Review1

pdf

School

University of Toronto *

*We aren’t endorsed by this school

Course

537

Subject

Mechanical Engineering

Date

Dec 6, 2023

Type

pdf

Pages

4

Uploaded by Rogers_123

Report
REVIEW 1 1. The Rockwell hardness index for steel is determined by pressing a diamond point into the steel and measuring the depth of penetration. For 16 randomly selected specimens of a certain type of steel, the Rockwell hardness index averaged 61 with a standard deviation of 8. The manufacturer claims that this steel has an average hardness index of at least 64. (a) Is there evidence that the manufacturer overstates the average hardness index for the steel? Carry out the appropriate test at the level of significance α=0.05: (b) Obtain a 95% confidence interval for the mean hardness. Compare the interval with the outcome of the test in part (a). 2. The strength of concrete depends, to some extent, on the method used for drying. Two different drying methods for concrete were used on seven different mixes, with each mix of concrete subjected to each drying method. The resulting strength-test measurements (in psi) are given below: Mix Method I Method II A 3,160 3,170 B 3,240 3,220 C 3,190 3,160 D 3,520 3,530 E 3,480 3,440 F 3,220 3,210 G 3,120 3,120 (a) Is there evidence of a difference between average strengths for the two drying methods? Carry out the appropriate test α=0.05 to answer the question. (b) Obtain a 95% two-sided confidence interval for the difference in the average strength of concrete for the two drying methods. Compare the interval with the outcome of the test in part (a). (c) Now suppose the Method 1 and Method 2 values were obtained for the same mix. Assume equal variances. Is there evidence that the methods produce different strength concrete? Use SPSS to answer the question. Group Statistics Method N Mean Std. Deviation Std. Error Mean Strength 1 7 3275.7143 158.51994 59.91491 2 7 3264.2857 156.50955 59.15505
3. In an effort to link cold environments with hypertension in humans, two preliminary experiments were conducted to investigate the effect of cold on hypertension in rats. The two experiments are described below. Part 1: Experiment 1 In the first experiment 114 rats were divided at random into two groups of 57 rats each. The two samples of rats were exposed to different environments. One sample of rats was held in a normal environment at 26 o C (control group). The other sample order was held in a cold 5 o C environment (treatment group). Blood pressure was measured and recorded for rats in both groups. The goal of experiment was to compare blood pressure of rats in the two groups. (a) Is the study an example of an observational study or a randomized experiment? What kind of inferences can be drawn from the data in the study? Provide brief explanations. (b) As the distributions of blood pressure for both groups were right skewed, the natural logarithm transformation of blood pressure was applied to each group. The side-by-side boxplots for the log- transformed data exhibited approximately the same spread for the two groups and the corresponding Q-Q plots did not indicate any departures from the normality assumption. The following descriptive statistics were obtained (group 1=26 o C, group 2=5 o C):
(c) Use the above output to estimate the additive effect of a 5 o C environment on log blood pressure. Moreover, obtain an estimate of the multiplicative effect of a 5 0 environment on blood pressure of the rats. (d) Do the data provide sufficient evidence that rats exposed to a 5 o C environment have a higher mean blood pressure than rats exposed to a 26 o environment? Answer the question by carrying out an appropriate test for the log-transformed data. In particular, define the null and alternative hypotheses; report the value of the test statistic and the p-value of the test. State your conclusions. Moreover, obtain a 95 % confidence interval for the additive effect of a 5 o C environment on log blood pressure. (e) Use the 95% confidence interval obtained in part (b) to compute by hand a 95% confidence interval for the multiplicative effect of a 5 o C environment on blood pressure (take the antilogarithms of the endpoints). Summarize the interval in one sentence. Part 2: Experiment 2 In the second experiment 20 rats were randomly selected and their blood pressure was recorded in the temperature of 26 o C. Then the blood pressure of those 20 rats was measured again at 5 o C environment. The mean of the blood pressure differences (the blood pressure at 5 o C minus the blood pressure at 26 o C) was 110.5 (mm Hg) and their standard deviation 14.4859 (mm Hg). (f) Do the data allow us to conclude that rats exposed to a 5 o C environment have higher blood pressure than rats hold in a normal temperature? Carry out the appropriate test in SPSS to answer the question. In particular, specify the null and alternative hypotheses, report the value of the test statistic, the distribution of the test statistic under the null hypothesis and the p-value of the test. State your conclusions. Descriptives 5.2431 2.683E-02 5.1893 5.2968 5.2379 5.2040 4.104E-02 .2026 4.91 5.68 .77 .3023 .462 .316 -.528 .623 5.6898 3.324E-02 5.6232 5.7564 5.6871 5.6490 6.298E-02 .2510 5.26 6.19 .93 .3847 .299 .316 -.696 .623 Mean Low er Bound Upper Bound 95% Confidence Interval for Mean 5% Trimmed Mean Median Variance Std. Deviation Minimum Maximum Range Interquartile Range Skew ness Kurtosis Mean Low er Bound Upper Bound 95% Confidence Interval for Mean 5% Trimmed Mean Median Variance Std. Deviation Minimum Maximum Range Interquartile Range Skew ness Kurtosis GROUP 1.00 2.00 LNPRESS Statistic Std. Error
Your preview ends here
Eager to read complete document? Join bartleby learn and gain access to the full version
  • Access to all documents
  • Unlimited textbook solutions
  • 24/7 expert homework help
Part 3: Summary (g) Compare the sources of variation in the data in each of the two experiments. Which of the two experiment designs is more suitable to evaluate the effect of a 5 o C environment on blood pressure of rats? Can we conclude that the low temperature is the cause of the observed hypertension in rats? Explain briefly.