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# Two types of insulation are applied to houses and then the temperatures are measured at midnight for each house

INSTRUCTIONS TO CANDIDATES

In total, five short answer questions and four multiple choice ques- tions.

Q1. (15 points) Two types of insulation are applied to houses and then the temperatures are measured at midnight for each house. The temperatures yielded from both methods are normally distributed with the same variance. The first method was applied to 15 houses, yielding a mean temperature of 3.7 and variance 2.8; the second method was also applied to 15 other houses, yielding a mean temperature of 4.75 and variance 0.18. Do hypothesis testing to confirm or reject that the two methods yield the same population mean at α = 0.05.

Q2. (15 points) Two types of insulation are applied to houses and then the temperatures are measured at midnight for each house. The first method was applied to 30 houses, yielding a mean temperature of 3.7 and variance 2.8; the second method was also applied to 30 other houses, yielding a mean temperature of 4.75 and variance 0.18. Do hypothesis testing to confirm or reject that the two methods yield the same population mean at α = 0.05.

Q3. (20 points) A new machine has been created to produce a specific type of computer components. The length of components follows a normal distribution. The machine must be stable enough to be considered non-defective, i.e., σ2 < 2. The experimenter wants to examine whether this machine is non-defective. She selects 20 products and calculates the variance is 1. Find the appropriate confidence interval or bounds at level 95% for the experimenter and make a conclusion.

Q4. (15 points) A new machine has been created to produce a specific type of computer components. The length of components follows a normal distribution. The machine must be stable enough to be considered non-defective, i.e., σ2 < 2. The experimenter wants to examine whether this machine is non-defective. She selects 20 products and calculates the variance is 1. Do an appropriate hypothesis test at 95% for the experimenter.

Q5. (15 points)

Does a baby's sleeping positionaffect the development of motor skill? In one study, published n the Archives of Pediatric Adolescent Medicine, 343 full-term innfants were examined at their 4-month checkups for various developmetal milestones, such as rolling over, grasping a rattle, reaching for an object, and so on. The baby's predominat sleep position-either prone( on the stomach) or supine (on the back) or side- was determined by a telephone interview with the parent. The sample result for 320 of the 343 infants for whom information ws recieved are shoen here:

Prone     Spine or Side

Number of Infants              121       199

Number That Roll Over       93         199

Q6. (20 points) Multiple choice. For each question, there is exactly one correct answer.

(1) Suppose X  follows an  F -distribution  with  df1 =  10, df2 =  8.  What  is the probability

P (X < 3.35)? A. 0.05

B. 0.95

C. 0.1

D. 0.9

(2) A sample is collected from a normal population and the 95% confidence interval on the mean is calculated to be [1.43, 2.78]. One of the following corresponds to the 90% confidence interval, determine which.

A. [1.38, 2.94]

B. [1.33, 2.59]

C. [1.67, 2.89]

D. [1.52, 2.69]

(3) Two catalysts are being analysed to determine how they affect the mean yield of a chemical process. We perform a hypothesis test on the difference of mean yields. Our aim is to show that catalyst 2 (with mean yield µ2) has greater mean yield than the catalyst 1 (with mean yield µ1). Determine which of the following would be an appropriate hypothesis test.

A.  H0 : µ1 − µ2 = 0 vs H1 : µ1 − µ2 < 0,

B.  H0 : µ1 − µ2 = 0 vs H1 : µ1 − µ2 > 0,

C.  H0 : µ1 − µ2 = 0 vs H1 : µ1 − µ2 =/ 0,

D. None of the above.

(4) You have a collection of data points x1 = 9, x2 = 3, x3 = 7, x4 = 4, x5 = 2 sampled from a normal distribution with mean µ and variance σ2. What is your estimate of σ2?

A. 4.56

B. 5.00

C. 2.92

D. 8.50

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