# 6 question for stat

Question 1:Two dice are rolled. Define the following events:

A = ”sum of two dice equals 4”

B = ”sum of two dice equals 10”

C = ”at least one of the dice shows a 1”

(a) [2 pts] List all the outcomes of event A.

(b) [3 pts] Calculate the probability P (A|C).

(c) [3 pts] Are A and C disjoint? What about B and C? Explain.

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Let A and B be two events such that P (A) = 0.5, P (B) = 0.5 and P (A ∪ B) = 0.75. Let A be the

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complement of A and B be the complement of B.

(a) [3 pts] Are A and B independent events? Explain.

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(b) [2 pts] Calculate P (A ∪ B ).

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(c) [2 pts] Calculate P [(A ∩ B ) ∪ (A ∩ B)].

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Question 2:

Let X be a continuous random variable with the following probability density function

3×2 −1 ≤ x ≤ 0

f (x) =

0

otherwise

(a) [3 pts] Check if f(x) is a legitimate probability density function. Show steps.

(b) [2 pts] Calculate the probability P (X = 0.5)?

(c) [2 pts] Calculate the expected value E(X).

(d) [3 pts] Let Y = 2X. Find the 87.5th percentile of Y.

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Question 3:

Let X and Y be two random variables with the following joint distribution

p(x, y)

y

1

2

x

1

0.3

0.1

2

0.1

0.5

(a) [2 pts] Calculate E( X

Y ).

(b) [2 pts] Calculate P (X = 1|Y = 2).

(c) [2 pts] Calculate the correlation between X and Y, Corr(X, Y ).

(d) [1 pt] Linear transformations of variables will NOT change the correlation. That is, Corr(aX + b, cY +

d) = Corr(X, Y ), where a, b, c, d are any non-zero values. Circle the correct answer.

(a) True

(b) False

(e) [1 pt] If X and Y are independent variables, then Corr(X, Y ) = 0; and vice versa, Corr(X, Y ) = 0 also

implies that X and Y are independent. Circle the correct answer.

(a) True

(b) False

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Question 4:

Suppose (0.3, 0.5) is a 95% confidence interval for p, the true population proportion.

(a) [2 pts] What was p̂, the sample proportion?

(b) [2 pts] What was the margin of error?

(c) [2 pts] What was the value of Zα/2 ?

(d) [3 pts] How large a sample would be needed to have the width of a 95 % CI to be 0.3?

(e) [3 pts] Would a 99% confidence interval constructed from the same sample be wider or narrower than

the interval given in the question? Explain.

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Question 5:

A plan for an executive traveler’s club has been developed by an airline on the premise that 8 % of its

current customers would qualify for membership. A random sample of 500 customers yielded 50 who would

qualify. Using this data, test at level 0.01 the null hypothesis that the company’s premise is correct against

the alternative that it is not correct.

(a) [1 pt] What is the null hypothesis for this test? Circle the correct answer.

(a) p = 0.1

(b) p ̸= 0.08

(c) p ̸= 0.1

(d) p = 0.08

(b) [1 pt] What is the alternative hypothesis for this test? Circle the correct answer.

(a) p = 0.1

(b) p ̸= 0.08

(c) p ̸= 0.1

(d) p = 0.08

(c) [2 pts] Calculate the value of the test statistic (keep 2 digits after decimal). Show steps.

(d) [3 pts] Calculate the P-value. Would you reject null hypothesis that the company’s premise is correct

at significance level α = 0.01? Explain.

(e) [3 pts] What is the probability that when the test is used, the company’s premise will be judged correct

when in fact 12 % of all current customers qualify?

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Question 6:

A study was carried out on the effectiveness of Reiki therapeutic touch to manage pain in cancer patients.

Pain was assessed twice for each of the 16 cancer patients: once before and once after Reiki touch therapy.

For each of the 16 patients, the difference between each pair of pain scores was recorded (difference = after

– before). The sample mean and sample standard deviation of the difference are -1.02 and 2.64, respectively.

Does the data suggest that Reiki touch therapy lower the true average pain score for the patients? Draw the

conclusion at 0.05 level of significance.

(a) [1 pt] State the null hypothesis (H0 ) for this test.

(b) [1 pt] State the alternative hypothesis (Ha ) for this test.

(c) [2 pts] Calculate the test statistic value.

(d) [3 pts] Would you reject the null hypothesis at significance level α = 0.05? Explain.

(e) [1 pt] If you incorrectly reject the null hypothesis, what type of error did you make? Circle the correct

answer.

(a) Type I error

(b) Type II error

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Question 7:

Olestra is a fat substitute approved by the FDA for use in snack foods. Because there have been anecdotal reports of gastrointestinal problems associated with olestra consumption, a randomized, double-blind,

placebo-controlled experiment was carried out to compare olestra potato chips to regular potato chips with

respect to GI symptom. Among 480 individuals in the TG control group, 17.8% experienced an adverse GI

event, whereas among the 540 individuals in the olestra treatment group, 16% experienced such an event.

(a) [3 pts] Calculate a 95% CI for the difference between the true percentages for the two treatments.

(b) [4 pts] If the true percentages for the two treatments were 17% and 23%, respectively, what sample

sizes (m = n) would be necessary to detect such a difference with probability 0.9 at the 0.05 significance

level? [hint: probability 0.9 is the power of the test]

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