The University of SydneySchool of Mathematics and Statistics

Quiz 1 Sample

MATH1005: Statistics 2013Lecturers: Nicola Armstrong, Emi Tanaka and Di Warren

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Please write your answers in the answer boxes. Each question is worth one mark.There are 15 marks in all.

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Working will not be marked. Marks will be awarded on the basis of answers only. Answers will only be marked if they are written in the correct answer box. Non-programmable, non-graphics calculators are permitted. Your tutor will provide you with working paper. No other paper is allowed. Working paper will be collected, but not marked. You have 40 minutes to complete the quiz. By signing, you have agreed not to talk about the quiz problems until the releaseof the results.

Quiz Sample

Copyright c 2013 The University of Sydney 1

2

1. Find the third quartile, Q3, of the 8 observations :

35 33 34 36 30 32 32 32

Answer:

2. For the 8 observations in question 1, find the mean. Answer:

3. How many outliers are among the 15 observations :

10.6 11.5 11.2 12.2 10.0 12.7 15.0 10.2 11.1 10.0 10.4 12.1 11.8 12.3 12.9

(Hint: Q1 = 10.4 and IQR = 1.9. An ordered stem-and-leaf plot is useful for the nextquestion).

Answer:

4. For the 15 observations in question 3, find the median. Answer:

5. A correlation coefficient of r = 0.1 is reported for a sample of pairs (xi, yi). Without anyfurther information this implies:

(a) the points (xi, yi) lie on a straight line of slope 0.1.

(b) the points (xi, yi) are scattered about a straight line of slope 0.1.

(c) with a probability of 10% the relationship between x and y is best described with astraight line.

(d) the relationship between x and y is more likely to be non linear

(e) none of these

Write the corresponding letter in the box.

Answer:

6. For a sample x1, x2, ..., x17 we have17

i=1 xi = 258 and17

i=1 x2i = 4098. Find the stan-

dard deviation to 2 decimal places.

Answer:

3

7. For the following data:

i 1 2 3 4 5 6 7 8 9 10 11 12

xi 13 8 10 12 8 14 14 14 15 8 7 13

yi 13.4 8.8 10.0 12.6 7.2 13.1 14.9 12.6 14.1 7.1 7.4 12.5

i xi = 136,

i yi = 133.7, Sxy = 89.13333, Sxx = 94.66667, Syy = 90.56917. Find the

coefficient of correlation r (to 3 d.p.) and the slope b of the linear regression of y on x(to 3 d.p.). BOTH answers need to be correct.

Answer:

r = b =

8. The success probability of a new strategy is 0.1. Use tables to find the probabilityof less than 2 successes in 10 independent repetitions of applying the new strategy.

Answer:

9. Two doping tests are available, test A and test B. If it is known that a certain bloodsample was taken from a doped athlete then test A detects it with probability 0.6 and testB detects it with probability 0.8 whereas the probability that both tests simultaneouslydetect it is 0.54. What is the probability that neither of the tests detects that the bloodsample is suspicious?

Answer:

10. The random variable X is described by the following probabilities.

i 1 2 3

P (X = i) 0.5 0.4 0.1

What is the expected value of X?

Answer:

11. For the random variable in Question 10, what is the expected value of (X 1)?Answer:

4

12. Suppose that X is the number of heads obtained by flipping independently 3 times a coinfor which the probability of observing a tail is 23 and for head

13 . What is the expected

value of X?

Answer:

13. For X in question 12, find var(X) to 4 d.p.Answer:

14. If X B(10, 0.35), then what is P (X = 6) to 4 d.p.?Answer:

15. If a random variable X has mean 1 and variance 2 then for Y = X + 3 what is the resultof var(Y ) + E(Y )?

Answer:

5

Formula sheet for MATH1005 Statistics

Calculation formulae:

For a sample x1, x2, . . . , xn

Sample mean x 1nn

i=1 xi

Sample variance s2 1n1

[ni=1 x

2i 1n

(ni=1 xi

)2].

For paired observations (x1, y1), (x2, y2), . . . , (xn, yn)

Sxyn

i=1 xiyi 1nn

i=1 xin

i=1 yi

Sxxn

i=1 x2i 1n

(ni=1 xi

)2Syy

ni=1 y

2i 1n

(ni=1 yi

)2r

SxySxxSyy

For the regression line

y = a+ bx:

bSxySxx

a y bx

Some probability results:

For any two events A and B P (A B) = P (A) + P (B) P (A B) andP (A B) = P (A)P (B|A)

If A and B are mutually exclusive (m.e.) P (A B) = 0 and P (A B) = P (A) + P (B)If A and B are independent P (A B) = P (A)P (B)

If X B(n, p), then :P (X = r) =

(nr

)pr(1 p)nr, r = 0, . . . , n, E(X) = np and var(X) = np(1 p)

Some test statistics and sampling distributions under appropriate assumptions andhypotheses:

X N(,

2

n

)X/n N (0, 1)

XS/n tn1

XYSp

1nx

+ 1ny

tnx+ny2, where

S2p = [(nx 1)S2x + (ny 1)S2y ]/(nx + ny 2)

i

(OiEi)2Ei

2 , for appropriate

6

7

Answers

1. 34.5

2. 33

3. 1

4. 11.5

5. (e)

6. 3.38

7. r = 0.963, b = 0.942.

8. 0.7361

9. 0.14

10. 1.6

11. 0.6

12. 1

13. 23

14. 0.0689

15. 6

8