Question

Suppose a coin is randomly tossed n = 400 times, resulting in X = 240 Heads. Answer each of the following; show all work!

(a) Calculate the point estimate, and the corresponding two-sided 95% confidence interval, for the true probability pi = P(Heads), based on this sample.

(b) Calculate the two-sided 95% acceptance region for the null hypothesis H0: pi = 0.5 that the coin is fair.

(c) Calculate the two-sided p-value (without correction term) of this sample, under the null hypothesis H0: pi = 0.5 that the coin is fair.

(d) Explain below how EACH of your answers in (a), (b), and (c) can be used to conduct a formal test of the null hypothesis, and arrive at a formal “reject or retain” conclusion at the alpha = .05 significance level. Also interpret in the context of this experiment.

• 95% Confidence Interval:

• 95% Acceptance Region:

• p-value:

• Conclusion:

• Interpretation:

Answer #1

PROBABILITY QUESTION
A fair coin is tossed n times. Sn is the # of heads
after tossed. Show that P(Sn ≥ 3n/4) ≤ e -n/8
.

A coin is tossed 54 times and 39 heads are observed. Would we
infer that this is a fair coin? Use a 97% level confidence interval
to base your inference.
The sample statistic for the proportion of heads is: (3
decimals)
The standard error in this estimate is: (3
decimals)
The correct z* value for a 97% level confidence interval
is: (3 decimals)
The lower limit of the confidence interval is: (3
decimals)
The upper limit of the confidence interval is: (3
decimals)
Based on...

A coin is tossed 73 times and 30 heads are observed. Would we
infer that this is a fair coin? Use a 97% level confidence interval
to base your inference.
The sample statistic for the proportion of heads is: (3
decimals)
The standard error in this estimate is: (3
decimals)
The correct z* value for a 97% level confidence interval
is: (3 decimals)
The lower limit of the confidence interval is: (3
decimals)
The upper limit of the confidence interval is: (3
decimals)
Based on...

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

You flip a fair coin N=100 times. Approximate the probability
that the proportion of heads among 100 coin tosses is at least
45%.
Question 4. You conduct a two-sided hypothesis test (α=0.05):
H0: µ=25. You collect data from a population of size N=100 and
compute a test statistic z = - 1.5. The null hypothesis is actually
false and µ=22. Determine which of the following statements are
true.
I) The two-sided p-value is 0.1336.
II) You reject the null hypothesis...

(SHOW YOUR WORK!!!) A Coin is tossed 1000 times and 570 heads
appear. At ? = .05, test the claim that this is not a biased
coin.
a.) State the null and alternative hypotheses.
b.) Verify that the requirements are met for conducting the
hypothesis test.
c.) Conduct the test of hypothesis by using a P-value.

Let p denote the probability that a particular coin will show
heads when randomly tossed. It is not necessarily true that the
coin is a “fair” coin wherein p=1/2. Find the a posteriori
probability density function f(p|TN ) where TN is the observed
number of heads n observed in N tosses of a coin. The a priori
density is p~U[0.2,0.8], i.e., uniform over this interval. Make
some plots of the a posteriori density.

Assume p represents the probability that a particular
coin will show heads when randomly tossed. Don't assume its true
that the coin is a “fair” coin wherein p=1/2. Determine
the a posteriori probability density function
f(p|TN) where
TN is the observed number of heads n
observed in N tosses of a coin. The a priori
density is p~U[0.2,0.8], i.e., uniform over this
interval. Create some plots of the a posteriori
density.

A fair coin is flipped 400 times. Let X be the number of heads
resulting, find P[190<= X <= 200]
a) About 34%
b) About 95%
c) About 68%
d) About 25%
e) About 50%

A fair coin is tossed 4 times, what is the probability that it
lands on Heads each time?
You have just tossed a fair coin 4 times and it landed on Heads
each time, if you toss that coin again, what is the probability
that it will land on heads?
Give examples of two independent events.
Dependent events are (sometimes, always, never) (choose one)
mutually exclusive.
If you were studying the effect that eating a healthy breakfast
has on a...

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