Assignment in Engineering Applications of Mathematical Statistics

Solve tasks from 1 to 6. In each case describe the statistical method/tool used and justify its choice.

Explain the results you obtained.

1. Find the data on daily temperatures, in the city where you originally study, in November 2014.

Use the descriptive statistics tools to characterize this set of data.

2. A machine is used for filing 1 kg canes with the liquid paint. Based on the experience, the distribution of the dispensed amounts of paint is normal, with the standard deviation =1.1. In order to check the mean dose of paint, 9 canes were randomly selected and their content was weighted.

The results obtained were: 100.3, 99.4, 97.9, 101.2, 102.0, 100.7, 98.5, 99.6, 100.9. Estimate the confidence interval for the mean amount of paint dispensed by the machine at the confidence level P= 0.99 and P= 0.95.

3. The acceleration of car was measured on two roads A and B, which have the same quality.

Verify the hypothesis that car acceleration is the same on both roads at the significance level of 0.05.

Measurement 1 2 3 4 5 6 7 8 9 10 11

acceleration on
road A [m/s^{2}]

4.2 4.3 3.9 4.5 3.8 3.6 3.9 4.3 4.5 3.6 4.2 acceleration on

road B [m/s^{2}]

4.8 5.2 4.2 5.1 4.7 4.3 4.9 5.3 5.6 5.2 4.8

4. A study was performed to determine whether men and women differ in their
repeatability in assembling components on printed circuit boards. Random samples of 25 men
and 21 women were selected and each subject assembled the units. The two sample standard
deviations of assembly time were 𝑠_{𝑚𝑒𝑛} = 0.98 min and 𝑠_{𝑤𝑜𝑚𝑒𝑛} = 1.02 min. Can we support
the claim that men and women differ in repeatability for this assembly task? Use 𝛼 = 0.01.

State the necessary assumptions about the underlying distribution of the data.

5. The average chemical yield of a process was investigated on a pilot plant. The influence of temperature, catalyst and catalyst concentration on the yield was examined. The results of experiment are shown in the table. Investigate the influence of temperature, catalyst and its concentration on the yield of reaction.

Temperature (◦C) Concentration (%) Catalyst Yield (g)

160 20 C1 59, 61

160 20 C2 50, 54

160 40 C1 50, 58

160 40 C2 46, 44

180 20 C1 74, 70

180 20 C2 81, 85

180 40 C1 69, 67

180 40 C2 79, 81

6. Stack-losses of ammonia 𝑌 were measured in course of 21 days of operation of a plant
for the oxidation of ammonia (NH3) to nitric acid (HNO3). The factors influencing stack losses
were: the flow of cooling air 𝑥_{1}, the temperature of cooling water 𝑥_{2}, and acid concentration
𝑥_{3}. The experimental data is given in the table.

𝑦 ^{42 } ^{37 } ^{37 } ^{28 } ^{18 } ^{18 } ^{19 } ^{20 } ^{15 } ^{14 } ^{14 } ^{13 } ^{11 } ^{12 } ^{8 } ^{7 } ^{8 } ^{8 } ^{9 } ^{15 } ^{15 }
𝑥_{1} ^{80 } ^{80 } ^{75 } ^{62 } ^{62 } ^{62 } ^{62 } ^{62 } ^{58 } ^{58 } ^{58 } ^{58 } ^{58 } ^{58 } ^{50 } ^{50 } ^{50 } ^{50 } ^{50 } ^{56 } ^{70 }
𝑥_{2} ^{27 } ^{27 } ^{25 } ^{24 } ^{22 } ^{23 } ^{24 } ^{24 } ^{23 } ^{18 } ^{18 } ^{17 } ^{18 } ^{19 } ^{18 } ^{18 } ^{19 } ^{19 } ^{20 } ^{20 } ^{20 }
𝑥_{3} ^{89 } ^{88 } ^{90 } ^{87 } ^{87 } ^{87 } ^{93 } ^{93 } ^{87 } ^{80 } ^{89 } ^{88 } ^{82 } ^{93 } ^{89 } ^{86 } ^{72 } ^{79 } ^{80 } ^{82 } ^{91 }

Fit a multiple regression model to this data. Discuss the obtained results. Use the regression
model to predict stack-loss of ammonia when 𝑥_{1} = 61, 𝑥2 = 20 and 𝑥3 = 85.

dr hab inż. Monika Maciejewska