If you toss a fair coin four times would you expect the proportion of heads (or
ID: 3180107 • Letter: I
Question
If you toss a fair coin four times would you expect the proportion of heads (or tails) always to be exactly 0.5 (yes or no)? (b, c, d, e) What other proportions would you probably find on carrying out this experiment a few more times? If you repeated this experiment again but, instead of tossing the coin only four times, you tossed it a very large number of times (a million, say), (f) what proportion of heads (or tails) would you expect to find this long series approaching increasingly closely? (g) What is the name of the 'law' that deals with proportions emerging from this last kind of experiment? (h) What kind of distribution is obtained by sampling from this ('binomial') distribution, assuming the sample meets the necessary criteria? (i, j) What are the 'necessary criteria' to be used for a sample from a binomial distribution to be distributed approximately Normally? What is the shape of the probability distribution obtained by plotting random numbers in a frequency diagram? What kind of distribution is obtained by sampling from this distribution (assuming the sample size is sufficiently large)? What is the name of the theorem that proves this important result? What is the recommended ('sufficiently large') minimum sample size to be used when the distribution of the population being investigated is either unknown or known not to be Normal? What is the relationship between the standard deviation, o, of a population and the standard deviation, s, of a sample of size n drawn from that population? (In words or in the form of an algebraic expression.)Explanation / Answer
A. tossing a coin probablity of either coming head or tail is=1/2 every time because there is only two possible outcome head or tail
but in 4 trial probablity of head or tail will not be equal
It will follow binomial distribution as there are two possible outcome only.
In this theorem, the samplesize remain fixed and it is denoted by n.
There are two mutually exclusive outcome in each trial,namely sucess and failure
The success denoted by p and failure denoted by q=1-p as p+q=1
probablity of p and q remain constant in each trial if this is not same binomial distribution will not apply.
by this distribution we can find r successs in n trial.p(r) success=ncr*q power n-r*p power r
p(2) head=4C2q4-2*p2
=6(1/25)2*(1/2)2
6*1/4*1/4
3/8 not exactly .5
Because the normal approximation is not accurate for small values of n, a good rule of thumb is to use the normal approximation only if np>10 and np(1-p)>10.
Mean and Variance of the Binomial Distribution
The binomial distribution for a random variable X with parameters n and p represents the sum of n independent variables Z which may assume the values 0 or 1. If the probability that each Z variable assumes the value 1 is equal to p, then the mean of each variable is equal to 1*p + 0*(1-p) = p, and the variance is equal to p(1-p). By the addition properties for independent random variables, the mean and variance of the binomial distribution are equal to the sum of the means and variances of the n independent Z variables, so meam mu=np,
variance sigma square=np(1-p)
Normal Approximations for Counts and Proportions
For large values of n, the distributions of the count X and the sample proportion P are approximately normal. This result follows from the Central Limit Theorem. The mean and variance for the approximately normal distribution of X are np andnp(1-p) identical to the mean and variance of the binomial(n,p) distribution. Similarly, the mean and variance for the approximately normal distribution of the sample proportion are p and (p(1-p)/n).