Respuesta :
Answer:
a) The probability mass function for the Binomial distribution is given as:
[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]
Where (nCx) means combinatory and it's given by this formula:
[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]
Let X the random variable of interest, on this case we know that:
[tex]X \sim Binom(n=80, p=0.613)[/tex]
b) [tex] \mu = n*p = 80*0.613= 49.04[/tex]
[tex]\sigma = \sqrt{np(1-p)}= \sqrt{80*0.613*(1-0.613)}= 4.356[/tex]
c) For this case we can use the following excel code:
"=BINOM.DIST(60,80,0.613,FALSE)"
And we got 0.00355
d) [tex] P(X>50) 1-P(X \leq 50)[/tex]
And we can use the following excel code:
"=1-BINOM.DIST(50,80,0.613,TRUE)"
And we got 0.3718
Step-by-step explanation:
Assuming the following questions:
a) What is the probability distirbution for X?
The binomial distribution is a "DISCRETE probability distribution that summarizes the probability that a value will take one of two independent values under a given set of parameters. The assumptions for the binomial distribution are that there is only one outcome for each trial, each trial has the same probability of success, and each trial is mutually exclusive, or independent of each other".
The probability mass function for the Binomial distribution is given as:
[tex]P(X)=(nCx)(p)^x (1-p)^{n-x}[/tex]
Where (nCx) means combinatory and it's given by this formula:
[tex]nCx=\frac{n!}{(n-x)! x!}[/tex]
Let X the random variable of interest, on this case we know that:
[tex]X \sim Binom(n=80, p=0.613)[/tex]
b) Using formula calculate the mean and standard deviation of X
By properties we have:
[tex] \mu = n*p = 80*0.613= 49.04[/tex]
[tex]\sigma = \sqrt{np(1-p)}= \sqrt{80*0.613*(1-0.613)}= 4.356[/tex]
c) Use your calculator to find the probability that DeAndre scored with 60 of these shots
For this case we can use the following excel code:
"=BINOM.DIST(60,80,0.613,FALSE)"
And we got 0.00355
d) Find the probability that DeAndre scores with more than 50 of these shots
We want this probability:
[tex] P(X>50) 1-P(X \leq 50)[/tex]
And we can use the following excel code:
"=1-BINOM.DIST(50,80,0.613,TRUE)"
And we got 0.3718
Using the normal approximation to the binomial, it is found that there is a 0.0668 = 6.68% probability that DeAndre scored on at most 42 shots.
In a normal distribution with mean [tex]\mu[/tex] and standard deviation [tex]\sigma[/tex], the z-score of a measure X is given by:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
- It measures how many standard deviations the measure is from the mean.
- After finding the z-score, we look at the z-score table and find the p-value associated with this z-score, which is the percentile of X.
- The binomial distribution is the probability of x successes on n trials, with p probability of a success on each trial. It can be approximated to the normal distribution with [tex]\mu = np, \sigma = \sqrt{np(1-p)}[/tex].
In this problem:
- DeAndre scored with 61.3% of his shots, hence [tex]p = 0.613[/tex]
- Suppose you choose a random sample of 80 shots, hence [tex]n = 80[/tex].
The mean and the standard deviation are given by:
[tex]\mu = np = 80(0.613) = 49.04[/tex]
[tex]\sigma = \sqrt{np(1 - p)} = \sqrt{80(0.613)(0.387)} = 4.3564[/tex]
The probability that he scored on at most 42 shots, using continuity correction, is [tex]P(X \leq 42 + 0.5) = P(X \leq 42.5)[/tex], which is the p-value of Z when X = 42.5, hence:
[tex]Z = \frac{X - \mu}{\sigma}[/tex]
[tex]Z = \frac{42.5 - 49.04}{4.3564}[/tex]
[tex]Z = -1.5[/tex]
[tex]Z = -1.5[/tex] has a p-value of 0.0668
0.0668 = 6.68% probability that DeAndre scored on at most 42 shots.
To learn more about the normal approximation to the binomial, you can take a look at https://brainly.com/question/15727396
