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Applied Science - Statistics, Exams of Statistics

Central University of Jammu and KashmirStatistics

Applied Science, Bachelor of Science, Environmental Science, Analytical Biochemistry, Standard Normal, Applied Mathematics, Horseshoe Bats, Sample Variance are some points from questions of this past exam paper.

Typology: Exams

2011/2012

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SPRING EXAMINATIONS 2009/2010
MODULE CODE: MA419/BC852
Module Statistics
Module Instances
1AM1 Higher Diploma in Applied Science (Microbiology)
3EV2 Bachelor of Science (Environmental Science) (Hons.)
1AS1 H.Dip. Applied Science (Analytical Biochemistry/Chemistry)
1CB1 Master of Science (Analytical Biochemistry/Chemistry)
External Examiner(s) Prof. Byron J.T. Morgan
Internal Examiner(s) Dr. Paul J. Wilson
Prof. John P. Hinde
Instructions: Answer any four questions
Tables of the standard normal, t, and χ2distributions are attached
Duration Three Hours
No. of Pages 15 pages
School Mathematics, Statistics and Applied Mathematics
Question One is on the next page
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SPRING EXAMINATIONS 2009/

MODULE CODE: MA419/BC

Module Statistics

Module Instances

1AM1 Higher Diploma in Applied Science (Microbiology)

3EV2 Bachelor of Science (Environmental Science) (Hons.)

1AS1 H.Dip. Applied Science (Analytical Biochemistry/Chemistry)

1CB1 Master of Science (Analytical Biochemistry/Chemistry)

External Examiner(s) Prof. Byron J.T. Morgan

Internal Examiner(s) Dr. Paul J. Wilson

Prof. John P. Hinde

Instructions: Answer any four questions

Tables of the standard normal, t, and χ

distributions are attached

Duration Three Hours

No. of Pages 15 pages

School Mathematics, Statistics and Applied Mathematics

Question One is on the next page

1. (a) The body lengths (in millimetres) of of forty randomly selected lesser horseshoe bats were

as follows:

i. Construct a stem–and–leaf plot using intervals of width 5 for the data above.

ii. Find the five number summary of the data.

iii. Draw a boxplot that illustrates the data. (There is no need to use graph paper, but

you may do so if you wish).

iv. It may be shown that the variance of the above data is 15.91 if the “population variance”

formula is used, and 16.24 if the “sample variance” formula is used. Interpret these

two values.

(b) A chemist wishes to ascertain the time taken for a chemical reaction to complete at various

temperatures. She runs the reaction 30 times at each of the temperatures of 10◦C, 20◦C,

30 ◦C and 40◦C and records the various reaction times. The results are summarised in the

multiple boxplot of Figure 1.

Figure 1: Multiple Box Plot

l

l

10 20 30 40

50

100

150

200

250

300

350

Time to Completion of Reaction

Temperature (Celsius)

seconds

Briefly comment upon how the temperature affects the reaction times. Your answer should refer

to the average and variation of the reaction times, reaction time, and any unusual values.

Question Two is on the next page

3. (a) A large company believes that it employs equal numbers of employees that are aged between

16 and 20; 21 and 35; 36 and 50; 51 and 60; and 61 and 66.

A director of the company doubts this, he randomly surveys 100 of the staff. The results

of this survey are shown in Table 1:

Table 1: Ages of Employees

Age 16–20 21–35 36–50 51–60 61–

Quantity 15 18 24 24 19

At a level of significance of α = 0.05, do the results of the survey provide significant evidence

that the director is correct?

(b) 300 adults were classified according to their sex and their views on genetically modified

food. The results are summarized in the Table 2.

Table 2: Views on Genetically Modified Food

In Favour No Opinion Against

Male 36 61 63

Female 26 45 69

Is there evidence, at α = 0.05, of a relationship between the sex of a person and his or her

views on genetically modified food?

This question is continued on the next page

(c) A researcher wishes to use a χ^2 goodness of fit test to determine whether the weights of

lab mice are normally distributed. She selects a random sample of one hundred such mice,

weighs them and records the results. It is found that the mean weight of the hundred mice

in the sample is 23.3 grammes, with a standard deviation of 1.9 grammes. She proceeds to

test that the weights are consistent with a normal distribution with this mean and standard

deviation by creating a frequency table of the data, calculating the expected counts, and

entering the data into minitab. The frequency table and the mintab output are shown in

Table 3.

