Sample Final Sem 1 0809 SSE2193/SSM3763

Answer All Questions

1. (a) (i) Suppose that samples of size n = 25 are selected at random from a nor-

mal population with mean 100 and standard deviation 10. What is the

probability that the sample mean falls in the interval from µX − 1.8σX to

µX + 1.0σX? [4 marks]

(ii) A normal population has mean 100 and variance 25. How large must the

random sample be if we want the standard error of the sample average to

be 1.5? [3 marks]

(b) In November 2008, The United States of America will be having its general

election to elect a president. A survey was conducted on two major states

to determine which of the two political parties will win the coming general

election. In the first state, 52% of the voters are Republicans, and 48% are

Democrats. In the second state, 47% of the voters are Republicans, and 53%

are Democrats. Suppose 100 voters are selected at random from each state.

What is the probability that the sample proportion of Republican voters in the

second state will be more than in the first state? [5 marks]

2. The machine shop in an aircraft manufacturing company deals with a variety of jobs.

The floor supervisor wants to estimate the proportion of nonconforming parts since

rework is affecting the production cost. A random sample of 200 parts shows that

six are nonconforming. Construct a 90% confidence interval for the true proportion

of nonconforming parts. [5 marks]

3. The current curfew hour of students residing in college of residences in UTM is at 12

midnight. The student council received complaints that this rule is not convenient

for the students. A survey was taken to investigate whether the support against

this ruling differs among male and female students. A random sample of 90 male

students were taken and 75 were against the ruling. Meanwhile 60 out of a sample

of 90 female students were against the ruling. Do these data indicate that there is a

significant difference in the support against the current curfew hour at 12 midnight

among male and female students? Use α = 0.01. [8 marks]

4. An analyst is investigating the time spent by shift A and shift B employees to

assemble products in a certain industry. He collected the following random data of

the times (in minutes) spent by 13 employees on product assembly in the industry.

Assembly time by shift A employees 20 19 17 15 25 20 16

Assembly time by shift B employees 27 28 26 29 19 20

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Sample Final Sem 1 0809 SSE2193/SSM3763

Assume that the assembly time is normally distributed.

(a) By letting σ2A and σ2

B as the variances of assembly times spent by shift A and

shift B employees respectively, help the analyst to construct a 98% confidence

interval for the ratio of σ2B to σ2

A. [5 marks]

(b) Show that at 2% level of significance the variability of assembly times is the

same between two shifts. Compare your answer in (a) with this statement and

comment. [6 marks]

(c) Use statement in (b) as your additional assumption, can you conclude that the

shift A employees assemble the products faster than their shift B counterparts

do at α = 0.01? [8 marks]

(d) Would you still make the same conclusion in part (c) if α increases by 2%?

[1 mark]

5. (a) The thickness of 194 sunscreen films (measured in 10−7 mm) produced by a

process are shown below:

Thickness (X) No. of films

X < 50 8

50 6 X < 55 26

55 6 X < 60 59

60 6 X < 65 67

65 6 X < 70 30

X > 70 4

It is claimed that the thickness of all sunscreen films follows normal distribu-

tion with mean and standard deviation of 60 × 10−7 mm and 5.5 × 10−7 mm

respectively. Thus after standardization,

Thickness, X Z =X − µ

σ

X < 50 Z < −1.82

50 6 X < 55 −1.82 < Z < −0.91

55 6 X < 60 −0.91 < Z < 0

60 6 X < 65 0 < Z < 0.91

65 6 X < 70 0.91 < Z < 1.82

X > 70 Z > 1.82

Find the probability of the standard normal random variable in the above table

and test whether the claim is true. Use α = 0.05 [10 marks]

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Sample Final Sem 1 0809 SSE2193/SSM3763

(b) A company operates three machines in three shifts each day. From production

records, the following table shows the number of breakdowns in three shifts

during the past years.

