Z}>\frac{{59.5-50}}{5}}\right)\\\text{Probability}=P\left(Z}>\frac{9.5}{5}}\right)\\\text{Probability}=P(Z}>1.9})[/tex]
The Normal distribution is symmetric.
Therefore, the probability of greater than [tex]1.9[/tex] is equal to the probability of less than [tex]1.9 [/tex].
[tex]P(Z>1.9})=1-P(Z<1.9)\\P(Z>1.9})=1-0.9722\\P(Z>1.9})=0.0278[/tex]
Hence, the probability that no more than [tex]25[/tex] were victims of e-mail fraud is [tex]\fbox{0.0278}[/tex].
Learn More:
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2. Learn more about domain https://brainly.com/question/3852778
Answer Details:
Grade: College Statistics
Subject: Mathematics
Chapter: Probability and Statistics
Keywords:
Probability, Statistics, Appliance, Apartment complex, Binomial distribution, Normal distribution, Normal approximation, Central Limit Theorem, Z-table, Mean, Standard deviation, Symmetric.
