Here are some practice questions to help with preparing for the final exam.

1. Here is the proportion of black pigmentation on the nose (0 to 1) and the age (years) for each
of 6 male lions that we will treat as a simple random sample:
proportion black 0.14 0.30 0.59 0.48 0.79 0.51
age 1.5 4.3 5.4 7.3 8.8 5.4

(a) Find the correlation, r, for these data.

(0) 0.049, (1) 0.118, (2) 0.131, (3) 0.302, (4) 0.390,
(5) 0.427, (6) 0.472, (7) 0.730, (8) 0.907, (9) 0.940
(b) Assuming a straight-line model, estimate the intercept (β0 ) of the line
age = (intercept β̂0 ) + (slope β̂1 ) × (proportion black).
(0) 0.049, (1) 0.118, (2) 0.023, (3) 0.082, (4) 0.733,
(5) 5.140, (6) 6.928, (7) 10.073, (8) 12.884, (9) 14.940
(c) Assuming a straight-line model, estimate the slope (β1 ) of the line
age = (intercept β̂0 ) + (slope β̂1 ) × (proportion black).
(0) 0.049, (1) 0.118, (2) 0.023, (3) 0.082, (4) 0.733,
(5) 5.140, (6) 6.928, (7) 10.073, (8) 12.884, (9) 14.940
(d) A larger sample of 32 lions led to this regression model:
age = 0.88 + 10.65×(proportion black).
What age is predicted by this model for a male lion whose nose is 40% black? (That is,
it has proportion 0.40 of black pigment.)
(0) 0.049, (1) 0.118, (2) 0.023, (3) 0.082, (4) 0.733,
(5) 5.140, (6) 6.928, (7) 10.073, (8) 12.884, (9) 14.940
(e) A larger sample of 32 lions led to this regression model:
age = 0.88 + 10.65×(proportion black).
Which correlation, r, is possible for the data that led to this model? Hint: You do not
have the data, so you cannot calculate r; but enough information is provided to answer
the question.
(0) -1, (1) -0.5, (2) 0, (3) 0.88, (4) 10.65,
(f) Returning to the sample of size 6 lions in this problem’s introduction, and supposing the
population of ages is normally distributed, find the center of a 95% confidence interval
for the population mean age.
(0) 0.23, (1) 0.47, (2) 2.01, (3) 2.51, (4) 2.64,
(5) 5.45, (6) 7.33, (7) 8.81, (8) 10.90, (9) 12.08
(g) Find the error margin of the same 95% confidence interval for the population mean age.
(0) 0.23, (1) 0.47, (2) 2.01, (3) 2.51, (4) 2.64,
(5) 5.45, (6) 7.33, (7) 8.81, (8) 10.90, (9) 12.08

1
2. Consider the Wilcoxon rank sum test. Suppose two independent simple random samples of
sizes nX and nY are drawn from populations X and Y that have the same shape. Suppose
there are no ties in the data.

(a) Suppose nX = 2 and nY = 3. Find RXmin = the minimum possible rank sum for the X
sample.
(0) 0, (1) 1, (2) 2, (3) 3, (4) 4, (5) 5, (6) 6, (7) −1, (8) −2, (9) −3,

(b) Suppose the X sample is 11, 14, and the Y sample is 13, 15, 12. Find RX = the rank
sum for the X sample.
(0) 0, (1) 1, (2) 2, (3) 3, (4) 4, (5) 5, (6) 6, (7) 7, (8) 8, (9) 9,

(c) Find UX = the value of the test statistic based on the X sample.
(0) 0, (1) 1, (2) 2, (3) 3, (4) 4, (5) 5, (6) 6, (7) 7, (8) 8, (9) 9,

(d) Suppose we are testing H0 : the two populations are identical vs. HA : the X population
is shifted relative to the Y population. Find the p-value corresponding to UX = 3.
1 2 3 4 5 6
(0) 10 = 0.01, (1) 10 = 0.2, (2) 10 = 0.3, (3) 10 = 0.4, (4) 10 = 0.5, (5) 10 = 0.6, (6)
7 10 11 12
10 = 0.7, (7) 10 = 1.0, (8) 10 = 1.1, (9) 10 = 1.2,

(e) Why is running this test at the usual level α = 0.05 unhelpful? (Hint: What is the
smallest possible p-value?)

