Describe a probabilistic model

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Statistical Inference Questions

A hardware company developed a new procedure to build processors. They want to analyse the effectiveness of this new procedure.

1. They built 100 processors with this new procedure and tested if the processors passed a basicquality test. The results are in the processor_pass data set (on moodle, 1 pass, 0 not). Give a probabilistic model which describes the setup well. Fit the model to the data set and report your estimate.

2. The company also applied an old procedure to produce another 100 processors and testedthe quality in the same way. The results are in old_pass. Estimate a second model for this new data and compare the estimates. Can the company make a strong point that the new procedure is superior in your opinion?

3. The company wants to produce many batches of processors and is interested in knowing theaverage rate of processors that pass the test out of 100. Develop a new model to determine the rate and infer the rate from the data (use again the processor_pass dataset).

4. There are different ways how the new procedure can be implemented. The company measured the quality of the processors on a scale from 0 to 10 for the different implementations and ordered the results according to how costly the procedures are (cheapest first). The data sets priceQualityX and priceQualityY contain their measurements. Analyse the data and make an informed discussion about how costs influence the quality of the processors. Do the costs change the quality? If so, how does the quality scale with the costs? Write down a model that explains well the dependence between price and quality.

Meteorologists measured the average daily temperature in Lancaster and Preston.

1. Describe a probabilistic model that can be used to model the temperature variation betweendifferent days. Use the datasets weather to infer the average daily temperature using your probabilistic model. The first column corresponds to Lancaster and the second to Preston.

2. The meteorologists determined the variance in daily temperature in Lancaster to be 4 and7 in Preston. They believe that the average temperature is 15i C in both cities. Test this hypothesis for them. Is there sufficient evidence to doubt their belief? Discuss.

3. Can you provide the meteorologists with an approximate 95 % confidence interval for thedaily temperature in Lancaster and Preston?

4. The meteorologists think that the temperature in Lancaster and Preston is correlated. Is thereany indication that they are right? Can you describe a way to test this hypothesis?

An airline keeps a data base of the number of passengers per day. The airline wants to make sure that most of the days there are enough seats for all passengers.

1. Give a generative model that can be used to help the airline decide how many flights it shouldhave per day (assume that each airplane can carry 100 passengers).

2. How would you adjust the model to the data? Write down the likelihood function and your model estimate.

3. The airline is currently guessing that on average 1300 passengers fly per day. Use the airline data set (on moodle) to test this hypothesis. Use a likelihood ratio test. Also plot the rejection region as in the lecture notes and show the test statistic in the plot.

4. To be sure that enough seats are available the airline flies 20 times a day. Can you give the airline a well justified approximation of the number of days a month (30 days) when there will be an insufficient amount of seats?

5. The airline observed that nearly every day many seats are empty. They like to reduce the number of flights while guaranteeing that with less than 5 % on a given day there are too few seats. Tell them the minimum number of flights they need per day to make sure that with 95 % probability there are enough seats available for all the passengers. Discuss your result.

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