Confidence intervals for the mean

Suppose you are a researcher in a hospital. You are experimenting with a new tranquilizer. You collect data from a random sample of 10 patients. The period of effectiveness of the tranquilizer for each patient (in hours) is as follows:

a. What is a point estimate for the population mean length of time. (Round answer to 4 decimal places)

b. Which distribution should you use for this problem?

• normal distribution
• t-distribution

c. Why?


d. What must be true in order to construct a confidence interval in this situation?

• The population must be approximately normal
• The population mean must be known
• The population standard deviation must be known
• The sample size must be greater than 30

e. Construct a 95% confidence interval for the population mean length of time. Enter your answer as an open-interval (i.e., parentheses) Round upper and lower bounds to two decimal places

f. Interpret the confidence interval in a complete sentence. Make sure you include units

g. What does it mean to be "95% confident" in this problem? Use the definition of confidence level.

• 95% of all simple random samples of size 10 from this population will result in confidence intervals that contain the population mean
• There is a 95% chance that the confidence interval contains the population mean
• The confidence interval contains 95% of all samples

h. Suppose that the company releases a statement that the mean time for all patients is 2 hours.

Is this possible?

• No
• Yes

Is it likely?

• No
• Yes

For this problem, carry at least four digits after the decimal in your calculations. Answers may vary slightly due to rounding.

The National Council of Small Businesses is interested in the proportion of small businesses that declared Chapter 11 bankruptcy last year. Since there are so many small businesses, the National Council intends to estimate the proportion from a random sample. Let p be the proportion of small businesses that declared Chapter 11 bankruptcy last year.

(a) If no preliminary sample is taken to estimate p, how large a sample is necessary to be 99% sure that a point estimate p̂ will be within a distance of 0.09 from p? (Round your answer up to the nearest whole number.)

? small businesses

(b) In a preliminary random sample of 30 small businesses, it was found that six had declared Chapter 11 bankruptcy. How many more small businesses should be included in the sample to be 99% sure that a point estimate p̂ will be within a distance of 0.090 from p? (Round your answer up to the nearest whole number.)

? more small businesses

1. A sample of 40 runners will be used to compare two new routines for stretching. The runners will be randomly assigned to one of the routines which they will follow for two weeks. Satisfaction with the routines will be measured using a questionnaire at the end of the two-week period. For the first routine, nine runners said that they were satisfied or very satisfied. For the second routine, six runners said that they were satisfied or very satisfied.

a) What is the test statistic?

b) What is the p-value?

The table below gives the number of hours five randomly selected students spent studying and their corresponding midterm exam grades. Using this data, consider the equation of the regression line, yˆ=b0+b1x, for predicting the midterm exam grade that a student will earn based on the number of hours spent studying. Keep in mind, the correlation coefficient may or may not be statistically significant for the data given. Remember, in practice, it would not be appropriate to use the regression line to make a prediction if the correlation coefficient is not statistically significant.

Hours Studying 1 3 4 5 6

Midterm Grades 65 66 67 74 97

There are 32 games played in the first round of the NCAA basketball tournament. Joe enters a pool where you choose the winner of each game. He knows nothing about basketball and picks his teams randomly by placing both team’s names on slips of paper. Puts them in a hat and draws one out. The team he pulls out is the team he chooses to win the game. Let X be the number of picks Joe gets correct out of the 32 first-round games.

A. Find ?(? = 15).

B. Find ?(? ≤ 12)

C. Find ?(13 ≤ ? ≤ 18).

D. Would you be surprised if Joe got more than 22 out of 32 correct using his method? Explain

2. Suppose that you are waiting for a friend to call you and that the time you wait in minutes has an exponential distribution with parameter λ = 0.1. (a) What is the expectation of your waiting time? (b) What is the probability that you will wait longer than 10 minutes? (c) What is the probability that you will wait less than 5 minutes? (d) Suppose that after 5 minutes you are still waiting for the call. What is the distribution of your additional waiting time? In this case, what is the probability that your total waiting time is longer than 15 minutes? (e) Suppose now that the time you wait in minutes for the call has a U(0, 20) distribution. What is the expectation of your waiting time? If after 5 minutes you are still waiting for the call, what is the distribution of your additional waiting time?

3. The arrival times of workers at a factory first-aid room satisfy a Poisson process with an average of 1.8 per hour. (a) What is the value of the parameter λ of the Poisson process? (b) What is the expectation of the time between two arrivals at the first-aid room? (c) What is the probability that there is at least 1 hour between two arrivals at the firstaid room? (d) What is the distribution of the number of workers visiting the first-aid room during a 4-hour period? (e) What is the probability that at least four workers visit the first-aid room during a 4- hour period?

it took 85 wsu students an average of 36 minutes to commute to campus( with a standard deviation of 3.5 minute ). at the 95% confidence level, construct a confidence interval within which lies the mean commute time of all WSU students.

Solve the problem.

The State Association of Retired Teachers has recently taken flak from some of its members regarding the poor choice of the association's name. The association's by-laws require that more than 60 percent of the association must approve a name change. Rather than convene a meeting, it is first desired to use a sample to determine if meeting is necessary. Suppose the association decided to conduct a test of hypothesis using the following null and alternative hypotheses:

H₀: p = 0.6
H_A: p > 0.6

Define a Type II Error in the context of this problem.

A.They conclude that more than 60% of the association wants a name change when, in fact, that is not true.

B.They conclude that exactly 60% of the association wants a name change when that is, in fact, true.

C.They conclude that more than 60% of the association wants a name change when that is, in fact, true.

