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)

Let x represent the number of mountain climbers killed each year. The long-term variance of x is approximately σ² = 136.2. Suppose that for the past 6 years, the variance has been s² = 107.1. Use a 1% level of significance to test the claim that the recent variance for number of mountain-climber deaths is less than 136.2. Find a 90% confidence interval for the population variance.

(a) What is the level of significance?


State the null and alternate hypotheses.

H_o: σ² = 136.2; H₁: σ² > 136.2H_o: σ² < 136.2; H₁: σ² = 136.2    H_o: σ² = 136.2; H₁: σ² ≠ 136.2H_o: σ² = 136.2; H₁: σ² < 136.2

(b) Find the value of the chi-square statistic for the sample. (Round your answer to two decimal places.)


What are the degrees of freedom?


What assumptions are you making about the original distribution?

We assume a uniform population distribution.We assume a exponential population distribution.    We assume a binomial population distribution.We assume a normal population distribution.

(c) Find or estimate the P-value of the sample test statistic.

P-value > 0.1000.050 < P-value < 0.100    0.025 < P-value < 0.0500.010 < P-value < 0.0250.005 < P-value < 0.010P-value < 0.005

(d) Based on your answers in parts (a) to (c), will you reject or fail to reject the null hypothesis?

Since the P-value > α, we fail to reject the null hypothesis.Since the P-value > α, we reject the null hypothesis.    Since the P-value ≤ α, we reject the null hypothesis.Since the P-value ≤ α, we fail to reject the null hypothesis.

(e) Interpret your conclusion in the context of the application.

At the 1% level of significance, there is insufficient evidence to conclude that the variance for number of mountain climber deaths is less than 136.2At the 1% level of significance, there is sufficient evidence to conclude that the variance for number of mountain climber deaths is less than 136.2    

(f) Find the requested confidence interval for the population variance. (Round your answers to two decimal places.)

lower limit
upper limit

Interpret the results in the context of the application.

We are 90% confident that σ² lies above this interval.We are 90% confident that σ² lies within this interval.    We are 90% confident that σ² lies outside this interval.We are 90% confident that σ² lies below this interval.

I believe that more patients are seen in a walk-in urgent care center leading up to lunch time, then slows down around late afternoon and then increases again approaching closing time. I recorded the number of patients seen every hour throughout the day. How do I apply the Goodness of Fit to test my hypothesis?

Level of Significance: α= 0.05

Time of day (Hour)	Number of patients
8	6
9	8
10	4
11	8
12	9
1	11
2	7
3	6
4	5
5	6
6	8
7	9

In a consumer research study, several Meijer and Walmart stores were surveyed at random and the average basket price was recorded for each. You wish to determine if the average basket price for Meijer is different from the average basket price for Walmart. It was found that the average basket price for 18 randomly chosen Meijer stores (group 1) was $49.451 with a standard deviation of $12.3146. Similarly, a random sample of 25 Walmart stores (group 2) had an average basket price of $56.847 with a standard deviation of $13.7821. Perform a two independent samples t-test on the hypotheses Null Hypothesis: μ₁ = μ₂, Alternative Hypothesis: μ₁ ≠ μ₂. What is the test statistic and p-value of this test? You can assume that the standard deviations of the two populations are statistically similar.

Question 14 options:

	1) Test Statistic: -1.813, P-Value: 0.9615
	2) Test Statistic: -1.813, P-Value: 0.0385
	3) Test Statistic: -1.813, P-Value: 1.9615
	4) Test Statistic: -1.813, P-Value: 0.077
	5) Test Statistic: 1.813, P-Value: 0.077

7% of Americans have an O negative blood type. A SRS of 400 Americans are surveyed. Using normal approximation to the binomial, what is the approximate probability that at least 34 of them have the o negative blood type?

Please show work!

A seed company is developing many strains of tomatoes by selective breeding. Trials of two similar but not identical strains with favorable qualities were done in two fields under similar conditions. The company would like to know if the population average weight of tomato for strain 2 (u2) is statistically significantly larger than the population average weight for stain 1 (u1). Consequently they picked at random 15 tomatoes from the field with strain 1 and 14 from the field with strain 2 and weighed them.

Weight grams:

strain 1

132, 68, 74, 93, 61, 81, 62, 68, 103, 72, 64, 104, 62, 86, 95.

strain 2

40, 88, 112, 127, 114, 124, 95, 125, 989, 86, 142, 130, 70, 81.

Does the strain 2 have significantly greater mean weight than for strain 1? Test this hypothesis at the alpha = 0.01 and 0.05 levels, using the two samples. Make the assumption that the weight is distributed normally in both populations with equal variances. You will be testing u1 vs u2.

1) Which diagram shows reject/ fail to reject regions for this problem?

2) what is the test statistic is use for question?

3) compute the sample means and standard deviations that you will need for 5 &6.

4) what is the test statistic value computed from the data in question 1 &3?

5) if the level of significance alpha is .01 state the critical values which you would use relevant to questions 1&4.

6) if the level of significance alpha is 0.05 state the critical value which you would use relevant to questions 1 and4.

The scores of students on the SAT college entrance examinations at a certain high school had a normal distribution with mean μ=548.8μ=548.8 and standard deviation σ=26.4σ=26.4.

(a) What is the probability that a single student randomly chosen from all those taking the test scores 555 or higher?
ANSWER:

For parts (b) through (d), consider a simple random sample (SRS) of 35 students who took the test.

(b) What are the mean and standard deviation of the sample mean score x¯x¯, of 35 students?
The mean of the sampling distribution for x¯ is:
The standard deviation of the sampling distribution for x¯ is:

(c) What z-score corresponds to the mean score x¯ of 555?
ANSWER:

(d) What is the probability that the mean score x¯ of these students is 555 or higher?
ANSWER:

consider setting a goal with a longer term. You have decided to save for your newborns college education and have determined that by the time the child reaches 18, you would like them to have $10,000. you will make monthly deposits into an account that compounds monthly at an APR of 5.8%. How much will you need to deposit every month to make your goal? what is your total investment?

MAXIMIZING PROFIT: CUSTOM GAMING PCs

You are the owner of a mid-sized computer manufacturing and assembling facility – what started as a small business building and customizing computers for friends and family has grown into a much larger operation, and you want to make sure you are using your labor hours in the most efficient way to maximize your profits. Your company manufactures and assembles three types of custom gaming PCs, and the amount of manufacturing and assembling times required for each model along with the profit you make on each unit are given below:

Model Name	Manufacturing Time (in hours)	Assembling Time (in hours)	Profit (per unit sold)
Dendrite Ice	2	2	$280
Neuron Pro	3	2	$320
Axon Glacier Pro	2	4	$400

Your labor budget each week is 1,000 hours of total manufacturing time, and 1,600 hours of assembly time. How many of each type of custom gaming PC should you task your teams with creating each week to maximize your profit?

1. Provide the solution to your completed tableau, listing the values for each variable (including any slack variables) and the objective function. Finally, state the solution in terms of our original problem: how many of each type of custom gaming PC should your teams manufacture each week in order to maximize your profits?

The following table exhibits the age of antique furniture and the corresponding prices. Use the table to answer the following question(s). (Hint: Use scatter diagram and the Excel Trendline tool where necessary).

No. Years	Value($)
78	925
91	1010
83	970
159	1950
134	1610
210	2770
88	960
178	2010
124	1350
72	888

What is the expected value for a 90 year-old piece of furniture?

a. $934.56

b. $1029.36

c. $1002.45

d. $1033.21