(0) The smallest possible p-value is 0.0, but this is unlikely, so we will only rarely reject
H0 .
(1) The smallest possible p-value is 0.1, which is greater than α = 0.05, so we can never
reject H0 .
(2) The smallest possible p-value is 0.1, which is greater than α = 0.05, so we will always
reject HA .
(3) The smallest possible p-value is 0.2, which is greater than α = 0.05, so we can never
reject H0 .
(4) The smallest possible p-value is 0.2, which is greater than α = 0.05, so we will always
reject HA .
(5) The smallest possible p-value is 0.3, which is greater than α = 0.05, so we can never
reject H0 .
(6) The smallest possible p-value is 0.3, which is greater than α = 0.05, so we will always
HA .
(8) The smallest possible p-value is 1.0, so we will always affirm H0 .
(9) The smallest possible p-value is 1.2, so we will always affirm H0 .

2
3. A simple random sample of three people are tested for strength in a one-hand bicep curl
with a dumbbell. Each person is measured for right-handed maximum weight curled and
left-handed maximum weight curled:
Person Right Left
Jack 40 36
Jill 31 28
Barry 50 45

(a) Suppose it is reasonable to assume normality. Find the observed value of the test statistic
for a test of whether there is a difference in population mean right and left arm strengths.
(0) 0.049, (1) 0.118, (2) 0.023, (3) 0.082, (4) 0.543,
(5) 4.273, (6) 6.928, (7) 10.073, (8) 12.884, (9) 14.940

(b) Find the observed value of the test statistic for a test of whether there is a difference in
population median right and left arm strengths.
(0) 0.000, (1) 0.023, (2) 0.543, (3) 1.000, (4) 1.538,
(5) 2.000, (6) 2.839, (7) 3.000, (8) 3.403, (9) 4.000

4. Each part of this question consists of a fragment of R code. Choose the best description, from
among these, of the output of that code fragment.

(0) Z, the test statistic for a Z test of H0 : µX = 0

(1) t, the test statistic for a Welch’s t test of H0 : µX − µY = 0 vs. HA : µX − µY 6= 0
(2) p-value for a bootstrap test of H0 : µX − µY = 0 vs. HA : µX − µY 6= 0
(3) center of a level 1 − α confidence interval for µX
(4) error margin of a level 1 − α confidence interval for µX

Here are the question parts.

(a) mean(x)
(b) (sum(t.hats < -abs(t.obs)) + sum(t.hats > abs(t.obs))) / B
(c) n = length(x); -qt(alpha/2, df=n-1) * sd(x) / sqrt(n)
(d) (mean(x) - 0) / (sigma / sqrt(length(x)))
(e) ((mean(x) - mean(y)) - 0) / sqrt(sd(x)^2 / length(x) + sd(y)^2 / length(y))

3
5. A study investigated the effect of exercise on attempted weight loss. It randomly selected
12 people from among a large company’s employees, and then randomly assigned 4 to do no
exercise program, 4 to do a mild walking program, and 4 to do an intensive bicycling program.
Here are the weight losses after six weeks of participants in the study (note that a negative
loss is a gain):
Exercise Weight loss (pounds)
None 1.5 −0.8 −0.3 0.0
Walk 0.7 1.7 3.0 1.3
Bike 4.5 4.0 3.7 2.7
Here is a partial ANOVA table for these data from R:

--------- Df Sum Sq Mean Sq F value Pr(>F)

program (b) 26.432 13.216 (d) 0.00113
Residuals 9 (c) 0.835

(a) State a suitable null hypothesis for this study’s ANOVA.

(0) H0 : exercise level and weight loss are independent.
(1) H0 : weight loss depends on exercise level
(2) H0 : Mnone = Mwalk = Mbike , where Mnone is the population median weight loss for
employees in no exercise program, with similar definitions for the other two M ’s.
2
(3) H0 : σnone 2
= σwalk 2 , where σ 2
= σbike none is the population variance of weight losses
for employees in no exercise program, with similar definitions for the other two σ 2 ’s.
(4) H0 : µnone = µwalk = µbike , where µnone is the population mean weight loss for
employees in no exercise program, with similar definitions for the other two µ’s.
(b) What is the value of the missing exercise program degrees of freedom?
(0) 0, (1) 1, (2) 2, (3) 3, (4) 4,
(5) 5, (6) 6, (7) 9, (8) 11, (9) 12,

(c) What is the value of the missing SSE?

