The probability that a radish seed will germinate is 0.6. Estimate the probability that of 130 randomly selected seeds, exactly 90 will germinate.
Note: I keep getting the answer wrong.
This is a binomial distribution with parameters N=130, p=0.6, r=90
so
P(X=90)=C(N,r)p^r(1-p)^(N-r)
=C(130,90)*0.6^90*0.4^40
=5.3347282074*10^33*1.0804695562359849*10^-20 * 1.2089258196146345*10^-16
=0.00697
But, I'm suppose to use the normal distribution approximation to the binomial distribution.
Glad that you mentioned that an approximation is required. The question asks for exactly 90 seeds, which is discrete.
Here's how I would proceed to approximate a discrete random variable from a continuous distribution.
"Exactly 90" is approximately equal to the random variable X=89.5 to 90.5.
We can generally approximate a binomial distribution by a normal distribution when np>5. Here np=130*0.6=78 > 5, so approximation will be reasonable.
The equivalent μ=np=78
σ
=√(npq)
=√(130*.6*(1-0.6))
=√(31.2)
=5.585696
Z(X=90.5)=(90.5-78)/5.585696=2.237859
Z(X=89.5)=(89.5-78)/5.585696=2.058830
Here, we are dealing with a small difference of two probabilities, so normal tables (on paper) by interpolation may or may not be adequate. I suggest you use a calculator with a Z function, or use a normal distribution calculator online, such as:
http://stattrek.com/online-calculator/normal.aspx
Using 5 digits, I get
P(X=90.5)=0.98738, and
P(X=89.5)=0.98024
(remember to use the respective Z-values when looking up probabilities)
Thus
P(89.5≤X≤90.5)
=0.98738-0.98024
=0.00714
(approximated using normal distribution)
(compared with value of 0.00697 using the binomial distribution).
No wonder I keep getting the wrong answer. Thank you so much for helping me how to get the answer. Now I know how to do the next problem which is similar to this problem.
You're most welcome.
I am glad things are working out.
Post if you have difficulties.
To estimate the probability that exactly 90 out of 130 radish seeds will germinate, we can use the binomial probability formula and calculate it step by step. The binomial probability formula is:
P(x) = (nCx) * p^x * (1-p)^(n-x)
Where:
P(x) is the probability of getting exactly x successes,
n is the number of trials,
x is the number of successful outcomes,
p is the probability of success in a single trial,
(1-p) is the probability of failure in a single trial,
nCx represents the number of ways to choose x objects from a set of n objects.
In this case, the probability of germination is 0.6 (p = 0.6), and the number of trials is 130 (n = 130). We want to know the probability of exactly 90 germinated seeds (x = 90). Let's plug these values into the formula:
P(90) = (130C90) * (0.6)^90 * (1-0.6)^(130-90)
To calculate (130C90), which represents the number of ways to choose 90 radish seeds out of 130, we can use the combination formula:
(130C90) = 130! / (90!(130-90)!)
Now, let's break down the calculation steps:
Step 1: Calculate (130C90):
(130C90) = 130! / (90! * (130-90)!)
= 130! / (90! * 40!)
Step 2: Calculate (0.6)^90:
(0.6)^90 = (0.6 * 0.6 * ... * 0.6) (90 times)
≈ 5.2214e-10 (rounded to the nearest decimal place)
Step 3: Calculate (1-0.6)^(130-90):
(1-0.6)^(130-90) = 0.4^40
≈ 1.0995e-16 (rounded to the nearest decimal place)
Step 4: Multiply the results of steps 1, 2, and 3:
P(90) = (130C90) * (0.6)^90 * (1-0.6)^(130-90)
≈ (130! / (90! * 40!)) * 5.2214e-10 * 1.0995e-16
Now, by plugging these values into a calculator, we can estimate the probability that exactly 90 out of 130 radish seeds will germinate.