a. all Hypothesis Tests must include all four steps, clearly labeled;

b. all Confidence Intervals must include all output as well as the CI itself

c. include which calculator function you used for each problem.

3. At a community college, the mathematics department has been experimenting with four different delivery mechanisms for content in their Statistics courses. One method is traditional lecture (Method I), the second is a hybrid format in which half the time is spent online and half is spent in-class (Method II), the third is online (Method III), and the fourth is an emporium model from which students obtain their lectures and do their work in a lab with an instructor available for assistance (Method IV). To assess the effective of the four methods, students in each approach are given a final exam with the results shown in the following table. Assume an approximate normal distribution for each method. At the 5% significance level, does the data suggest that any method has a different mean score from the others?

A psychic claims to be able to predict the outcome of coin flips before they happen. Someone who guesses randomly will predict about half of coin flips correctly. In 100 flips, the psychic correctly predicts 57 flips. Do the results of this test indicate that the psychic does better than random guessing? The hypotheses are Ho:p=0.50 Ha:p>50 where p is the proportion of correct coin flip predictions by the psychic.

1. Give the test statistic for this problem.

Group of answer choices

a.Z = 1.40

b.Z = -9.43

c.Z = -1.40

2. What is the P value

a P-value = 0.2843

b P-value = 0.919

c P-value = 0.081

d P-value = 0.162

3. Give the conclusion based on the P-value above. Use a 5% level of significance.

a. Someone who guesses randomly will predict about half of coin flips correctly (about 50 out of 100). In 100 flips, the psychic correctly predicts 57 flips, a 14% increase. Therefore we can conclude that the psychic's predictions are not due to random chance.

b The psychic correctly predicted 57% of the coin flips (57 out of 100). A person who guesses randomly will correctly predict about 50% (half) of the 100 coin flips. So the psychic's predictions are significantly better.

c The psychic’s prediction of the outcomes of coin flips is not significantly better than we would expect with random guessing.

PLEASE DO NOT COPY OTHERS ANSWER, THANK YOU!

Let x1, x2, · · · , xn ∈ {0, 1}.

(a) (10 points) Consider the equation x1 + x2 + · · · + xn = 0 mod 2. How many solutions does this equation have? Express your answer in terms of n. For example, if n = 2, x1 + x2 = 0 has 2 solutions: (x1, x2) = (0, 0) and (x1, x2) = (1, 1).

(b) (5 points) Consider the equations x1 + x2 + · · · + xn = 0 mod 2 x1 + x2 + · · · + x10 = 0 mod 2 for n ≥ 10. How many solutions are there satisfying both equations?

A researcher is interested in determining whether there is a correlation between number of packs of cigarettes smoked per day and longevity (in years). n=10.

Longevity

PLEASE DO NOT COPY OTHERS ANSWER, THANK YOU!

Alice and Bob play the following game: in each round, Alice first rolls a single standard fair die. Bob then rolls a single standard fair die. If the difference between Bob’s roll and Alice's roll is at most one, Bob wins the round. Otherwise, Alice wins the round.

(a) (5 points) What is the probability that Bob wins a single round?

(b) (7 points) Alice and Bob play until one of them wins three rounds. The first player to three wins is declared the winner of the series. What is the probability that Bob wins the series?

(c) (7 points) In a single series, what is the expected number of wins for Bob?

(d) (6 points) In a single series, how many more games is Alice expected to win than Bob? That is, what is the expected value of the number of wins for Alice minus the number of wins for Bob?

(e) (5 points) In a single series, what is the variance of the expected number of wins for Bob?

1. Solve this problem by hand.

A thermal interaction between two components requires that both components have similar temperature. One random sample for each component was obtained. The data is summarized in the table below:

1. Calculate x1,s1,x2,and s2
2. Is there evidence to support that the two components has the same temperature. Assume α=0.05 . Assume that both populations are normally distributed with equal unknown variances
3. Construct two sided 95% CI for part a

1. National Bank (see previous problem: pasted down below) is considering adding a second teller to the lunch-time situation to alleviate congestion. If the second teller is added, find the following:
   1. The average teller utilization
   2. The probability that there are 0 customers in the system
   3. The average number of customers in line
   4. The average time a customer waits before it seeing the teller
   5. The average time a customer spends in the service system
   6. If the tellers are paid $15/hour, is it worth adding the second teller (Hint: answer the question by only looking at the single hour by comparing 1 vs. 2 people)

To help answer the question, here is the "previous problem":

1. National Bank currently employs a single teller to assist customers over their lunch breaks. The typical arrival rate of customers is 11 people per hour where the teller can service people at a rate of 12 customers per hour. Assuming the standard assumptions of queuing models are met, find the following:
   1. The utilization of the teller
   2. The probability that there are 0 customers in the system, 8 customers in the system.
   3. The average number of customers in line
   4. The average time a customer waits before it seeing the teller
   5. The average time a customer spends in the service system

(A-Grade) The operations manager of a large production plant would like to estimate the mean amount of time a worker takes to assemble a new electronic component. Assume that the population standard deviation of time for this assembly is 3.6 minutes.