(0) 0.049, (1) 0.118, (2) 0.023, (3) 0.093, (4) 0.543,
(5) 4.720, (6) 6.289, (7) 7.515, (8) 12.841, (9) 15.828

(d) What is the value of the missing F statistic?

(0) 0.049, (1) 0.118, (2) 0.023, (3) 0.093, (4) 0.543,
(5) 4.720, (6) 6.289, (7) 7.515, (8) 12.841, (9) 15.828

4
 (e) Supposing the ANOVA assumptions are met, what conclusion do you draw in the context
of the problem? Choose the best answer.
(0) Do not reject H0 . The data are not strong evidence exercise level and weight loss
are not independent.
(1) Reject H0 . The data are strong evidence exercise level and weight loss are not
independent.
(2) Do not reject H0 . The data are not strong evidence median weight loss is not the
same for all three populations.
(3) Reject H0 . The data are strong evidence median weight loss is not the same for all
three populations.
(4) Do not reject H0 . The data are not strong evidence weight loss variance is not the
same for all three populations.
(5) Reject H0 . The data are strong evidence weight loss variance is not the same for all
three populations.
(6) Do not reject H0 . The data are not strong evidence mean weight loss is not the
same for all three populations.
(7) Reject H0 . The data are strong evidence mean weight loss is not the same for all
three populations.
(f) Consider an ANOVA test for a data set with three sample means: 11, 12, and 13. For
which three corresponding sample standard deviations would the p-value be smallest?
(Hint: No calculation is necessary.)
(0) 0.11, 0.12, 0.13
(1) 0.11, 0.11, 0.11
(2) 11, 12, 13
(3) 11, 11, 11

6. A survey asked several thousand teens “What do you think are the chances you will be married
in the next ten years?” Here is a contingency table of the responses by biological sex:
Female Male Total
Almost no chance 119 103 222
Some chance, but probably not 150 171 321
A 50-50 chance 447 512 959
A good chance 735 710 1445
Almost certain 1174 756 1930
Total 2625 2252 4877

(a) Under H0 : “Biological sex and perceived chance of marriage are independent”, find the
approximate expected count of females who respond “Almost certain.”
(0) 102, (1) 108, (2) 891, (3) 1039, (4) 1174,
(5) 2084, (6) 3809, (7) 4771, (8) 12841, (9) 15828

5
(b) The expected count for males who respond “A 50-50 chance” is 442.8. The chi-square
statistic is a sum of ten terms. The term in the chi-square statistic for males who respond
“A 50-50 chance” is .
(0) 10.2, (1) 10.8, (2) 89.1, (3) 103.9, (4) 117.4,
(5) 208.4, (6) 380.9, (7) 477.1, (8) 1284, (9) 1583

(c) Find the degrees of freedom for the chi-square test for this contingency table.
(0) 0, (1) 1, (2) 2, (3) 3, (4) 4,
(5) 5, (6) 6, (7) 7, (8) 8, (9) 10

(d) Software gives a chi-square statistic of 69.8 for the whole table. Find the P -value.
(0) p-value < .001
(1) .001 < p-value < .01
(2) .01 < p-value < .05
(3) .05 < p-value < .10
(4) .10 < p-value < .25
(5) .25 < p-value
(e) What conclusion do you draw in the context of the problem?

6
7. Mark each statement as either

(0) false, or (1) true

(a) In a hypothesis test, the p-value is the probability that H0 is true.

(b) If Z ∼ N (0, 1), and P (Z < z) = 0.4207 for some number z, then P (Z < −z) = 0.5793.

(c) Increasing the confidence level from 90% to 95%, while keeping everything else the same,
increases a confidence interval’s margin of error.

(d) If events A and B are independent, then P (A and B) = P (A) · P (B).

(e) In a hypothesis test with significance level α = 0.05 and power 1 − β = 0.80, the
probability H0 is rejected when H0 is true is 0.80.

(f) A researcher who randomly samples 50 students at a Badger football game to test
whether the distribution of the class (freshman, sophomore, junior, senior, graduate)
of students at the game is the same as the distribution (from the registrar) across all
UW students should use a chi-squared test for independence.

(g) In a Wilcoxon Rank Sum test on a sample 10, 20, 40 from population A and a sample
30, 50, 60 from population B, the observed value of the test statistic calculated from the
A sample ranks is 1.

