Question

Find the probability of more than 30 heads in 50 flips of a fair
coin by using the normal approximation to the binomial
distribution.

a) Check the possibility to meet the requirements to use normal
approximation (show your calculation)

b) Find the normal parameters of the mean(Mu) and standard deviation from the binomial distribution.

c) Apply normal approximation by using P(X>30.5) with continuity correction and find the probability from the table of standard normal distribution.

Answer #1

n = 50, p = 0.5

**a)**

the normal approximation works best when p is close to 0.5 and
it becomes better and better when we have a larger sample size
*n*. This can be summarized in a way that the normal
approximation is reasonable if both
and
as well.

as p = 0.5 which is close to 0.5

and n*p = 50*0.5 = 25 >=10

and n*(1-p) = 50*(1-0.5) = 25 >= 10

as all the requirements are satisfied normal approximation will be a good estimate for binomial

**b)**

mu = n*p = 50*0.5 = 25 ( mean of binomial distribution)

sigma2 = n*p*(1-p) = 50*0.5*0.5 = 12.5 ( variance of binomial distribution )

sigma = (sigma2)1/2 = (12.5)1/2 = 3.535534

**c)**

= 0.05989746

Suppose a fair coin (P[heads] = ½) is flipped 50 times. What is
the probability of obtaining 30 or fewer heads using the normal
approximation to the binomial with the continuity correction
factor?
Use Minitab or some other software package to obtain the your
probability answer. Round your answer to two decimal points.

The number of heads in 100 flips of a fair coin is approximately
Normally distributed. To estimate the chance of getting between 49
and 51 heads (inclusive), what would the endpoints of the interval
be after a continuity correction?

Find the probability of getting 8 or more heads in 10 flips of a
coin

Gracefully estimate the probability that in 1000 flips of a fair
coin the number
of heads will be at least 400 and no more than 600.

suppose i flip a coin n=100 times and i obtain heads x=44 times.
assuming the coin is fair, calculate P(x>44) using the normal
approximation with continuity correction. x=44 significantly
low

As in the previous problem, a fair coin is flipped 28 times. If
X is the number of heads, then the distribution of X can be
approximated with a normal distribution, N(14,2.6), where the mean
(μ) is 14 and standard deviation (σ) is 2.6. Using this
approximation, find the probability of flipping 18 or 19 heads. You
may use the portion of the Standard Normal Table below.
z1.21.31.41.51.61.71.81.92.02.12.20.000.88490.90320.91920.93320.94520.95540.96410.97130.97720.98210.98610.010.88690.90490.92070.93450.94630.95640.96490.97190.97780.98260.98640.020.88880.90660.92220.93570.94740.95730.96560.97260.97830.98300.98680.030.89070.90820.92360.93700.94840.95820.96640.97320.97880.98340.98710.040.89250.90990.92510.93820.94950.95910.96710.97380.97930.98380.98750.050.89440.91150.92650.93940.95050.95990.96780.97440.97980.98420.98780.060.89620.91310.92790.94060.95150.96080.96860.97500.98030.98460.98810.070.89800.91470.92920.94180.95250.96160.96930.97560.98080.98500.98840.080.89970.91620.93060.94290.95350.96250.96990.97610.98120.98540.98870.090.90150.91770.93190.94410.95450.96330.97060.97670.98170.98570.9890

For a fair coin toss the probability of Heads is, of course
50%.
a.) What is the standard deviation for
the sampling distribution for the sample proportion of Heads in a
sample size of n=100?
Answer: ____________.
b.) How large a sample would be
required in order to get a standard deviation for less than
0.01?
Answer: ________________.

Use the normal approximation of the binomial to find the
probability of getting between 12 to 16 heads in 36-coin flips.

Deriving fair coin flips from biased coins: From coins with
uneven heads/tails probabilities construct an experiment for which
there are two disjoint events, with equal probabilities, that we
call "heads" and "tails".
a. given c1 and c2, where c1 lands heads up with probability 2/3
and c2 lands heads up with probability 1/4, construct a "fair coin
flip" experiment.
b. given one coin with unknown probability p of landing heads
up, where 0 < p < 1, construct a "fair...

Fair Coin? A coin is called fair if it
lands on heads 50% of all possible tosses. You flip a game token
100 times and it comes up heads 42 times. You suspect this token
may not be fair.
(a) What is the point estimate for the proportion of heads in
all flips of this token? Round your answer to 2 decimal
places.
(b) What is the critical value of z (denoted
zα/2) for a 99% confidence interval?
Use the...

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