A fair coin is tossed until the first head occurs. Do this experiment T = 10; 100; 1,000; 10,000 times in R, and plot the relative frequencies of this occurring at the ith toss, for suitable values of i. Compare this plot to the pmf that should govern such an experiment. Show that they converge as T increases. What is the expected number of tosses required? For each value of T, what is the sample average of the number of tosses required?

Describe the algorithm to generate random numbers from an arbitrary discrete distribution with finite number of outcomes.

• What is the level of significance? • What are Type I and Type II errors? • Interpreting and determining p-values • What is the relationship between sample size and power? • Understand the difference between a p-value and a confidence interval—strengths and weaknesses

What are the percentages of scores that fall between:

a) -1 and 0/0 and 1

b) -1 and -2/1 and 2

c) In the tails of a normal distribution.

• Define each type of variable: dichotomous, ordinal, categorical, continuous • Define the following study designs: Randomized controlled trial, prospective cohort study, case-control study, crossover study. • Define in dependent versus independent samples.

The following table is protein concentration in three groups of trypsin secretion.

<=50	51-1000	>1000
1.7	1.4	2.9
2	2.4	3.8
2	2.4	4.4
2.2	3.3	4.7
4	4.4	5
5	4.7	5.6
6.7	6.7	7.4
7.8	7.6	9.4
4	9.5	10.3

Perform ANOVA test BY HAND to test if there is a significant difference among these groups

7. The following data were collected in a clinical trial to compare a new drug to a placebo for its effectiveness in lowering total serum cholesterol. New Drug (n=75) Placebo (n=75) Total Sample (n=150) Mean (SD) Total Serum Cholesterol 182.0 (24.5) 206.3 (21.8) 194.15 (23.2) % Patients with Total Cholesterol < 200 78.0% 65.0% 71.5% a) Generate the 95% confidence interval for the difference in mean total cholesterol levels between treatments b) Generate a 95% confidence interval for the difference in proportions of all patients with total cholesterol < 200. c) How many patients would be required to detect the difference in proportions observed in the current study with a confidence interval of 95%.  

High daily temperatures and coffee sales (in hundreds of dollars) were recorded at a local Starbucks for eight randomly selected days. Results are listed in the table below. Is here a correlation between temperature and coffee sales? Use α=.05. What is the best predicted sales total for a day with a high temperature of 60° F?
x(° F) 32 39 51 60 65 72 78 81
y(hundreds of dollars) 26.2 24.8 19.7 20.0 13.3 13.9 11.4 11.2

r:
critical value:
Significant linear correlation? Yes or No?
Regression equation:
Best predicted time on 60° day:
What proportion of variation in sales is due to temperature?

The probability is 0.3 that a traffic fatality involves an intoxicated or​ alcohol-impaired driver or nonoccupant. In eight traffic​ fatalities, find the probability that the​ number, Y, which involve an intoxicated or​ alcohol-impaired driver or nonoccupant is a. exactly​ three; at least​ three; at most three. b. between two and four​, inclusive. c. Find and interpret the mean of the random variable Y. d. Obtain the standard deviation of Y.

Please answer form 6-14

I. Consider the random experiment of rolling a pair of dice. Note: Write ALL probabilities as reduced fractions or whole numbers (no decimals).

1) One possible outcome of this experiment is 5-2 (the first die comes up 5 and the second die comes up 2). Write out the rest of the sample space for this experiment below by completing the pattern:

1-1	2-1
1-2
1-3
1-4
1-5
1-6

2) How many outcomes does the sample space contain? _____________

3) Draw a circle (or shape) around each of the following events (like you would to circle a word in a word search puzzle). Label each event in the sample space with the corresponding letter. Event A has been done for you.

A: Roll a sum of 3.
B: Roll a sum of 7.
C: Roll a sum of at least 10.

D: Roll doubles.
E: Roll snake eyes (two 1’s). F: First die is a 4.

4) Find the following probabilities:
P(A) = _________ P(B) = _________ P(C) = _________

P(D) = _________ P(E) = _________ P(F) = _________

5) The conditional probability of B given A, denoted by P(B|A), is the probability that B will occur when A has already occurred. Use the sample space above (not a special rule) to find the following conditional probabilities:

P(D|C) = _________ P(E|D) = _________ P(D|E) = _________ P(A|B) = _________ P(C|F) = _________

6) Two events are mutually exclusive if they have no outcomes in common, so they cannot both occur at the same time.