1. After observing 120 workers assembling similar devices, the manager noticed that their average time was 16.2 minutes. Construct a 92% confidence interval for the mean assembly time.

2. How many workers should be involved in this study in order to have the mean assembly time estimated up to 15 seconds with 92% confidence?

3. In a second study a sample 102 workers had standard deviation of 2.78 minutes, with 95% confidence level, construct a confidence interval for the population standard deviation.

The company has a new process for manufacturing large artificial sapphires. In a trial run, 12 sapphires are produced. The mean weight for these 12 gems is x = 6.75 carats, and the sample standard deviation s = 0.33 carats. Find the 95% confidence interval for the population mean weight of artificial sapphires.

True/False

Type T (for True) or F (for False) for each statement for both Discrete and Continuous random variables.

1. X cannot map the same sample point to two different numbers.

• Discrete:

• Continuous:

2. The sample space (domain) can be either discrete or continuous.

• Discrete:

• Continuous:

3. The range of X is countable.

• Discrete:
  • Continuous:

4. The area under the F(x) function is 1.

• Discrete:

• Continuous:

5. The cdf F(x) is non-decreasing.

• Discrete:

• Continuous:

6. The cdf F(x) is differentiable everywhere.

• Discrete:

• Continuous:

7. The pf/pdf f(x) is continuous.

• Discrete:

• Continuous:

8. The pf/pdf f(x) is between 0 and 1 (inclusive).

• Discrete:

• Continuous:

9. E[g(X)] = g(E[X]).

• Discrete:

• Continuous:

10. Variables need to be independent for the CLT to apply.

• Discrete:

• Continuous:

Identify the Distribution

Select the Distribution that best fits the definition of the random variable X in each case.

1. Each hurricane independently has a certain probability of being classified as "serious." A climatologist wants to study the effects of the next 5 serious hurricanes. X = the number of non-serious hurricanes observed until the data is collected.

2. Ten percent of Netflix users watch a particular show. A survey asks 25 independent viewers whether they watch this show. X = the number who say yes.

3. The number of car accidents at a particular intersection occur independently at a constant rate with no chance of two occurring at exactly the same time. X = the number of accidents on a Thursday.

4. Potholes along a road occur independently at a constant rate with no chance of two occurring at exactly the same place. X = the distance between consecutive potholes.

5. Buses arrive at a certain stop EXACTLY every 15 minutes. You show up at this bus stop at a random time. Let X = your waiting time until the next bus.

6. A soccer player has a certain probability p of being injured in each game, independently of other games. X = the number of games played before the player is injured.

7. Proportions of individuals with tree blood types in a population are 0.2, 0.3 and 0.5 respectively. We select randomly 50 individuals from a large population. What is the joint distribution of the number of individuals in the sample with the first and second blood type, respectively?

8. A designer is working on a new ergonomic chair, and they want it to work best for average height people, so they measure the heights of all 50 people working in their office. Let X = the average height.

9. In Lotto 6/49 a player selects a set of six numbers (with no repeats) from the set{1, 2, ..., 49}. In the lottery draw, six numbers are selected at random. Let X = the first number drawn.

10. A tank contains 10 tropical fish, 2 of which are a rare species. Five fish are removed from the tank. X = the number of rare fish left in the tank.

Assuming all of the distributions are normal, find the x for each of the cases:

a. P(Z < k) = 0.92

b. P(Z > k) = 0.72

c. P(−1 < Z < k) = 0.60

d. (A-Grade) P(k < Z < 1.7) = 0.57

e. (A-Grade) P(Z = k) = 0.00

Fast Facts Financial (FFF), Inc. provides credit reports to lending institutions which are

evaluating applicants for home mortgages, vehicle, home equity, and other loans.

A pressure faced by FFF Inc. is that several competing credit reporting companies are

able to provide reports in about the same average amount of time, but are able to promise

a lower time than FFF Inc - the reason being that the variation in time required to

compile and summarize credit data is smaller than the time required by FFF.

FFF has identified & implemented procedures which they believe will reduce this

variation. If the historic standard deviation is 2.3 days, and the standard deviation for a

sample of 25 credit reports under the new procedures is 1.8 days, then test the appropriate

hypothesis at the alpha = .05 level of significance.

Hint: Use chi-square distribution, assume measured trait is normal, and evidence should be insufficient for rejecting null hypothesis.

Z is a standard normal random variable, then k is ...

a. P(Z < k) = 0.92

b. P(Z > k) = 0.72

c. P(Z ≤ k) = 0.26

d. (A-Grade) P(−1 < Z < k) = 0.60

e. (A-Grade) P(k < Z < 1.7) = 0.57

f. (A-Grade) P(Z = k) = 0.00