Table 3: Weights of Mice

weight in grams < 20 20 − 22 22 − 24 24 − 26 > 26

Observed Count 7 15 35 32 9

Expected Count 4. 12 20. 57 39. 70 27. 86 7. 77

Chi-Square Goodness-of-Fit Test for Observed Counts in Variable: Weight

Contribution

Category Observed Expected to Chi-Sq

N DF Chi-Sq P-Value

1 cell(s) (20.00%) with expected value(s) less than 5.

i. What are the null and alternative hypotheses here?

ii. Show how the “expected count” of 20.57 for the interval 20–22 is calculated.

iii. Given that the researcher used the values of the mean and standard deviation of her

sample to estimate those of the normal distribution, the number of degrees of freedom

stated in the above output is in fact incorrect, what should the correct number of

degrees of freedom be?

iv. Based upon the P-value of 0.406 a decision would be made to not reject the null

hypothesis. If the correct number of degrees of freedom were used, would our decision

be altered? Justify your answer.

Question Four is on the next page

(c) Nine people had their blood sugar level recorded before (Glucose1) and after (Glucose2)

undertaking a strict diet. To investigate whether this diet will affect blood glucose levels,

the data was analysed using a paired t-test. The minitab output is presented in Figure 3.

Figure 3: Blood Sugar Levels Before and After Diet

Results for: Bloodsugar2.MTW

Paired T-Test and CI: Glucose1, Glucose

Paired T for Glucose1 - Glucose

N Mean StDev SE Mean

Glucose1 9 90.67 26.72 8.

Glucose2 9 89.44 27.13 9.

Difference 9 1.22 6.08 2.

95% CI for mean difference: (-3.45, 5.89)

T-Test of mean difference = 0 (vs not = 0): T-Value = 0.60 P-Value = 0.

i. State the null and alternative hypotheses that are being tested.

ii. Carefully interpret the output “P-Value = 0.563”, explaining why on the basis of this

P-Value we should not reject the null hypothesis.

iii. Carefully interpret the output “95% CI for mean difference: (-3.45, 5.89)”.

iv. In general, does non-rejection of a null hypothesis mean that the null hypothesis is

true? If not, how should non-rejection of the null hypothesis be interpreted?

Question Five is on the next page

5. (a) In relation to statistical regression, what is meant by the terms

i. residual,

ii. least squares regression line?

iii. What is meant by the term correlation? Your answer should include brief explanations

of what is meant by strong/weak and positive/negative correlation.

iv. Does the existence of a strong correlation between two variables necessarily indicate

that one is causing the other? Illustrate you answer with an example.

(b) i. Researchers wish to determine whether it is possible to estimate the age of a given

species of bear based upon the overall length of the bear, the girth of the bear’s chest,

its weight, and several other such measurements.

Given that the length of a bear is easily determined from photographs, the researchers

first investigate whether it is possible to estimate the age of a bear given the bears

length. Data relating to the age and length of 83 bears are analysed using linear

regression in minitab and the output and “normal probability” and “residuals versus

fits” plots of Figure 4 obtained.

This question is continued on the next page

Figure 5: Regression Analysis of log(Age) versus Length

Regression Analysis: log(Age) versus Length

The regression equation is log(Age) = - 0.178 + 0.0615 Length

Predictor Coef SE Coef T P

Constant -0.1777 0.2852 -0.62 0.

Length 0.061499 0.004717 13.04 0.

S = 0.439212 R-Sq = 67.7% R-Sq(adj) = 67.3%

Assume the researchers decide to proceed with the analysis of Figure 5.

A. Would you have any reservations about this decision? Justify your answer.

B. Based upon the regression equation of Figure 5, what is the expected age of a bear

that was 56 inches long?

C. It may be shown that a 95% confidence interval for the expected age of a bear that

is 56 inches long is 23.68–29.02 months. Carefully interpret this interval.

D. In Figure 5 the p-value associated with the coefficient of “length” in the regression

is 0.000. Briefly explain the meaning of this p-value. Is there any reasonable chance

that in fact there is no relationship between the length and age of a bear?

E. The five number summary of the lengths of the bears upon which the regression

analysis was based is

. From the regression equation the expected value of log age for a bear that is 20

inches long is 1.0522, corresponding to an expected actual age of exp(1.0522) = 2.9.

Why should we be wary of this expected age?

This question is continued on the next page

(c) The researchers decide to investigate whether it would be worthwhile performing a multiple

regression analysis to estimate the age of a bear. Using the “best subsets” option in minitab

the following output for log(Age) versus Sex, Head Length, Head Weight, Neck

Girth, Length, Chest Girth and Weight is obtained.

Best Subsets Regression: log(Age) versus Sex, Head.L, ...