Type of Breakdowns

Shift A B C

1 38 18 18

2 34 22 16

3 18 16 20

Test the hypothesis using α = 0.05 that the types of breakdown are not related

to the shift. [8 marks]

6. A manufacturer of paper used for making envelopes is interested in improving the

tensile strength of the product. A product engineer thinks that tensile strength is a

function of the hardwood concentration in the pulp. The product engineering team

decides to investigate the following levels of hardwood concentration, 0.5%, 5.5%,

10.5%, and 15.5%. In their pilot plant they run six test specimens at each concen-

tration level. All 24 specimens are tested randomly in a tensile tester laboratory.

The data from this experiment has been partially analysed and the following incom-

plete ANOVA table is obtained. Help the team complete the analysis by stating

the appropriate hypotheses, reconstructing and completing the table, and comment

whether or not the tensile strength is a function of the hardwood concentration in

the pulp at α = 0.05.

Source of Sum of Degrees Mean Squares Fcalcvariation Squares of freedom

Hardwood

concentration 362.79

Error

Total 492.96 23

[8 marks]

7. (a) Below is a summary for a set of data from a response variable Y and a predictor

variable X.∑X = 195,

∑Y = 467,

∑X2 = 6775,∑

Y 2 = 38061,∑XY = 14320, n = 6.

(i) Based on the above, find the estimated equation of the linear regression

line, Y = α + βX. [6 marks]

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Sample Final Sem 1 0809 SSE2193/SSM3763

Figure 1: Plot of peak hour demand versus total monthly energy usage for the electric

utility data.

(ii) Calculate the Pearson product moment correlation coefficient for the above

data and interpret the result. [2 marks]

(b) An electric utility company is interested in developing a model relating peak

hour demand (y, in kilowatts) to the total monthly energy usage during the

month (x, in kilowatts). Data for a small sample of 10 residential customers

are observed. Figure 1 shows the scatter plot of the two variables. The

following questions relate to the observations in Figure 1.

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Sample Final Sem 1 0809 SSE2193/SSM3763

(i) Describe the general relationship in relating the peak hour demand to

the total monthly energy usage during the month amongst the residential

customers. [1 mark]

(ii) The fitted regression line for the data is given by

Y = −1.206 + 0.004X.

What can you infer from the value of the estimated slope coefficient?

[1 mark]

(iii) The following table shows an extraction of the Excel output for the esti-

mates of unknown regression coefficients for the data.

Coefficients Standard Error t-Stat p-Value

Intercept −1.206 0.241 −5.000 0.001

Energy usage 0.004 0.0002 17.945 9.53E − 08

Based on the table above, is there any evidence of the existence of linear

relationship between the peak hour demand and the total monthly energy

usage during the month at 1% significance level? [2 marks]

8. (a) Fluoride emissions (parts per million) from a chemical plant are monitored

routinely. The following data are 12 observations based on air specimens taken

during one month of production:

151 148 143 146 154 145 155 144 142 153 152 147

Can you claim that the fluoride impurity level is randomly below and above

the mean of 150 ppm? Use α = 0.05. [5 marks]

(b) A 15 randomly chosen consumers were asked to rate two engineering products

on a scale of 1 to 5 (5 being the highest). For each consumer, the first rating

in the pair is of product A, and the second is of product B. The data are:

(3, 4), (4, 2), (2, 5), (1, 5), (4, 5), (3, 3), (3, 5), (2, 5),

(5, 4), (4, 4), (3, 4), (3, 5), (3, 4), (2, 2), (4, 5)

Use the sign test to test the hypothesis, at 0.05 level of significance, that

product B is favored over product A. [6 marks]

(c) An uninterrupted power supply manufacturer claims that the continuous sup-

ply of electric power from a momentary power interruption of brand X is iden-

tical with that of brand Y . To test this claim, the following determinations of

continuous supply of electric power, in minutes, were recorded:

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Sample Final Sem 1 0809 SSE2193/SSM3763

Brand X 26.3 28.1 25.9 27.7 26.3 25.8 28.5

Brand Y 27.7 23.2 25.8 24.1 23.5

Use the Mann-Whitney test with α = 0.1 to test whether the claim is valid.

[6 marks]

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