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FIN10002!FINANCIAL!STATISTICS!REVISION! !

Descriptive Statistics 1.Determine the mode, median and mean of the following data. 16

9

19

10

16

16

4

18

19

10

8

2

20

8

17

2. The owner of a new Indian restaurant is wondering how its prices compare with others in the local area. Use the following price information on a sample of Rogan Josh main dishes to write a short report to the restaurant owner about the central tendency of the data. OUTLET

PRICE

OUTLET

PRICE

Naan Place

$14.50

Spice of India

$14.90

Vindaloo Palace

$13.90

Tasty Biryani

$14.50

Chole Now

$10.90

Jimmy’s

$11.90

Sounds of India

$13.50

House of Spice

$11.50

Café India

$12.90

Malai Village

$11.50

Kings

$14.50

Lassi Palace

$14.50

Metro

$13.90

Tasty Tandoori

$12.50

Saffron Plaza

$11.50

Second Avenue

$10.50

3. A data set contains the following seven values. 6 a. b. c. d. e.

2

4

7

8

3

5

Calculate the range. Calculate the population standard deviation. Calculate the interquartile range. Calculate the z-score for each value. Calculate the coefficient of variation.

4. A wine industry association reported in its magazine that a particular wine was being marketed by online wine distributors with an average market price of $125 and standard deviation of $12, with the distribution of prices being approximately bell shaped. One boutique wine distributor is concerned by this report as they are charging $50 per bottle for this particular wine. Between what two price points would approximately 68% of prices fall? Between what two numbers would 95% of the prices fall? Between what two values would 99.7% of the prices fall? Write a short report informing the distributor whether the current price being charged is comparable to others. 5. An NRMA report stated that the average age of a car in Australia is 10.5 years. Suppose the distribution of ages of cars on Australian roads is approximately bell shaped. If the standard deviation is 2.4 years, between what two values would 95% of the car ages fall?

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FIN10002!FINANCIAL!STATISTICS!REVISION! !

Probability 6. If X is the set of events that lists the seven days of a week, what is the probability that any single day chosen from X will be: a. b. c. d.

a Monday not a Monday a weekday a weekend day?

7. A white goods manufacturer is sourcing parts for its air conditioning units. Management are currently putting together a timeline for the next stages of manufacturing. To do so they consider that one set of parts relating to the motor are coming from Germany and one set of parts relating to the casing will come from the United States. Past experience suggests that there is an 8% probability that the German parts will arrive late due to disruptions in shipping. The probability of parts arriving late from the United States is deemed to be 5%. a. What is the probability that parts from Germany will not be delayed? b. What is the probability that parts from the United States will not be delayed? c. Assume that the events in the two previous questions are independent. What is the probability that the parts from Germany will arrive late given that parts from the United States arrived late? 8. A hardware store determines that 70% of its customers do not use the selfcheckout system to make their purchases. It also determines that 80% of its customers pay by cash. Among those using the self-checkout system, however, only 60% pay by cash. a. Use this information to determine the probability that a customer uses the selfcheckout system and pays by cash. b. If use of the self-checkout system and payment by cash are independent, what would the probability in part (a) be?

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FIN10002!FINANCIAL!STATISTICS!REVISION! ! 9. use the values in the contingency table to solve the equations given.

A

B

C

D

a. b. c. d.

E

F

G

15

12

8

11

17

19

21

32

27

18

13

12

P (G | A) =___ P (B | F) =___ P (C | E) =___ P (E | G) =___

10. Consider the following results of a survey asking, ‘Have you visited a museum in the past 12 months?’ and ‘Do you have a child less than 10 years of age?’ VISITED MUSEUM IN LAST YEAR

CHILD UNDER 10

Yes

No

Total

YES

NO

TOTAL

160

80

240

40

120

160

200

200

400

Is the variable ‘Museum visitor’ independent of the variable ‘Child under 10’? Why or why not?

