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Second Semester Examination – ECON1310 Quantitative Economic & Business Analysis A

PART A

Answer ALL questions in Part A.

A1. The most important continuous probability distribution is the normal distribution.

a.         What  are the parameters of the normal  distribution? What are some important properties of the normal distribution? (3 marks)

b.         Describe the standard normal distribution. Why is it useful in statistical work? (3 marks)

c.         Following their production,  industrial  generator  shafts are tested for static and dynamic balance, and the necessary weight is added to predrilled holes in order to bring each shaft within balance specifications. From past experience, the amount of weight added to a shaft has been normally distributed, with an average of 35 grams and a variance of 86 grams squared.

(i)        Find the probability that a randomly selected shaft will require at least 50 grams. (2 marks)

(ii)       The  worst  8%  of output  will  be  scrapped  and  recycled. What weight

cutoff should be used in deciding which shafts should be scrapped? (2 marks)

(iii)      If a random sample of 20 shafts is selected, find the probability that the mean weight that must be added will exceed 38 g. (3 marks)

A2. a.         A new cheese product is to be test marketed by giving a free sample to

randomly selected customers who are asked to state whether or not they like the product. With a 98% confidence level and a target sampling error of 0.05 or less, what sample size would you recommend if preliminary estimates are that 35% of the population will like the product? (2 marks)

b.        A  statistical  report  states that  a 90% confidence interval for the mean cost of business travel in a company is [$1050, $1336]. If there were 20 people in the sample determine the population standard deviation. (2 marks)

A4. A  research  analyst  for  an  oil  company  wanted  to  develop  a model to  predict  fuel consumption  (in miles per gallon) based on highway  speed  (in miles per hour). An experiment was designed in which a test car is driven at speeds ranging from 10 mph to 75 mph. The following simple linear regression output was obtained:

ANOVA

	df	SS
Regression	1	17.370
Residual	26	834.629
Total	27	851.999

	Coefficients	Standard Error
Intercept	12.746	2.499
Speed	0.039	0.053

a.         State the estimated equation. (1 mark)

b.         Interpret the coefficient of speed. (1 mark)

c.         Determine the standard error of the estimate. Explain what this measures. (2 marks)

d.         Calculate the Coefficient of Determination and interpret. (2 marks)

e.         Test whether  speed  is  a significant predictor of petrol consumption at the  5% level of significance. (5 marks)

f.          Comment on whether or not you think this is an adequate model. (1 mark)

PART B

Answer ALL questions in Part B.

B1. Given a normal distribution with µ = 13, σ = 2.5, what is the probability that X > 14.2?

a.         0.684

b.         0.48

c.         0.316

d.         none of the above.

B2. Wages in a particular industry average $11.90 per hour and the standard deviation is 40 cents. If the wages are assumed to be normally distributed, what must the wage be if only 10% of the workers can earn more?

a.         $12.41

b.         $63.10

c.         $13.18

d.         $11.39

B8. A random sample of customers buying petrol was selected. From this sample the 95% confidence interval estimate for the mean amount of petrol purchased per customer for the city was calculated to be between 55 and 78 litres. This means that

a.         95% of all customers bought an amount of petrol in the given range.

b.         There is a 95% probability that the sample mean lies in this range.

c.         If another sample was taken there is a 95% chance that the confidence interval for the mean would be between 55 and 78 litres.

d.         The proprietor can be  95% confident that city customers purchase between 55 and 78 litres of petrol on average.

B9. Strategic business planning is required for the successful transfer of control between    generations in family-owned companies. A survey of 17 family firms whose annual     turnover exceeded one million dollars found that 28%  had no strategic business plan.  Assuming that simple random sampling was employed, estimate with 95% confidence the proportion of family-owned companies operating without strategic business plans. The upper limit of this estimate is

a.         0.4934

b.         0.3032

c.         0.2134

d.         0.4591

B10. A  manufacturing  company  was  interested  in  determining  whether  different  training methods have an effect on speed of work (hence productivity) of their  assembly line employees. From independent samples from two training methods the 90% confidence interval for the difference between the mean assembly times (in minutes) was calculated to be  2.02 < µ1 - µ2 < 12.51. We could conclude from this with 90% confidence that

a.         training method 2 is better than training method 1.

b.         training method 1 is better than training method 2.

c.         the minimum average assembly time from training method  1 is 2.02 minutes.

d.         we are unable to say which training method is better.

B11. If n = 24 and α = 0.05, then the critical value of t for testing the hypotheses H0 : µ = 38

H1 : µ < 38 is:

a.         2.069

b.         1.714

c.         -1.714

d.         -2.069

B12. The approval process for selling life insurance is complex. The ability to deliver

approved policies to customers in a timely manner is critical to the profitability of the    firm. A random sample of 27 policies was selected and the processing time in days was recorded. The PhStat output for testing whether the average processing time is different from 30 days is presented below.

t Test for Hypothesis of the Mean
Null Hypothesis µ=	30
Level of Significance	0.05
Sample Size	27
Sample Mean	43.8889
Sample Standard Deviation	25.2835
p-Value	0.0084

Which of the following is TRUE?

a.         the null hypothesis cannot be rejected.

b.         a one tail test has been conducted with degrees of freedom 26.

c.         the  test statistic is calculated to be 2.056.

d.         can conclude processing time is not 30 days.

B13. The Road Safety Council suggests that more than 35% of all road accidents causing    injury have a basis in alcohol abuse. To test this, the appropriate hypotheses would be

a.         H0 :      p > .35            H1 : p < .35

b.         H0 :      p < .35 H1 : p > .35

c.         H0 :      p = .35            H1 : p ≠ .35

d.         H0 :      p = .35            H1 : p > .35

B14. In a hypothesis test

a.         the null hypothesis is always assumed to be true until proved otherwise.

b.         the alternative hypothesis is always assumed to be true until provedotherwise.

c.         the alternative hypothesis is accepted unless there is sufficient evidence to say otherwise.

d.         the null hypothesis is assumed to be false until proved otherwise.

B19. Suppose that in a right-tailed test you compute the value of the Z test statistic to be 2.12. What is the p-value?

a.         0.9830

b.         0.0340

c.         0.4915

d.         0.0170

B20. Given the null hypothesis H0 : µ ≤ 350, and the decision rule

‘Reject H0 if Zcalc > 1.85’, a type II error will occur when

a.         µ = 360,           Zcalc  = 2.01

b.         µ = 360,           Zcalc  = 1.67

c.         µ = 340,           Zcalc  = 2.01

d.         µ = 340,           Zcalc = 1.67

B21.    Which of the following is TRUE for simple linear regression?

a.         The variance of the error distribution is a function of the independent variable.

b.         The regression slope coefficient is different from zero.

c.         The covariance between any pair of error terms is equal to one.

d.         The population values of the dependent variable are normally distributed.

B22.    In simple linear regression, if there is no linear relationship between X andY, then:

a. β 1 = 0

b.         MSE = 1

c. β0 = 0

d.         r  = –1