Response is log(Age) 83 cases used

C

H H N L h W

e e e e e e

a a c n s i

S d d k g t g

Mallows e... t. h

Vars R-Sq R-Sq(adj) Cp S x L W G h G t

1 67.7 67.3 64.5 0.43921 X

1 64.6 64.2 78.4 0.45998 X

2 76.6 76.0 27.0 0.37629 X X

2 75.9 75.3 29.9 0.38154 X X

3 81.3 80.6 8.2 0.33866 X X X

3 81.1 80.4 9.0 0.34039 X X X

4 82.1 81.2 6.6 0.33332 X X X X

4 82.0 81.1 6.8 0.33384 X X X X

5 82.7 81.6 6.0 0.33000 X X X X X

5 82.6 81.4 6.4 0.33096 X X X X X

6 82.9 81.6 6.8 0.32961 X X X X X X

6 82.8 81.4 7.5 0.33105 X X X X X X

7 83.1 81.6 8.0 0.33001 X X X X X X X

Explain why, based upon the values of R-Sq(Adj), Mallow’s Cp and S above the models

with the five predictors Sex, Head Length, Head Weight, Length and Weight would be

considered the best model, but why the researchers may opt to go for the model with the

three predictors Sex, Neck Girth and Length.

Probability

Table entry for is the z

probability lying below.

z

TABLE A Standard normal probabilities

z

Tables

Probability
Table entry for is the z

probability lying below.

z

TABLE A Standard normal probabilities ( Continued)

Probabilityp

(x^2 )*

Table entry for p is the point ( X^2 ) with probability p lying above it. TABLE F X 2 distribution critical values p