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FIN10002!FINANCIAL!STATISTICS!REVISION! !

11. The following probability matrix contains a breakdown of the age and gender of general practitioners working in Australia. AGE (YEARS)

GENDER

Male

Female

Total

< 35

35–44

45–54

> 54

TOTAL

0.036

0.129

0.194

0.261

0.620

0.054

0.122

0.130

0.074

0.380

0.090

0.251

0.324

0.335

1.000

What is the probability that one randomly selected general practitioner: a. b. c. d. e.

is 35–44 years old is both female and 45–54 years old is male or is 35–44 years old is less than 35 years old or more than 54 years old is female if they are 45–54 years old?

12. A study of tweeting behaviour revealed that among 36- to 45-year-olds, the number one tweeted topic related to family, with 28% making some reference of it among those tweets that could be categorised in a mutually exclusive manner. Other tweets were categorised as referring to the arts (22%), entertainment (18%), education (17%), and sports (11%), while 4% referred to some other topic. If a tweet by a 36- to 45-year-old is randomly selected and able to be categorised in the same way, determine the probabilities of the following. a. b. c. d.

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The tweet is about family. The tweet is not about sports. The tweet is about the arts or entertainment. The tweet is neither about family nor about entertainment.

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FIN10002!FINANCIAL!STATISTICS!REVISION! !

Normal Distribution 13. Using μ = 25 and σ = 3, calculate the corresponding z-scores for each of the following values of x. a. b. c. d.

16 20 24 28

14 Using μ = 100 and σ2 = 25, calculate the corresponding z-scores for each of the following values of x. a. b. c. d.

85 92 103 113

15. Consider the average home mortgage in New Zealand of $283 000 where the standard deviation of the mortgages is $50 000 and home mortgages are normally distributed. a. What proportion of home loans are more than $250 000? b. What proportion of home loans are between $250 000 and $300 000? c. If a home loan is known to be more than $250 000, what is the probability that it is less than $280 000 16. According to the Department of Building and Housing, the average rent for a one-bedroom Auckland City apartment was $314 with a standard deviation of $88. Given the rents are normally distributed, what is the probability that a student can find a one-bedroom apartment that rents for: a. less than $200 per week b. more than $400 per week c. between $280 and $350 per week? 17. The weight of a medium size loaf of homemade bread in Australia, using highgrade baking flour and a bread-making machine, has been found to be normally distributed with an average weight of 750 g and standard deviation of 10 g. What is the probability that a homemade loaf of bread will be: a. less than 745 g b. more than 795 g c. between 725 g and 785 g?

18. What is the difference between a continuous random variable and discrete random variable

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FIN10002!FINANCIAL!STATISTICS!REVISION! !

Sampling 19 A population has a mean of 150 and a standard deviation of 21. If a random sample of 49 is taken, what is the probability that the sample mean is: a. b. c. d. e.

greater than 154 less than 153 less than 147 between 152.5 and 157.5 between 148 and 158?

20 A population is normally distributed, with a mean of 14 and a standard deviation of 1.2. What is the probability of each of the following? a. Taking a sample of 26 and obtaining a sample mean of 13.7 or more b. Taking a sample of 15 and getting a sample mean of more than 15.7 21 A new estate contains 1500 houses. A sample of 100 houses is selected randomly and evaluated by a real estate agent. If the mean appraised value of a house for all houses in this area is $300 000, with a standard deviation of $10 500, what is the probability that the sample average is greater than $303 000? 22 The monthly mobile phone bill for all customers at a large telecommunications company has been found to be normally distributed with a mean of $145.55 per month and standard deviation of $15.22 per month. What value would be exceeded by 65% of sample means if the sample size was 40? 23 Use the following information to construct the confidence intervals specified to estimate μ. a. 99% confidence for 𝑥"= 35, σ = 5.2 and n = 55 b. 95% confidence for 𝑥"= 85, σ = 6.3 and n = 35

𝑥"= 6.4, σ = 0.8 and n = 42 85% confidence for 𝑥"= 22.3, σ = 2.1, N = 600 and n = 72

c. 90% confidence for d.

