6.3.5. Consider Example 6.3.4. (a) Show that we can write S∗ = 2T − n, where T = #{Xi > θ0}. (b) Show that the scores test for this model is equivalent to rejecting H0 if T < c1 or T > c2. (c) Show that under H0, T has the binomial distribution b(n, 1/2); hence, determine c1 and c2 so that the test has size α. (d) Determine the power function for the test based on T as a function of θ.

Data taken from "Validating Medical Equipment Repair and Maintenance Metrics...."

Participant/ Service Cost (millions) / Acquisition Cost (millions)

A 1.7,  49

B 2.96,  58

C 2.02,  48

D 1.5, 23

E 2.57, 45

F 8.3, 131

a) In using data in context mentioned above, which data is the independent variable (ie. x-axis) and why?

b) What is the predicted service cost for an acquisition cost of $90million along with its associated 95% prediction (confidence) interval?

c) Should you consider using this model to predict service costs of very inexpensive medical equipment purchases (eg. acquisition costs of much less than $1million.) ?

Question 2 (Modified from Sleuth 7.27) Black wheatears, Oenanthe leucura, are small birds of Spain and Morocco. Males of the species demonstrate an exaggerated sexual display by carrying heavy stones to nesting cavities. Different males carry somewhat different sized stones, prompting a study of whether larger stones may signal a higher health status. M. Soler et al. (1999) calculated the average stone mass (grams) carried by each of 21 male black wheatears, along with T-cell response measurements reflecting their immune systems’ strengths. The data are in ex0727.

(a) (1 point) Make a scatter plot of Mass (X) versus Tcell (Y ) including the estimated regression line.

(b) (2 points) Fit the linear model using the lm() function to regress Tcell on Mass (i.e., model the mean of Tcell as a function of Mass). Use the summary() function to view more information about the estimated regression model. Provide an interpretation for the p-values of the regression coefficients.

(c) (1 point) Construct 90% confidence intervals for the regression parameters using the confint() function.

(d) (1 point) Estimate the mean T-cell measurement for a new bird that is observed to carry stones averaging 4 grams in weight by using the predict() function. Construct a 95% confidence interval for mean T-cell measurement for that new bird.

(e) (1 point) Construct a 95% prediction interval for T-cell measurement for the new bird in part (d). How does the prediction interval compare to the confidence interval from part (d)?

A technician compares repair costs for two types of microwave ovens (type I and type II). He believes that the repair cost for type I ovens is greater than the repair cost for type II ovens. A sample of 61 type I ovens has a mean repair cost of $80.58. The population standard deviation for the repair of type I ovens is known to be $17.46. A sample of 64 type II ovens has a mean repair cost of $73.27. The population standard deviation for the repair of type II ovens is known to be $14.59. Conduct a hypothesis test of the technician's claim at the 0.1 level of significance. Let μ1 be the true mean repair cost for type I ovens and μ2 be the true mean repair cost for type II ovens.

Step 1 of 5 :  State the null and alternative hypotheses for the test.

Step 2 of 5 : Compute the value of the test statistic. Round your answer to two decimal places.

Step 3 of 5 : Find the p-value associated with the test statistic. Round your answer to four decimal places.

Step 4 of 5 : Make the decision for the hypothesis test: Reject Null Hypothesis or Fail to Reject Null Hypothesis

Step 5 of 5 : State the conclusion of the hypothesis test: There is sufficient evidence to support the claim or There is not sufficient evidence to support the claim.

Problem Set 1: The paired-samples t test Research Scenario: Children who experience chronic pain as a result of medical procedures are the focus of a psychiatrist’s study. Specifically, the psychiatrist wants to measure whether a new program helps decrease feelings of chronic pain in the short-term. She measures children’s self-reports of pain levels before treatment on a standardized scale with a range of 0-10, with 10 being the most severe. She then administers the new program, and measures children’s pain levels after treatment. The data are contained in the table below. Does the new treatment decrease self-reported levels of chronic pain? Using this table, enter the data into a new SPSS data file and run a paired-samples t test to test the claim that the new program decreases self-reported levels of chronic pain. Follow the directions below the table to complete the homework.

Write an APA-style Results section based on your analysis. Include your boxplot as an APA-style figure as demonstrated in the APA writing presentation.

Problem Set 2: The paired-samples t test

Research Scenario: A social worker in a rural school district is interested in how a new program is affecting the number of days that high school students are truant from school (i.e. absent without permission from parents, doctor, etc.). She takes a sample of 15 students from school records and notes the number of days truant in the year before the program and the year after the program. The data are listed in the table below.

Using this table, enter the data into a new SPSS data file and run a paired-samples t test to test the claim that there is a difference in the number of truant days before and after the program was implemented. Follow the directions below the table to complete the homework.

Write an APA-style Results section based on your analysis. Include your boxplot as an APA-style figure as demonstrated in the APA writing presentation.

find the area under the standard normal distribution curve between z=1.22 and z=2.22

Consider randomly selecting a student at a large university, and let A be the event that the selected student has a Visa card and B be the analogous event for MasterCard. Suppose thatP(A) = 0.7and P(B) = 0.3.

