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1) There are 100 mathematics majors and 200 engineering majors in a group of 500 students

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1) There are 100 mathematics majors and 200 engineering majors in a group of 500 students. If a student is randomly selected, answer the following questions:   (a)What is the probability that the student is not an engineering major?   (b)What is the probability that the student is a mathematics major?     (c) What is the probability that the student is neither a mathematics major nor an engineering major?

2) To assess the quality of a certain product of a company, the controller of the company uses a sample of size 3. His choices for rating are as follows:   i. Above the average,    ii. Average,     iii. Below the average.Answer the following questions:     (a) State the sample space of the rating.      (b)Define an event that he rates only one item above the average. Calculate the probability of this event.    (c)Define an event that he rates at least two items above the average. Calculate the probability of this event.           (d)Define an event that he rates at most one item above the average. Calculate the probability of this event.

3) Assume that the readings at freezing on a batch of thermometers are normally distributed with a mean of 0°C and a standard deviation of 1.00°C. A single thermometer is randomly selected and tested. FindP₂₈, the 28-percentile. This is the temperature reading separating the bottom 28% from the top 72%.

4) Summarize the dataset. Explain how you can use the summary statistic to detect an outlier. Is there any outlier in the dataset? 2. Support your answer in 1 by plotting a boxplot. Describe your boxplot on how it can support your answer in 1. 3. Explain how you can determine the skewness of the data from the summary statistics. 4. Support your answer in 3 by plotting a histogram. Describe your histogram on how it can support your answer in 3. 5. Plot a suitable graph to test the normality of the data. Explain the result that you get. 6. Null hypothesis, h0: The value of y is equal to 340. Alternative hypothesis, h1: The value of y is different from 340. Run a suitable test for the dataset and based on the p-value, make a conclusion.

5) 40 points) A manufacturer of hard safety hats for construction workers is concerned about the variation of the forces its helmets transmit to wearers when subjected to an external force. For simplicity, we assume that the measurements of the forces of n helmets in an experiment that transmit to wearers, X1,..., Xn , are a random sample from N(0,s2 ) , where s2 is unknown. Consider testing this simple hypotheses H0 :s s2 = 02 vs. H1 :s s2 = 12 , where the known constants satisfy s s02 < 12 and parameter space is ?={ss02, 12} (You already know that this statement of hypotheses is equivalent