D.They conclude that exactly 60% of the association wants a name change when, in fact, that is not true.

why is A wrong? which is correct?

. A web-based algorithm classifies emails as spams or no-spam at a success rate of 70% of detecting a spam. A. Find a 95% confidence interval for the number of spam emails expected within a sample of 100 emails. B. Find a 95% confidence interval for the accuracy (standard deviation) of the detected number of spams within the 100 emails. C. Determine the probability that at least 85 emails out of 100 emails are spams. D. If for the 100 emails above, there was indeed actually 85 spam emails, test the claim that the same algorithm can still detect at least 85 emails with a 95% confidence.

a) 10 children are to be divided into an A team and a B team of 5 each. The A team will play in one league and the B team in another. How many different divisions are possible? Answer: (10!/5!5!)

b) 10 children at a playground divide themselves into two teams of 5 each. How many different divisions are possible? Answer: (10!/5!5!2!)

I don't understand why in b we divide by 2!, can someone explain it in detail? It would be better to visualize. Thank you!

Scenario

Office Equipment, Inc. (OEI) leases automatic mailing machines to business customers in Fort Wayne, Indiana. The company built its success on a reputation of providing timely maintenance and repair service. Each OEI service contract states that a service technician will arrive at a customer’s business site within an average of 3 hours from the time that the customer notifies OEI of an equipment problem.

Currently, OEI has 10 customers with service contracts. One service technician is responsible for handling all service calls. A statistical analysis of historical service records indicates that a customer requests a service call at an average rate of one call per 50 hours of operation. If the service technician is available when a customer calls for service, it takes the technician an average of 1 hour of travel time to reach the customer’s office and an average of 1.5 hours to complete the repair service. However, if the service technician is busy with another customer when a new customer calls for service, the technician completes the current service call and any other waiting service calls before responding to the new service call. In such cases, after the technician is free from all existing service commitments, the technician takes an average of 1 hour of travel time to reach the new customer’s office and an average of 1.5 hours to complete the repair service. The cost of the service technician is $80 per hour. The downtime cost (wait time and service time) for customers is $100 per hour.

OEI is planning to expand its business. Within 1 year, OEI projects that it will have 20 customers, and within 2 years, OEI projects that it will have 30 customers. Although OEI is satisfied that one service technician can handle the 10 existing customers, management is concerned about the ability of one technician to meet the average 3-hour service call guarantee when the OEI customer base expands. In a recent planning meeting, the marketing manager made a proposal to add a second service technician when OEI reaches 20 customers and to add a third service technician when OEI reaches 30 customers. Before making a final decision, management would like an analysis of OEI service capabilities. OEI is particularly interested in meeting the average 3-hour waiting time guarantee at the lowest possible total cost.

Managerial Report

Develop a managerial report summarizing your analysis of the OEI service capabilities. Make recommendations regarding the number of technicians to be used when OEI reaches 20 and then 30 customers, and justify your response. Include a discussion of the following issues in your report:

1. What is the arrival rate for each customer?
2. What is the service rate in terms of the number of customers per hour? (Remember that the average travel time of 1 hour is counted as service time because the time that the service technician is busy handling a service call includes the travel time in addition to the time required to complete the repair.)
3. Waiting line models generally assume that the arriving customers are in the same location as the service facility. Consider how OEI is different in this regard, given that a service technician travels an average of 1 hour to reach each customer. How should the travel time and the waiting time predicted by the waiting line model be combined to determine the total customer waiting time? Explain.
4. OEI is satisfied that one service technician can handle the 10 existing customers. Use a waiting line model to determine the following information: (a) probability that no customers are in the system, (b) average number of customers in the waiting line, (c) average number of customers in the system, (d) average time a customer waits until the service technician arrives, (e) average time a customer waits until the machine is back in operation, (f) probability that a customer will have to wait more than one hour for the service technician to arrive, and (g) the total cost per hour for the service operation.
5. Do you agree with OEI management that one technician can meet the average 3-hour service call guarantee? Why or why not?
6. What is your recommendation for the number of service technicians to hire when OEI expands to 20 customers? Use the information that you developed in Question 4 (above) to justify your answer.
7. What is your recommendation for the number of service technicians to hire when OEI expands to 30 customers? Use the information that you developed in Question 4 (above) to justify your answer.
8. What are the annual savings of your recommendation in Question 6 (above) compared to the planning committee's proposal that 30 customers will require three service technicians? (Assume 250 days of operation per year.) How was this determination reached?

Your foreman claims that tree planting is a job for young people. He further claims that 3/4 of tree planters are below the age of 21. You think he's exaggerating and the proportion of tree planters under 21 is not nearly that high. You gain access to a page from the personnel files that has the birthdates for 50 tree planters and count 32 who are under 21. Do you have enough evidence (at ?=.05α=.05) to conclude that your foreman is wrong and that the proportion of tree planters under 21 is less than 3/4?

Explain the importance of understanding the dispersion of a set of scores in addition to the average score. Give one example from this simulation and one example you can envision.

6. Suppose that you choose a student at random, and let the random variable “H” represent the student’s studying time (in hours) last week. Suppose that “H” has a uniform distribution on the interval (25, 35). If you randomly select 10 students and that each person’s studying time follows this same probability distribution, independently from person to person, determine the probability that at least two of the students studied for more than 32 hours last week.  

When examining a plot of means for a two-way ANOVA, what should you conclude if the two lines completely overlap and have zero slope?

please do not guess, if you do not know carry on. Please show work, thank you :)