Question

Compare the mean and standard deviation for the Coin variable (question 2) with those of the mean and standard deviation for the binomial distribution that was calculated by hand in question 5. Explain how they are related in a short paragraph of several complete sentences.

Mean from question #2: 4.543

Standard deviation from question #2: 1.521

Mean from question #5: 5

Standard deviation from question #5: 1.581

Answer 1

Comparisons and explanations:

Let us first understand the formula for mean and standard deviation for binomial distribution

Mean and standard deviation of binomial distribution is given as

Mean=

Standard deviation=

Mean and Standard deviation in both question 2 and 5 are close in value.

As we know that when we will  flip the coin It will  represent binomial distribution (P=1/2=0.5)  

because appearance of tails and heads are independent from each other with equal probability of 0.5.

But mean value of variable coin from the sample is slightly lower comparing to the one from binomial distribution (4.543 and 5.000 respectively).

The same thing can be  observed for standard deviation (1.491 and 1.581 respectively).

This difference can be explained by the small size of the sample .

Whose mean values are given as(4.543 and 5) & Standard deviation values are given as(1.521 & 1.581)

It shows that the values are similar and if a few variables were to change, they would/could be the same

Here the thing which we can vary the number of samples

It will mainly depend up on the sample size

If the sample size is increased, mean and standard deviation is expected to match with the respective parameters of binomial distribution, because the last one represents infinitely large sample.

and binomial distribution is given as

Here

n=Number of samples or number of trials

p=probability of success desired

q=1-p=probability of failure

x=no of success desired

Conclusion:

Mean and standard deviation are nearly same .Because same number of trials  and probability of success is same (P=0.5)

They are a bit different as the one from the data file is based on actual experience

while the mean and standard deviation here are based on theoretical outcomes.