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Column - Applications of Statistics - Exam, Exams of Statistics

This is the Exam of Applications of Statistics which includes Confidence Interval, Definition, Population, Interval Related, Margin, Certain, Drug Causes, Drowsiness, Patients etc. Key important points are: Column, Confidence Interval, Mean Lifetime, Sentence Explanation, Standard Deviation, Lifetime, Median Lifetime, Measurements, Variable, Brand Considered

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DEPARTMENT OF MATHEMATICS & STATISTICS
STAT 2593
FINAL EXAMINATION
DECEMBER 1991 TIME: 21
2Hours
THERE ARE SEVEN PROBLEMS, WITH TOTAL VALUE 100 MARKS. PROBLEM 7 IS TO
BE ANSWERED ON A SEPARATE SHEET OF PAPER. MAKE SURE THAT YOU PUT YOUR
NAME ON THIS SHEET AND HAND IT IN.
1. The data in column C1 are measurements on lifetime, in years of normal operation, of a
certain brand of light bulbs.
MTB> print c1
C1
2.75 3.75 3.00 3.25 3.25 3.25 3.25 3.50 3.50 3.75
3.75 3.75 3.75 3.75 4.00 4.00 4.25 4.50 4.75 5.50
MTB > describe c1
N MEAN MEDIAN TRMEAN STDEV SEMEAN
C1 20 3.713 3.750 3.667 0.675 0.151
MIN MAX Q1 Q3
C1 2.750 5.500 3.250 4.000
(a) Give a 90% confidence interval for the mean lifetime for these light bulbs.
(b) Can a consumer group claim that the bulbs last, on average, less than 3.8 years? (Use
your answer to (a) and give a one sentence explanation.)
(c) Give a 90% confidence interval for the standard deviation of lifetime for these bulbs.
(d) Is the median lifetime less than 3.8 years? (Use the sign test.)
2. The data in column C2 are measurements on lifetime of a different brand of light bulb. Is
lifetime of this second brand less variable than lifetime of the brand considered in problem 1?
(Test.)
MTB > describe c2
N MEAN MEDIAN TRMEAN STDEV SEMEAN
C1 15 2.700 2.500 2.692 0.474 0.122
MIN MAX Q1 Q3
C1 2.000 3.500 2.500 3.000
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DEPARTMENT OF MATHEMATICS & STATISTICS

STAT 2593

FINAL EXAMINATION

DECEMBER 1991 TIME: 2 12 Hours

THERE ARE SEVEN PROBLEMS, WITH TOTAL VALUE 100 MARKS. PROBLEM 7 IS TO

BE ANSWERED ON A SEPARATE SHEET OF PAPER. MAKE SURE THAT YOU PUT YOUR

NAME ON THIS SHEET AND HAND IT IN.

1. The data in column C1 are measurements on lifetime, in years of normal operation, of a

certain brand of light bulbs.

MTB> print c

C 2.75 3.75 3.00 3.25 3.25 3.25 3.25 3.50 3.50 3. 3.75 3.75 3.75 3.75 4.00 4.00 4.25 4.50 4.75 5.

MTB > describe c

N MEAN MEDIAN TRMEAN STDEV SEMEAN C1 20 3.713 3.750 3.667 0.675 0.

MIN MAX Q1 Q C1 2.750 5.500 3.250 4.

(a) Give a 90% confidence interval for the mean lifetime for these light bulbs.

(b) Can a consumer group claim that the bulbs last, on average, less than 3.8 years? (Use

your answer to (a) and give a one sentence explanation.)

(c) Give a 90% confidence interval for the standard deviation of lifetime for these bulbs.

(d) Is the median lifetime less than 3.8 years? (Use the sign test.)

2. The data in column C2 are measurements on lifetime of a different brand of light bulb. Is

lifetime of this second brand less variable than lifetime of the brand considered in problem 1?

(Test.)

MTB > describe c

N MEAN MEDIAN TRMEAN STDEV SEMEAN C1 15 2.700 2.500 2.692 0.474 0.

MIN MAX Q1 Q C1 2.000 3.500 2.500 3.

3. Records from a small New Brunswick county indicate that in 1969, there were a total of 115

live births, 55 of which were male. Records for 1989, for the same county, show 237 live

births, 105 of which were male.

(a) Was the proportion of male births in 1989 significantly different from the proportion of

male births in 1969? (Test.)

(b) Pool the data from 1969 and 1989 and find a 95% confidence interval for the proportion

of live births who are male. Is this proportion significantly different from 50%?

4. Name the distribution of each of the following random variables. In each case, specify which

parameter values are known, and which are unknown.

(a) Number of defectives in a sample of size 3, selected from a large shipment.

(b) Number of defectives in a sample of size 3, selected from a shipment of 10 items.

(c) Number of rolls of a pair of fair die until snake-eyes (

) appear.

(d) Waiting time to the next telephone call, at an exchange that averages 4 calls/minute.

(e) Waiting time to the second telephone call, at the same exchange.

5. The data in columns C1 and C2 are airborne bacteria counts (number of colonies/f 3 ) in front

of the elevators on two floors of each of eight government buildings. For each building, the

count was taken on a floor which had carpet in front of the elevators, and on another floor

which did not have carpeting.

(a) Three different analyses of the data are given on the next page. Which one is appropri-

ate? Why?

(b) Does there seem to be a difference in the average bacteria count in carpeted and uncar-

peted areas? Explain briefly how you used the computer output to answer this question.

(c) Suppose you learned later that the uncarpeted floors were on the ground floor and

the carpeted floors were on the tenth floor. Would you be able to assess the effect of

carpeting? (Give reason.)

6. Suppose that the true weight of 1 kg loaves of “Ma’s Tasty Bread” are normally distributed

with mean 1.0l kg and standard deviation 0.02 kg.

(a) What proportion of loaves are underweight (i.e less than 1 kg)?

(b) Three loaves are randomly selected. What is the probability that

(i) their average weight is less than 1 kg?

(ii) they all weigh less than 1 kg?

  1. ANSWER THIS QUESTION ON THE SHEET PROVIDED. DON’T FOR-

GET TO WRITE YOUR NAME ON THE SHEET AND HAND IT IN.

The data in columns C1 and C2 show the results of measurements of the electrical resistance

in (ohms/cm) × 10 −^6 of platinum, at several temperatures in degrees Kelvin. There is also

output from the MINITAB command REGRESS.

(a) Plot resistance vs temperature on the paper provided.

(b) Sketch in the regression line. Show calculations for two points on the line.

(c) Give a 95% confidence interval for slope of the regression line.

(d) Give a 95% confidence interval for mean resistance at 250◦K.

(e) Would you be surprised to get a resistance reading of 7.0 (ohms/cm) × 10 −^6 at 250◦K?

(Explain briefly.)

MTB > print c1 c

ROW Temp Resist 1 100 2 2 200 9 3 300 12 4 400 16 5 500 21

MTB > regress c2 1 c1; SUBC> predict 250.

The regression equation is Resist = - 1.50 + 0.0450 Temp

Predictor Coef Stdev t-ratio p Constant -1.500 1.133 -1.32 0. Temp 0.045000 0.003416 13.17 0.

s = 1.080 R-sq = 98.3% R-sq(adj) = 97.7%

Analysis of Variance

SOURCE DF SS MS F p Regression 1 202.50 202.50 173.57 0. Error 3 3.50 1. Total 4 206.

Fit Stdev.Fit 95% C.I. 95% P.I. 9.750 0.512 ( 8.119, 11.381) ( 5.945, 13.555)

NAME: ............................................................................... I.D.#: .....................

QUESTION 7