(h) zα/2 < tn−1,α/2 for each n > 2 and each α such that 0 < α < 0.5.

(i) In a test of H0 : µX − µY = 0 against HA : µX − µY 6= 0 from two independent simple

random samples from populations X and Y , if σX2 and σ 2 are known to be very different,
Y
then a two-sample t test using a pooled variance estimate is appropriate.

(j) For a simple random sample X1 , . . . , Xn from a population

P with mean µ and standard
deviation σ, the population standard deviation of X̄ = n1 ni=1 Xi is √σn

(k) Doubling the largest value in a large sample of positive measurements increases the
sample standard deviation.

(l) Doubling the largest value in a large sample of positive measurements increases the
interquartile range.

(m) Doubling the largest value in a large sample of positive measurements increases the
sample mean.

(n) A bootstrap test relies on many samples of size n from the population.

7
8. Consider these statistical methods:

(0) Z test for µ (for one sample or differences from paired data)
(1) t test for µ (for one sample or differences from paired data)
(2) bootstrap test for µ (for one sample or differences from paired data)
(3) 2-sample t test for µX − µY
(4) Welch’s t test for µX − µY
(5) bootstrap test for µX − µY
(6) ANOVA
(7) χ2 test for goodness of fit
(8) χ2 test for independence
(9) correlation and linear regression

For each question, below, write the digit corresponding to the most appropriate method,
above. It is ok to use a method more than once.

(a) Fred interviews a random sample of 20 students in his Intro Botany class and asks
each student, “How many minutes did you study for the final exam?” and “How many
points did you earn on the exam?” He makes a scatter plot of (minutes, points) for
the 20 students and notices that the points are, more-or-less, along a line. What is the
relationship between minutes of study time and points earned?
(b) Lisa randomly selects 20 cars in the large UW parking “Lot 60.” For each car, she
measures the front-left tire pressure and records the difference between her measurement
and the pressure specified by the manufacturer. (Note: The specified pressure is not a
measurement). She makes a normal QQ plot of these differences and notices that the
points are, more-or-less, along a line. For the Lot 60 population, is the average front left
tire under-inflated by more than 5 psi?
(c) Andre counts the number of each grade of evergreen tree seedling in the box of 50 he
bought from a nursery. The nursery advertised that it would select trees in his box
randomly from a population with 50% graded “best,” 30% graded “good,” and 20%
graded “satisfactory.” Are the counts evidence that his 50 seedlings weren’t randomly
taken from the promised population?
(d) Diamond randomly selects 10 students on each floor of her 4-floor dorm and asks the 40
students how much sleep they got the night before. Is there a difference in population
average sleep times across the 4 floors?
(e) Hao randomly selects 50 right-handed students from his dorm. He counts the number of
times each student can bounce a ping-pong ball on a paddle before it hits the ground.
He tests each student twice, once holding the paddle in the right hand and once in the
left. Is there a difference between the two population mean numbers of bounces?

9. Suppose the army has packages of 200 doses of a pain-killing drug in dispensaries all over
the country. The drug’s label says its mean opiod equivalance is 2.0, meaning it is 2.0

8
 times stronger than morphine. The population of opiod equivalances in each package is
approximately normal with standard deviation close to 0.1.
The drug can degrade over time if stored incorrectly. Instructions for testing a package are
sent to each dispensary. They describe how to take a simple random sample of n doses from
a package (for some n with 0 < n < 200), measure the opiod equivalence of those n doses,
and test H0 : µ = 2.0 vs. HA : µ 6= 2.0, where µ is the package mean, at significance level α.

(a) Assign the correct probability, one of α, 1 − α, βµA , and 1 − βµA , where βµA is the
probability of a type II error when µ = µA , to each situation described below.
ˆ The package has mean 2.0, but the test asserts the mean is different. The
package is discarded unnecessarily.
ˆ The package has some mean µA different than 2.0, and the test asserts it
has a mean different than 2.0. The defective package is discarded.
ˆ The package has mean 2.0, and the test does not assert the mean is different.
The package (now containing 200 − n doses) is marked as tested and put back into
use.
ˆ The package has some mean µA different than 2.0, and the test does not
assert it has a mean different than 2.0. The defective package (now containing only
200 − n doses) is marked as tested and put back into use.
(b) Supposing the true mean for a package is 1.9, find the power for a test at level .03 with
sample size 16.
(c) Supposing the true mean for a package is 1.9, what sample size n is required for the test
at level .03 to have power .67?
(d) A 97% confidence interval for µ for a particular package is found to be (1.8, 1.9). At
significance level α = 0.03, what conclusion would you draw about H0 for this package?