Tables

 - .00 .01 .02 .03 .04 .05 .06 .07 .08. z
  • 3.4 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003 .0003.
  • 3.3 .0005 .0005 .0005 .0004 .0004 .0004 .0004 .0004 .0004.
  • 3.2 .0007 .0007 .0006 .0006 .0006 .0006 .0006 .0005 .0005.
  • 3.1 .0010 .0009 .0009 .0009 .0008 .0008 .0008 .0008 .0007.
  • 3.0 .0013 .0013 .0013 .0012 .0012 .0011 .0011 .0011 .0010.
  • 2.9 .0019 .0018 .0018 .0017 .0016 .0016 .0015 .0015 .0014.
  • 2.8 .0026 .0025 .0024 .0023 .0023 .0022 .0021 .0021 .0020.
  • 2.7 .0035 .0034 .0033 .0032 .0031 .0030 .0029 .0028 .0027.
  • 2.6 .0047 .0045 .0044 .0043 .0041 .0040 .0039 .0038 .0037.
  • 2.5 .0062 .0060 .0059 .0057 .0055 .0054 .0052 .0051 .0049.
  • 2.4 .0082 .0080 .0078 .0075 .0073 .0071 .0069 .0068 .0066.
  • 2.3 .0107 .0104 .0102 .0099 .0096 .0094 .0091 .0089 .0087.
  • 2.2 .0139 .0136 .0132 .0129 .0125 .0122 .0119 .0116 .0113.
  • 2.1 .0179 .0174 .0170 .0166 .0162 .0158 .0154 .0150 .0146.
  • 2.0 .0228 .0222 .0217 .0212 .0207 .0202 .0197 .0192 .0188.
  • 1.9 .0287 .0281 .0274 .0268 .0262 .0256 .0250 .0244 .0239.
  • 1.8 .0359 .0351 .0344 .0336 .0329 .0322 .0314 .0307 .0301.
  • 1.7 .0446 .0436 .0427 .0418 .0409 .0401 .0392 .0384 .0375.
  • 1.6 .0548 .0537 .0526 .0516 .0505 .0495 .0485 .0475 .0465.
  • 1.5 .0668 .0655 .0643 .0630 .0618 .0606 .0594 .0582 .0571.
  • 1.4 .0808 .0793 .0778 .0764 .0749 .0735 .0721 .0708 .0694.
  • 1.3 .0968 .0951 .0934 .0918 .0901 .0885 .0869 .0853 .0838.
  • 1.2 .1151 .1131 .1112 .1093 .1075 .1056 .1038 .1020 .1003.
  • 1.1 .1357 .1335 .1314 .1292 .1271 .1251 .1230 .1210 .1190.
  • 1.0 .1587 .1562 .1539 .1515 .1492 .1469 .1446 .1423 .1401.
  • 0.9 .1841 .1814 .1788 .1762 .1736 .1711 .1685 .1660 .1635.
  • 0.8 .2119 .2090 .2061 .2033 .2005 .1977 .1949 .1922 .1894.
  • 0.7 .2420 .2389 .2358 .2327 .2296 .2266 .2236 .2206 .2177.
  • 0.6 .2743 .2709 .2676 .2643 .2611 .2578 .2546 .2514 .2483.
  • 0.5 .3085 .3050 .3015 .2981 .2946 .2912 .2877 .2843 .2810.
  • 0.4 .3446 .3409 .3372 .3336 .3300 .3264 .3228 .3192 .3156.
  • 0.3 .3821 .3783 .3745 .3707 .3669 .3632 .3594 .3557 .3520.
  • 0.2 .4207 .4168 .4129 .4090 .4052 .4013 .3974 .3936 .3897.
  • 0.1 .4602 .4562 .4522 .4483 .4443 .4404 .4364 .4325 .4286.
  • 0.0 .5000 .4960 .4920 .4880 .4840 .4801 .4761 .4721 .4681.
  • T-
    • .00 .01 .02 .03 .04 .05 .06 .07 .08. z
  • 0.0 .5000 .5040 .5080 .5120 .5160 .5199 .5239 .5279 .5319.
  • 0.1 .5398 .5438 .5478 .5517 .5557 .5596 .5636 .5675 .5714.
  • 0.2 .5793 .5832 .5871 .5910 .5948 .5987 .6026 .6064 .6103.
  • 0.3 .6179 .6217 .6255 .6293 .6331 .6368 .6406 .6443 .6480.
  • 0.4 .6554 .6591 .6628 .6664 .6700 .6736 .6772 .6808 .6844.
  • 0.5 .6915 .6950 .6985 .7019 .7054 .7088 .7123 .7157 .7190.
  • 0.6 .7257 .7291 .7324 .7357 .7389 .7422 .7454 .7486 .7517.
  • 0.7 .7580 .7611 .7642 .7673 .7704 .7734 .7764 .7794 .7823.
  • 0.8 .7881 .7910 .7939 .7967 .7995 .8023 .8051 .8078 .8106.
  • 0.9 .8159 .8186 .8212 .8238 .8264 .8289 .8315 .8340 .8365.
  • 1.0 .8413 .8438 .8461 .8485 .8508 .8531 .8554 .8577 .8599.
  • 1.1 .8643 .8665 .8686 .8708 .8729 .8749 .8770 .8790 .8810.
  • 1.2 .8849 .8869 .8888 .8907 .8925 .8944 .8962 .8980 .8997.
  • 1.3 .9032 .9049 .9066 .9082 .9099 .9115 .9131 .9147 .9162.
  • 1.4 .9192 .9207 .9222 .9236 .9251 .9265 .9279 .9292 .9306.
  • 1.5 .9332 .9345 .9357 .9370 .9382 .9394 .9406 .9418 .9429.
  • 1.6 .9452 .9463 .9474 .9484 .9495 .9505 .9515 .9525 .9535.
  • 1.7 .9554 .9564 .9573 .9582 .9591 .9599 .9608 .9616 .9625.
  • 1.8 .9641 .9649 .9656 .9664 .9671 .9678 .9686 .9693 .9699.
  • 1.9 .9713 .9719 .9726 .9732 .9738 .9744 .9750 .9756 .9761.