24. A random sample of size 70 is taken from a population that has a variance of 49. The sample mean is 90.4. What is the point estimate of μ? Construct a 94% confidence interval for μ. 25 The following data are selected randomly from a population of normally distributed values. 20.5

24.8

22.3

23.7

22.2

27.8

27.6

19.7

21.0

24.3

23.1

20.6

20.5

35.3

Construct a 90% confidence interval to estimate the population mean. !

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FIN10002!FINANCIAL!STATISTICS!REVISION! ! 26. According to Runzheimer International, organisations are investing an average of $1436 each year per mobile device. A researcher randomly selects 28 organisations and calculates the average amount they invested during the year on mobile devices to be $1295. The sample standard deviation is calculated to be $245. Construct a 99% confidence interval for the population mean amount invested by organisations each year on mobile devices using these sample data. Assume the data are normally distributed in the population. Does the $1436 figure, reported by Runzheimer International, fall within the confidence interval calculated using the researcher’s sample data? What does this tell you?

Hypothesis testing 27. It is possible that the sampling error will lead to an error in the conclusion regarding the null hypothesis. What errors can occur and explain the difference between them. 28. For each of the following, state the null and alternative hypotheses. a. The mean length increased to above 15 cm. b. The population proportion is the same as the historical value of 50%. c. The average height of a 32-year-old male in a community is claimed to be at most 180 cm. d. The proportion of healthy children in a community is believed to be less than 75%. e. The average weight of people in a community is 85 kg. 29. For each of the hypothesis statements in problem 28, is a two-tailed test or onetailed test needed? If it is a one-tailed test, indicate if it is a left-tailed or right-tailed test. 30. From the information given, indicate if a correct decision, a Type I error or a Type II error was made. a. b. c. d.

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H0 : μ = 1.5 litres. The decision was to not reject H0 and μ is actually 1.5 litres. H0 : μ = 1.5 litres. The decision was to not reject H0 and μ is actually 1.6 litres. H0 : μ = 1.5 litres. The decision was to reject H0 and μ is actually 1.5 litres. H0 : μ = 1.5 litres. The decision was to reject H0 and μ is actually 1.6 litres.

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FIN10002!FINANCIAL!STATISTICS!REVISION! ! 31Use the data given to test the following hypotheses. H0: μ = 18 𝑥= 16.7

n = 31

Ha: μ ≠ 18 σ = 4.2

α = .05

b. Use the p-value to reach a statistical conclusion. c. Using the critical value method, what are the critical sample mean values? 32 Use the data given to test the following hypotheses. Assume the data are normally distributed in the population. H0: μ ≥ 7.48 𝑥 = 6.91

n = 24

Ha: μ < 7.48 σ = 1.21

α = .01

33 The average annual income for Australian childcare workers five years ago was $38 100. It is believed this average annual income has essentially remained the same since then. To test this hypothesis, a random sample of 22 Australian childcare workers was selected and the average annual salary found to be $40 200. Taking the population standard deviation for annual income to be $2500 per year, using α = .01, and assuming annual incomes of the workers are normally distributed, test the researcher’s hypothesis. 34 A random sample of size 20 is taken, resulting in a sample mean of 16.45 and a sample standard deviation of 3.59. Assuming that x is normally distributed, use this information and α = .05 to test the following hypotheses. H0: μ = 16 Ha: μ ≠ 16 35 A random sample of 51 items is taken, with 𝑥= 58.42 and s 2 = 25.68. Use these data to test the following hypotheses, assuming you want to take only a 1% risk of committing a Type I error and that x is normally distributed. H0: μ ≥ 60 Ha: μ < 60

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FIN10002!FINANCIAL!STATISTICS!REVISION! !