(a)Could it be the case thatP(A ∩ B) = 0.5?

Why or why not?

(b) From now on, suppose thatP(A ∩ B) = 0.2.

What is the probability that the selected student has at least one of these two types of cards?

(c)What is the probability that the selected student has neither type of card?

(d)Describe, in terms of A and B, the event that the selected student has a Visa card but not a MasterCard.

Calculate the probability of this event.

(e) Calculate the probability that the selected student has exactly one of the two types of cards.

1. Insurance companies track life expectancy information to assist in determining the cost of life insurance policies. Life expectancy is a statistical measure of average time a person is expected to live, based on a number of demographic factors. Mathematically, life expectancy is the mean number of years of life remaining at a given age, assuming age-specific mortality rates remain at their most recently measured levels. Last year the average life expectancy of all the Life Insurance policyholders in Ontario at age 65 was 22.3 years (meaning that a person reaching 65 last year was expected to live, on average, until 87.3). The insurance company wants to determine if their clients now have a longer average life expectancy, so they randomly sample some of their recently paid policies. The insurance company will only change their premium structure if there is evidence that people who buy their policies are living longer than before. The sample data is provided in the excel file. Answer the following questions. Results should be support by excel output.

a. Construct a 95% and 99% confidence intervals for the true average life expectancy. Use t-distribution and Descriptive Statistics function from Data Analysis. Interpret each Confidence interval and comment on the difference between the 95% and 99% interval.

b. Write the null and alternative hypotheses for this test:

c. In this context, describe a Type I error possible. How might such an error impact Life Insurance company’s decision regarding the premium structure?

d. What is the value of the t-test statistic?

e. What is the associated P-value?

f. State the conclusion using α = 0.05. Do it using both P-value and critical value.

Please answer it on excel. Thank you.

A campaign manager for a political candidate released a series of advertisements criticizing their opponent in an upcoming election. Their opponent previously had the support of 45% of voters, and the campaign manager wants to sample voters to test if support for their opponent has decreased. The manager surveyed 200 voters and the survey shows 40% support. Use .03 level of significance

(Data from Devore and Peck) Recent studies have looked at the duration of waiting time between a diagnosis and recommendation of surgery until the actual surgery. In the case of potentially lifethreatening conditions, the waiting time on average is hopefully of short duration. For a (assume random) sample of 539 patients recommended for a heart bypass procedure, the sample mean waiting time was 19 days, and the sample standard deviation was 10 days (i.e. x 19,s 10,n  539 ). We will assume the distribution of waiting times is normal with unknown population mean and variance. Z and t Critical values for 90%: z*=1.645, t*(200)=1.653, t*(369.5)=1.649, t*(538)=1.648, t*(739)=1.647 (You will not need all these). Chi-square table (provided).

a. Compute a 90% confidence interval for the population mean waiting time.

b. Compute a 95% confidence interval for the population standard deviation of waiting time, assuming s=10, but changing the sample size to be 26.

c. Compute a 95% lower confidence bound for the population standard deviation of waiting time, again assuming s=10 for a sample size of 26. (3)

V. Nimania plc builds water treatment facilities throughout the world. One contract it has concerns an installation in an area
prone to outbreaks of a dangerous disease. The company has to decide whether or not to vaccinate the employees who will be working there. Vaccination will cost £200,000, which will be deducted from the profit it makes from the venture. The company expects a profit of £1.2m from the contract but if there is an outbreak of the disease and the workforce has not been vaccinated, delays will result in the profit being reduced to £0.5m. If the workforce has been vaccinated and there is an outbreak of the disease, the work will progress as planned but disruption to infrastructure will result in their profit being reduced by £0.2m. Advise the company using:

(a) the maximax decision rule

(b) the maximin decision rule

(c) the minimax regret decision rule

(d) the equal likelihood decision rule

Consider the following time series data.

.

(a)  Use a multiple regression model with dummy variables as follows to develop an equation to account for seasonal effects in the data. Qtr1 = 1 if Quarter 1, 0 otherwise; Qtr2 = 1 if Quarter 2, 0 otherwise; Qtr3 = 1 if Quarter 3, 0 otherwise.

If required, round your answers to three decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300) If the constant is "1" it must be entered in the box. Do not round intermediate calculation.

.

(b) Use a multiple regression model to develop an equation to account for trend and seasonal effects in the data. Use the dummy variables you developed in part (b) to capture seasonal effects and create a variable t such that t = 1 for Quarter 1 in Year 1, t = 2 for Quarter 2 in Year 1,… t = 12 for Quarter 4 in Year 3.

If required, round your answers to three decimal places. For subtractive or negative numbers use a minus sign even if there is a + sign before the blank. (Example: -300)

.

(c)  Is the model you developed in part (b) or the model you developed in part (d) more effective?

Express numerical answers in decimal form and round to 3 decimal places as needed (unless otherwise stated).
[8] 1) The homework scores (out of 10 points) for a sample of 9 students are listed:
1, 7, 7, 8, 9, 9, 10, 10, 10
a) Find the five number summary (whole numbers, in order).
Five number summary: , , , ,