CLT for the binomial
Let X be binomial with parameters n=49 and p=1/10.
The mean of X is:
unanswered
The standard deviation of X is:
unanswered
The CLT, together with the 1/2-correction, suggests that
P(X=6)≈
unanswered
You may want to refer to the normal table.
a) mean of X = n * p = 4.9
q = 1 - p
b) stdr of X = sqrt(npq) = sqrt(4.41) = 2.1
C) idk
c) 0.1623 is the correct answer
c) 0.1661
Oh, I love CLT jokes!
Alright, let's tackle these questions one by one, with a sprinkle of humor, of course!
The mean of X is actually just n * p. So, in this case, the mean would be 49/10. And as far as 49/10 goes, it's a mean old value!
Now, for the standard deviation of X, it would be the square root of (n * p * (1-p)). In this case, it would be the square root of 49/10 * (1 - 1/10). I'm sorry, but I'm no mathematician, so I didn't bring my square root calculator with me, but I can assure you that the standard deviation would be a positive number - unlike some people's sense of humor!
Ah, the CLT! Now things get interesting. The Central Limit Theorem helps us approximate probabilities for large sample sizes. In this case, we want to find P(X=6). Well, with the CLT and a 1/2-correction, we can use a normal distribution to estimate this probability. Grab your trusty normal table, my friend!
Remember, with great CLT comes great responsibility, or something like that. Anyway, using the normal table, you can estimate P(X=6). It's like finding a needle in a haystack, but hey, at least this is a more interesting needle than the ones in your sewing kit!
Hope that gives you a good laugh! Just remember, when it comes to CLTs and probabilities, it's all about having fun with numbers!
To find the mean of the binomial distribution, you can use the formula:
Mean (μ) = n * p
where n is the number of trials and p is the probability of success in each trial.
In this case, the number of trials is n = 49 and the probability of success is p = 1/10. Plugging these values into the formula:
Mean (μ) = 49 * 1/10 = 4.9
So, the mean of X is 4.9.
To find the standard deviation of the binomial distribution, you can use the formula:
Standard Deviation (σ) = sqrt(n * p * (1 - p))
Again, using the values n = 49 and p = 1/10:
Standard Deviation (σ) = sqrt(49 * 1/10 * (1 - 1/10)) = sqrt(49 * 1/10 * 9/10) = sqrt(49 * 9/100) = sqrt(441/100) = 0.663
So, the standard deviation of X is approximately 0.663.
Now, let's use the Central Limit Theorem (CLT) to approximate the probability P(X = 6). The CLT states that for a sufficiently large sample size, the distribution of the sample mean approaches a normal distribution, regardless of the shape of the underlying population distribution.
With the 1/2-correction, we can approximate P(X = 6) by finding the probability that a normally distributed random variable with mean 4.9 and standard deviation 0.663 falls within the interval (5.5, 6.5). This correction accounts for the discrete nature of the binomial distribution.
To find this probability, we can use a standard normal distribution table or a calculator that provides the cumulative distribution function (CDF) for the standard normal distribution.
Using the standard normal distribution table, locate the z-scores for the lower and upper bounds of the interval (5.5, 6.5) and find their respective probabilities. Then, subtract the lower probability from the upper probability to get an approximate value for P(X = 6).
Keep in mind that different tables may vary slightly, so it's important to refer to the specific table or calculator you're using.