  • 2.0 .9772 .9778 .9783 .9788 .9793 .9798 .9803 .9808 .9812.
  • 2.1 .9821 .9826 .9830 .9834 .9838 .9842 .9846 .9850 .9854.
  • 2.2 .9861 .9864 .9868 .9871 .9875 .9878 .9881 .9884 .9887.
  • 2.3 .9893 .9896 .9898 .9901 .9904 .9906 .9909 .9911 .9913.
  • 2.4 .9918 .9920 .9922 .9925 .9927 .9929 .9931 .9932 .9934.
  • 2.5 .9938 .9940 .9941 .9943 .9945 .9946 .9948 .9949 .9951.
  • 2.6 .9953 .9955 .9956 .9957 .9959 .9960 .9961 .9962 .9963.
  • 2.7 .9965 .9966 .9967 .9968 .9969 .9970 .9971 .9972 .9973.
  • 2.8 .9974 .9975 .9976 .9977 .9977 .9978 .9979 .9979 .9980.
  • 2.9 .9981 .9982 .9982 .9983 .9984 .9984 .9985 .9985 .9986.
  • 3.0 .9987 .9987 .9987 .9988 .9988 .9989 .9989 .9989 .9990.
  • 3.1 .9990 .9991 .9991 .9991 .9992 .9992 .9992 .9992 .9993.
  • 3.2 .9993 .9993 .9994 .9994 .9994 .9994 .9994 .9995 .9995.
  • 3.3 .9995 .9995 .9995 .9996 .9996 .9996 .9996 .9996 .9996.
  • 3.4 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997 .9997. - Tables T- z - df .25 .20 .15 .10 .05 .025 .02 .01 .005 .0025 .001. Tail probability - 1 1.32 1.64 2.07 2.71 3.84 5.02 5.41 6.63 7.88 9.14 10.83 12. - 2 2.77 3.22 3.79 4.61 5.99 7.38 7.82 9.21 10.60 11.98 13.82 15. - 3 4.11 4.64 5.32 6.25 7.81 9.35 9.84 11.34 12.84 14.32 16.27 17. - 4 5.39 5.99 6.74 7.78 9.49 11.14 11.67 13.28 14.86 16.42 18.47 20. - 5 6.63 7.29 8.12 9.24 11.07 12.83 13.39 15.09 16.75 18.39 20.51 22. - 6 7.84 8.56 9.45 10.64 12.59 14.45 15.03 16.81 18.55 20.25 22.46 24. - 7 9.04 9.80 10.75 12.02 14.07 16.01 16.62 18.48 20.28 22.04 24.32 26. - 8 10.22 11.03 12.03 13.36 15.51 17.53 18.17 20.09 21.95 23.77 26.12 27. - 9 11.39 12.24 13.29 14.68 16.92 19.02 19.68 21.67 23.59 25.46 27.88 29. - 10 12.55 13.44 14.53 15.99 18.31 20.48 21.16 23.21 25.19 27.11 29.59 31. - 11 13.70 14.63 15.77 17.28 19.68 21.92 22.62 24.72 26.76 28.73 31.26 33. - 12 14.85 15.81 16.99 18.55 21.03 23.34 24.05 26.22 28.30 30.32 32.91 34. - 13 15.98 16.98 18.20 19.81 22.36 24.74 25.47 27.69 29.82 31.88 34.53 36. - 14 17.12 18.15 19.41 21.06 23.68 26.12 26.87 29.14 31.32 33.43 36.12 38. - 15 18.25 19.31 20.60 22.31 25.00 27.49 28.26 30.58 32.80 34.95 37.70 39. - 16 19.37 20.47 21.79 23.54 26.30 28.85 29.63 32.00 34.27 36.46 39.25 41. - 17 20.49 21.61 22.98 24.77 27.59 30.19 31.00 33.41 35.72 37.95 40.79 42. - 18 21.60 22.76 24.16 25.99 28.87 31.53 32.35 34.81 37.16 39.42 42.31 44. - 19 22.72 23.90 25.33 27.20 30.14 32.85 33.69 36.19 38.58 40.88 43.82 45. - 20 23.83 25.04 26.50 28.41 31.41 34.17 35.02 37.57 40.00 42.34 45.31 47. - 21 24.93 26.17 27.66 29.62 32.67 35.48 36.34 38.93 41.40 43.78 46.80 49. - 22 26.04 27.30 28.82 30.81 33.92 36.78 37.66 40.29 42.80 45.20 48.27 50. - 23 27.14 28.43 29.98 32.01 35.17 38.08 38.97 41.64 44.18 46.62 49.73 52. - 24 28.24 29.55 31.13 33.20 36.42 39.36 40.27 42.98 45.56 48.03 51.18 53. - 25 29.34 30.68 32.28 34.38 37.65 40.65 41.57 44.31 46.93 49.44 52.62 54. - 26 30.43 31.79 33.43 35.56 38.89 41.92 42.86 45.64 48.29 50.83 54.05 56. - 27 31.53 32.91 34.57 36.74 40.11 43.19 44.14 46.96 49.64 52.22 55.48 57. - 28 32.62 34.03 35.71 37.92 41.34 44.46 45.42 48.28 50.99 53.59 56.89 59. - 29 33.71 35.14 36.85 39.09 42.56 45.72 46.69 49.59 52.34 54.97 58.30 60. - 30 34.80 36.25 37.99 40.26 43.77 46.98 47.96 50.89 53.67 56.33 59.70 62. - 40 45.62 47.27 49.24 51.81 55.76 59.34 60.44 63.69 66.77 69.70 73.40 76. - 50 56.33 58.16 60.35 63.17 67.50 71.42 72.61 76.15 79.49 82.66 86.66 89. - 60 66.98 68.97 71.34 74.40 79.08 83.30 84.58 88.38 91.95 95.34 99.61 102. - 80 88.13 90.41 93.11 96.58 101.9 106.6 108.1 112.3 116.3 120.1 124.8 128.
    • 100 109.1 111.7 114.7 118.5 124.3 129.6 131.1 135.8 140.2 144.3 149.4 153.
  • T- p