10. A couple is considering developing an energy plantation on their land by planting trees for
firewood. They visit an existing firewood supplier and purchase air dried hickory logs and oak
logs that they decide to treat as independent simple random samples from the local forests.
They cuts and pack the logs into many 1-cubic foot boxes for weighing. Here are the resulting
weights in pounds:
tree sample mean sample standard deviation sample size
hickory 51 3 40
oak 44 2.5 154
QQ plots are compatible with normal populations. Consider a test to decide whether these
data strong evidence, at significance level 0.05, that cubic feet of wood from hickory logs are
heavier than those from oak logs.

(a) What hypotheses are suitable?

(b) What assumptions are required?
(c) What is the value of the test statistic?

9
 (d) Find the p-value corresponding to a test statistic value of 2.326 for the alternative
hypothesis constructed using “>”. (This may not be the alternative hypothesis or test
statistic value you found in the previous steps.)

(e) Do you reject H0 ? yes / no (circle one)

11. A neuroscientist gathered a sample, that we will treat as a simple random sample, of 80
Drosophila Melanogaster (fruitflies) and found that 55 reacted when prodded with a needle
heated at 41◦ C. If more than 62% of the flies react, she will need to recalibrate the heated
stimulus.

(a) What hypotheses should she test to find out whether recalibration is necessary?

(b) Choose an appropriate test and find the p-value.

(c) Make decision at the α = .05-level in the context of the problem.

12. The sunny south wall of a house was covered with boxelder bugs. A researcher enclosed the
wall in plastic to capture all the bugs. The lengths of a simple random sample of 10 bugs
were measured in mm: 10.9, 11.2, 11.6, 12.2, 12.2, 12.5, 12.6, 12.9, 13.1, 13.9. For these data,
x̄ ≈ 12.31 and s ≈ 0.90. Here is a QQ plot:
Normal Q−Q Plot
14.0
13.5
13.0
Sample Quantiles

12.5
12.0
11.5
11.0

−1.5 −1.0 −0.5 0.0 0.5 1.0 1.5

Theoretical Quantiles

(a) Is it plausible that the population of lengths is normally distributed? Why or why not?

(b) Suppose the population of lengths is normal. Find a 95% confidence interval, expressed
as “center ± error margin,” for the unknown population mean length.

(c) Find the test statistic and p-value for a test of H0 : M = 11 vs. HA : M > 11, where M
is the population median length. Draw a conclusion using significance level 0.05.
(Here are the data again: 10.9, 11.2, 11.6, 12.2, 12.2, 12.5, 12.6, 12.9, 13.1, 13.9.)

13. For each of the following statements, indicate whether the statement in bold is true or false
by circling the appropriate choice.

10
 (a) Suppose we wish to use the bootstrap to make a confidence interval for the mean of a
population with an unknown distribution. We take a sample from the population of size
∗
n. We use the statistic t̂ = x̄√s−x̄
∗ , where x̄ is the sample mean of the original sample, x̄
∗
n
is the sample mean of the resampled data, and s∗ is the sample standard deviation of
the resampled data. Suppose we resample B times. The larger the value of B, the
more like a normal the distribution of t̂ will be, because of the central limit
theorem. TRUE FALSE

(b) The p-value is the probability that the null hypothesis is true. TRUE FALSE

(c) When performing a linear regression, the y variable (response variable) must
be normally distributed in order for the t-test for the slope to be valid. TRUE
FALSE

(d) A p-value smaller than α calls for rejecting the null hypothesis. TRUE
FALSE

14. A certain chemical reaction was run three times at each of three temperatures. The yields,
expressed as a percent of a theoretical maximum, were as follows:

Temperature (◦ C) Yields Row Mean

70 81.2 82.6 77.4 80.4
80 93.3 88.9 86.0 89.4
90 87.8 89.2 88.5 88.5
Overall Mean = 86.1

Consider testing H0 : µ70◦ = µ80◦ = µ90◦ . Suppose the assumptions of ANOVA are met.

(a) Find the treatment sum of squares.

(b) The error sum of squares is about 42.48. Find the error mean square.

