It is known that approximately 20% of the population is colour blind. In a sample of 270 people, use the normal approximation to find probability that:
a)at least 90 people are colour blind
b)exactly 50 people are colour blind
I already posted this question and my solution. However, I am confused how do I find the a) and b) using the normal aaproximation.
My solution is:
n=270
p=0.2
np=54
np(1-p)=54(1-0.2)=43.2
z=(90-54)/6.57=5.479
P[x>90] ?
p[x=50] ?
Please explain how to find p[x>90] and p[x=50]
Thank you in advance.
The proper way to calculate P(x≥90) is to sum the binomial probabilities from 90 to 270. Using the computer, it should not be a problem. Using a calculator will need a little patience.
For P(x≥90), np>10, npq>10 and with a sample of 270 people (n>>40), you can use the normal approximation.
You have already calculated μ and σ² so that should not be a hurdle.
However, you need to do the continuity correction, which means that you would calculate P(x>89.5) using the normal distribution. The difference of the upper tail is about 50% with or without the correction, but both numbers are very small (around 10^(-8)).
For P(x=90), the binomial distribution would be required.
P(x,n,p)
=C(n,x)*p^x*(1-p)^(n-x)
Thanks. However, I've found z-score value and it's greater than 3.
z=(89.5-54)/6.57=5.4
If I can't find the value in the Z table, how I'm supposed to find the area value? I know I'm supposed to subtract the area value from 1 to find the probability for x>=90.
Please explain. Thank you very much.
Quite true. If you do not have access to software nor calculator, this might be a problem using paper tables.
You can try the following:
1. following link gives probabilities for z=0,1,2...50
but does not give fractional z values.
2. if possible, find a calculator that gives the normal distribution, or software. You can download (free) the R software at the following link:
https://www.r-project.org/
It is a serious and respected software, and is not too early to start using it if you are anywhere close to seeing statistics in your future career.
3. I created an abridged table for Z=3.0 to 7.9 for your reference.
The horizontal line below "3" is for probabilities for z=3.0, 3.1, 3.2...3.9 etc.
4. Abandon the normal approximation, and calculate P(90)+P(91)...P(270) using the binomial distribution.
Consolation: You can stop adding as soon as P(x) is much smaller than P(90).
In fact, if you sum from P(90) to P(100), you will get 4 significant digits of accuracy.
Here's the table of P(Z') for Z=3.0 to 7.9 in steps of 0.1
3
1.3498E-3 9.676E-4 6.8713E-4 4.8342E-4 3.3692E-4 2.3262E-4 1.591E-4 1.0779E-4 7.2348E-5 4.8096E-5
4
3.1671E-5 2.0657E-5 1.3345E-5 8.5399E-6 5.4125E-6 3.3976E-6 2.1124E-6 1.3008E-6 7.9332E-7 4.7918E-7
5
2.8665E-7 1.6982E-7 9.9644E-8 5.7901E-8 3.332E-8 1.8989E-8 1.0717E-8 5.9903E-9 3.3157E-9 1.8175E-9
6
9.8658E-10 5.3034E-10 2.8231E-10 1.4882E-10 7.7688E-11 4.0159E-11 2.0557E-11 1.042E-11 5.2309E-12 2.6001E-12
7
1.2798E-12 6.2372E-13 3.0109E-13 1.4388E-13 6.8167E-14 3.1974E-14 1.4876E-14 6.8833E-15 3.1086E-15 1.3322E-15
Note: P(Z)=1-P(Z')
Which sentence correctly describes a data set that follows a normal distribution with a standard deviation of 4 and a mean of 14?
68% of the data points lie between 10 and 14.
68% of the data points lie between 8 and 12.
68% of the data points lie between 10 and 18.
68% of the data points lie between 10 and 16
To find the probability of at least 90 people being color blind (P[x ≥ 90]), you can use the normal approximation.
First, calculate the z-score for the value of 90 using the formula:
z = (x - μ) / σ
Here,
x = the value we are interested in (90)
μ = mean (np, which is 54)
σ = standard deviation (√(np(1-p)), which is √(54*(1-0.2)) ≈ 6.57)
z = (90 - 54) / 6.57 ≈ 5.479
Now, to find P[x ≥ 90], you need to find the area under the normal curve to the right of z-score 5.479. Since the normal distribution is a symmetric curve, this probability is equivalent to finding P[z ≥ 5.479].
Using a standard normal distribution table or a calculator, you can find the probability corresponding to a z-score of 5.479. The table or calculator will give you an area, which is the probability.
Next, to find the probability that exactly 50 people are color blind (P[x = 50]), you need to use the normal approximation and treat it as a continuous distribution.
Since the normal distribution is continuous, the probability of getting an exact value (like 50) is infinitesimally small. However, to get an estimate, you can calculate the probability of being within a very narrow range around 50.
You can calculate P[49.5 < x < 50.5] as follows:
Step 1: Calculate the z-scores for the lower and upper limits of the range:
Lower z-score: (49.5 - 54) / 6.57 ≈ -0.688
Upper z-score: (50.5 - 54) / 6.57 ≈ -0.482
Step 2: Use a standard normal distribution table or calculator to find the probability associated with each z-score.
P[49.5 < x < 50.5] = P[-0.688 < z < -0.482]
Subtract the probability associated with the lower z-score from the probability associated with the upper z-score to obtain the required probability.
Note: Keep in mind that the normal approximation assumes that the sample size is large enough (np ≥ 5 and n(1-p) ≥ 5) for the approximation to hold. In this case, np = 54 which is greater than 5, so the normal approximation is valid for this problem.