Are C and E mutually exclusive? ___________
Find the probability of rolling a sum of at least 10 and snake eyes on the same roll, using the

sample space (not a special rule).
P(C and E) = __________

Find the probability of rolling a sum of at least 10 or snake eyes, using the sample space. P(C or E) = __________

7) Special case of Addition Rule: If A and B are mutually exclusive events, then P(A or B) = P(A) + P(B)

Use this rule to verify your last answer in #6:
P(C or E) = P(C) + P(E) = ________ + ________ = _________

8) Are C and F mutually exclusive? __________ Using sample space, P(C or F) = _________ 9) Find the probability of rolling a “4” on the first die and getting a sum of 10 or more, using the

sample space.
P (C and F) = ________

10) General case of Addition Rule: P(A or B) = P(A) + P(B) – P(A and B) Use this rule to verify your last answer in #8:

P(C or F) = P(C) + P(F) – P(C and F) = ________ + ________ − ________ = _________

11) Two events are independent if the occurrence of one does not influence the probability of the other occurring. In other words, A and B are independent if P(A|B) = P(A) or if P(B|A) = P(B).

Compare P(D|C) to P(D), using the sample space: P(D|C) = ________ . P(D) = ________ .
Are D and C independent? _________
When a gambler rolls at least 10, is she more or less likely to roll doubles than usual? ___________ Compare P(C|F) to P(C), using the sample space: P(C|F) = ________ . P(C) = ________ .

Are C and F independent? __________
12) Special case of Multiplication Rule: If A and B are independent, then P(A and B) = P(A) · P(B).

Use this rule to verify your answer to #9:
P(C and F) = P(C) • P(F) = ________ · ________ = ________ .

13) Find the probability of rolling a sum of at least 10 and getting doubles, using the sample space. P(C and D) = ________ .

14) General case of Multiplication Rule: P(A and B) = P(A) · P(B|A). Use this rule to verify your answer to #13:

P(C and D) = P(C) • P(D|C) = ________ · ________ = ________ .

Please show your work - the answer is d) but I'm not sure why. Thank you!

The time it takes to complete a Sta220 term test is normally distributed with a mean

of 100 minutes with standard deviation of 14 minutes. How much time should be

allowed if we wish to ensure that at least 9 out of 10 students (on average) can

complete it? (round to the nearest minute)

A) 115

B) 116

C) 117

D) 118

E) 119

*Please provide r studio code/file*

1) Find the equation of the best fit line using least squares
linear fit of x,y:
set.seed(88)
x <- 1:100
y <- jitter(1.5*x+8,amount=10)

2) For question 1, Draw the P=0.95 prediction intervals for y
when x=1:150

3) For question 1, Find the equation of the best fit line using
median-based linear fit of x,y.

4) For question 3, draw the P=0.95 prediction interval for y
# when x=1:150

*Please provide r studio file/code*

Question:

Test the equality of means of populations X,Y,Z using ANOVA:
set.seed(88)
dta <- data.frame(v = c(2+2*rnorm(100),
3+3*rnorm(100),
4+4*rnorm(100)),
id = rep(c("x","y","z"),c(100,100,100)))

Please answer all parts if possible.

Let X ~ Geometric (p) where 0 < p <1

a) Show explicitly that this family is “very regular,” that is, that R0,R1,R2,R3,R4 hold.

R 0 - different parameter values have different functions.

R 1 - parameter space does not contain its own endpoints.

R 2. - the set of points x where f (x, p) is not zero and should not depend on p.

R 3. One derivative can be found with respect to p.

R 4. Two derivatives can be found with respect to p.

b) Find the maximum likelihood estimator of p, call it Yn for this problem.

c) Is Yn unbiased? Explain.

d) Show that Yn is consistent asymptotically normal and identify the asymptotic normal variance.

e) Variance-stabilize your result in (d) or show there is no need to do so.

f) Compute I (p) where I is Fisher’s Information.

g) Compute the efficiency of Yn for p (or show that you should not!).

Question:

Test the mean of population X for equality to zero (mu=0)
using the sample x and t-test at a significance level 0.05

set.seed(88)
x <- rt(150,df=2)