Chi-Squared 36 The human resource manager of a large IT company collected data for days of the week on which employees were absent from work. The manager randomly selects 150 of the absences, which are shown in the table below. Use α = .05 to determine whether the data indicate whether absences during the various days of the week are equally likely. DAY OF WEEK

Monday

Tuesday

Wednesday

Thursday

Friday

Total

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OBSERVED FREQUENCY

42

18

24

27

39

150

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FIN10002!FINANCIAL!STATISTICS!REVISION! ! 37 Use a chi-square goodness-of-fit test to determine whether the following observed frequencies represent a uniform distribution. Let α = .01. CATEGORY

FO

1

20

2

17

3

13

4

14

5

18

6

21

7

19

8

18

38. Four investment companies were asked in a survey to rate the effect on their asset values of a 1 percentage point fall in stock market prices. The effect on asset values was rated using a 3-point scale, with 1 for no effect, 2 for a slight effect and 3 for a moderate effect. Actual (observed) counts of the responses are shown in the following table. Is the effect on asset values independent of the company? Let α = .01. EFFECT ON ASSET VALUES COMPANY

NONE

SLIGHT

MODERATE

TOTAL

A

60

27

7

94

B

71

22

14

107

C

57

38

15

110

D

54

37

18

109

242

124

54

420

Total

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FIN10002!FINANCIAL!STATISTICS!REVISION! ! 39 The following is a 3 × 2 contingency table for annual farm profit and the age of the farmer for a randomly selected sample of farmers in Fiji. Use an appropriate test to determine whether annual farm profit is related to a farmer’s age. Comment on the results of your test. a= 0.05 ANNUAL FARM PROFIT

AGE CATEGORY

< $2000

$2000 OR MORE

TOTAL

Less than 40 years

29

18

47

40−49 years

51

38

89

50 years or more

82

80

162

Total

162

136

298

Regression and Correlation 40 An economist for the state government of Mississippi recently collected the data contained in the table below on the percentage of people unemployed in the state at randomly selected points in time over the past 25 years and the interest rate of Treasury bills offered by the federal government at that point in time Interest Rates (x) 8.1 9.8 8.2 8.4 7.9 8.3 10.2 10 8.4 8.5 10.2 10.1 8.8 8.3 10.1 10.7 11.4 7.8

Unemployment %(y) 4.4 7.8 4.7 5 3.9 5.1 9.5 8.8 4.5 5 10.2 9 6 5 9.5 13.4 15 4

The economist used Excel to produce the following: !

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FIN10002!FINANCIAL!STATISTICS!REVISION! !

a. How did the economist produce the excel output? b. Using the output, what is the simple regression model to determine whether the variation in unemployment % can be explained by the interest rate c. What is the regression slope and what does it mean? d. What are the appropriate hypotheses to test? Using a significance level of 0.05. e. Determine the rejection region for the test statistic t and determine if the null hypothesis is rejected or accepted. f. Based on the estimated regression model, what percentage of the total variation in unemployment % can be explained by interest rate?

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FIN10002!FINANCIAL!STATISTICS!REVISION! ! 41. The Public Utility Commission in a southern state is interested in describing the relationship between household monthly utility bills and the size of the house. A recent study of 30 randomly selected household resulted in the following regression results:

Based on the information provided, indicate what, if any, conclusions can be reached about the relationship between utility bill and the size of the house in square feet.

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FIN10002!FINANCIAL!STATISTICS!REVISION! ! 42. Although the Jordan Banking System, a smaller regional bank, generally avoided the subprime mortgage market and consequently did not take money from the Federal Troubled Asset Relief Program (TARP), its board of directors has decided to look into all aspects of revenues and costs. One service the bank offers is free cheque account, and the board is interested in whether the costs of this service are offset by revenues from interest earned on the deposits. One aspect in studying cheque accounts is to determine whether changes in average cheque account balance can be explained by knowing the number of cheques written per month. The sample is in the table below:

a. Develop the least squares re...