(c) The F statistic is about 10.4. Find the p-value (providing a small range is ok) and make
a conclusion in the context of the problem.

11
15. At an assembly plant for hospital beds, monitoring of weld quality yields the following data:

Number of Welds
High Quality Moderate Quality Low Quality Total
Day Shift 467 191 42 700
Evening Shift 445 171 34 650
Night Shift 254 129 17 400
Total 1166 491 93 1750

(a) The observed count for the Day Shift / Low Quality cell is 42. What is the expected
count under “H0 : Shift and Quality are independent”?

(b) What is the contribution to the X 2 statistic from this cell?

(c) Find the degrees of freedom for the chi-square test for this contingency table.

(d) Software gives a chi-square statistic of 5.76 for the whole table. Find the P -value.

p-value =

(e) What conclusion do you draw in the context of the problem?

12
16. A roller coaster holds 60 riders. It’s maximum safe total rider weight is 12,000 pounds. The
weights of adult U.S. men have mean 194 and standard deviation 68 pounds. If a random
sample of 60 men ride the coaster, what is the probability the maximum safe weight will be
exceeded?

17. The average numbers of defectives produced by two press machines are given below:

Population Observation
Machine 1 10.5 7.1 5.4 8.3 4.2 7.5 3.6 5.4 5.4 5.2 10.3 9.5
Machine 2 5.4 9.9 5.9 5.3 6.2 6.3 9.6 10 9.4 7.9 6.8 8.5

Here is R code to put these data in vectors:

one = c(10.5,7.1,5.4,8.3,4.2,7.5,3.6, 5.4,5.4,5.2,10.3,9.5)

two = c(5.4 ,9.9,5.9,5.3,6.2,6.3,9.6,10 ,9.4,7.9, 6.8,8.5)

We are interested in whether Machine 1 is better than Machine 2 in the sense of the number
of defectives.

(a) Assume the two populations are normal and the two population variances are equal. Let
µ1 be the population mean for Machine 1 and µ2 be the population mean for Machine
2. Perform the test at α = 0.01.

(b) Find the 99% confidence interval for µ1 − µ2 .

13
18. Mehta and Deopura (1995) studied the mechanical properties of spun PET-LCP blend fibers.
They believe that the modulus (the response) depends on the percent of PET in the blend.
The data is as follows:

Percent PET (xi ) Modulus (yi )

100 2.12
97.5 2.26
95 2.57
90 3.26
80 3.46
50 4.54
0 8.5

(a) Assuming a straight-line model, compute estimates of the intercept (β0 ) and slope (β1 ).
(b) What is the equation of the line relating the modulus (the response) and the percent of
PET in the blend?
(c) Test whether the true model slope is 0 using a t-test at α = 0.05. Make a conclusion in
the context of the problem.
(d) If the percent of PET is 85, what does the model predict will be the modulus?

14
19. Consider these graphical representations of samples from two different populations:

QQPlot: A Histogram: A

12
●

1.0
●
●● ●
●
●●
●

10
●
●

Sample Quantiles
●●
●●

0.0
●●

8
Frequency
●●●
●●●●
●
●
●●●

6
●●
●●●
●●
−1.0
●●
●●

4
●●
●●●
●

2
●
−2.0

● ●

0
−2 −1 0 1 2 −2 −1 0 1

Theoretical Quantiles Data

QQPlot: B Histogram: B

25
●
6

20
5
Sample Quantiles

Frequency
4

15
●●
●
3

●
10
●
●
●
2

●●
●●●●●●
●●●
5

●●
●●●
1

●●●
●
●●●●●●●●
●●●●●●
● ● ● ●●●●
0
0

−2 −1 0 1 2 0 1 2 3 4 5 6 7

Theoretical Quantiles Data

Answer the following questions:

a) Which histogram (A or B) pertains to QQPlot A? Why?

b) Describe Histogram B in terms of skewness.

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20. Consider two distinct coins which were found backstage after a magic show. You wish to test
whether these two coins have the same probability of landing on Heads with α = .02. To this
end, you flip the first coin 24 times, giving 6 heads, and the second coin 30 times, giving 8
heads. Answer the following questions:

a) Write down null and alternative hypotheses which reflect this situation in terms of π1
and π2 , the true probabilities of heads for the first and second coin, respectively.
b) Find an appropriate test statistic and p-value make a decision.

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