Question

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 H0.

III) You fail to reject H0 and make a Type II error.

The answer for the first is 84% and the second is II and III only. I just want to know why its those. Thank you

Answer #1

N = 100

P[ head ] = 50% = 0.5 ( theoritical probability )

variance of p = p*(1-p)/n = 0.5*0.5/100 = 0.0025

standard deviation, SD = sqrt(variance ) = 0.05

we need to find probability of p being greater than 45%, where p follows N(0.5,0.0025)

P[ p > 0.45 ] = P[ ( p - 0.5 )/0.05 > ( 0.45 - 0.5 )/0.05 ] = P[ Z > -1 ] = 1 - P[ Z < -1 ] = 1 - 0.16 = 0.84

Probability of getting at least 45% heads in 100 trials is 45%

4. H0 : mu = 25

H1 : mu = 22 ( actually false )

N = 100

alpha = 0.05

z = -1.5

p value for two sided = 0.8664 ( from normal table )

reject H0 if p < alpha , here p ( 0.8664 ) > alpha ( 0.05 ), we failed to reject H0 and accept H1

and since, it is given that h1 is false, we accepted false H1 hence, made type II error.

The only correct option is III

If you flip a fair coin, the probability that the result is
heads will be 0.50. A given coin is tested for fairness using a
hypothesis test of H0:p=0.50H0:p=0.50 versus
HA:p≠0.50HA:p≠0.50.
The given coin is flipped 240 times, and comes up heads 143
times. Assume this can be treated as a Simple Random Sample.
The test statistic for this sample is z=
The P-value for this sample is
If we change the significance level of a hypothesis test from 5%...

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...

You are interested in testing if a coin is fair. That is, you
are conducting the hypothesis test:
H0 : P(Heads) = 0.5
Ha : P(Heads) ≠ .5
If you flip the coin 1,000 times and obtain 560 heads. Calculate
the P-value of this test, and decide whether or not to reject the
null hypothesis at the 5% significance level.

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. If the coin lands heads, you roll a fair
six-sided die 100 times. If the coin lands tails, you roll the die
101 times. Let X be 1 if the coin lands heads and 0 if the coin
lands tails. Let Y be the total number of times that you roll a 6.
Find P (X=1|Y =15) /P (X=0|Y =15) .

I flipped a coin 49 times and got heads only 18 times. I feel
like the coin is biased. I run a 1 sample z test proportion with
the null hypothesis set to .5 (50%) to represent a fair coin.
What's the p-value of the test?
Would you reject the null hypothesis or fail to reject the null
hypothesis?

A fair coin is tossed for n times independently. (i) Suppose
that n = 3. Given the appearance of successive heads, what is the
conditional probability that successive tails never appear? (ii)
Let X denote the probability that successive heads never appear.
Find an explicit formula for X. (iii) Let Y denote the conditional
probability that successive heads appear, given no successive heads
are observed in the first n − 1 tosses. What is the limit of Y as n...

You have a coin that you suspect is not fair (i.e., the
probability of tossing a head (PH) is not equal to the probability
of tossing a tail (PT) or stated in another way: PH ≠ 0.5). To
test yoursuspicion, you record the results of 25 tosses of the
coin. The 25 tosses result in 17 heads and 8 tails.
a) Use the results of the 25 tosses in SPSS to construct a 98%
confidence interval around the probability of...

Suppose you flip a fair coin 10 times. What is the
probability of the last two flips both being heads if you know that
the first eight flips were heads?

An experimenter flips a coin 100 times and gets 32 heads. Test
the claim that the coin is fair against the two-sided claim that it
is not fair at the level α=.01.
a) Ho: p = .5,
Ha: p ≠ .5; z = -3.60; Reject
Ho at the 1% significance level.
b) Ho: p = .5,
Ha: p < .5; z = -3.86;
Reject Ho at the 1% significance level.
c) Ho: p = .5,
Ha: p ≠ .5; z...

ADVERTISEMENT

Get Answers For Free

Most questions answered within 1 hours.

ADVERTISEMENT

asked 15 minutes ago

asked 24 minutes ago

asked 27 minutes ago

asked 42 minutes ago

asked 43 minutes ago

asked 47 minutes ago

asked 52 minutes ago

asked 53 minutes ago

asked 1 hour ago

asked 2 hours ago

asked 2 hours ago

asked 2 hours ago