Please answer ALL questions.
Question 1 [20 marks]
Consider the following probabilities:
P(A^c )=0.4, P(B)=0.48 and P(A∩B^c )=0.12
Find P(A|B^c )
Find P(B^C |A)
Are A and B independent? Explain your answer.
Question 2 [20 marks]
An investor owns a portfolio consisting of two mutual funds, A and B, with 40% invested in A.
Measures Fund A Fund B
Expected value 11 4
Variance 73 43
Derive the expected value for the portfolio return?
Derive the standard deviation for the portfolio return?
Question 3 [30 marks]
A bit more than half of the German highways do not have a speed limit and you can go as fast as you want. However, the recommended maximum speed is 130 km/h. One passage on the A2 between the Ruhr area and Berlin has lifted its former speed limit of 130 km/h. The highway police want to measure whether car drivers exploit the lifting of the strict speed limit and whether cars drive faster than 130 km/h. The police measure the speed of 100 cars. The average speed is 160 km/h with a standard derivation of 22km/h. The highway police would like to determine if the average speed is greater than the recommended speed:
█(H_0&:μ≤130@H_A&: μ >130)
Calculate the value of the test statistic.
Find the p-value.
At α=10%, can the police conclude that the average speed is greater than the recommended speed? Represent the results on a bell-shaped graph. Mark clearly the mean, the test statistic, the rejection region, and the non-rejection region.
Question 4 [30 marks]
Consider a normal population with a known population variance of 20. A random sample results in x ̅=50.
Compute the 95% confidence interval for μ if the sample x ̅ were obtained from a sample of 15 observations.
Compute the 95% confidence for μ if the sample x ̅ were obtained from a sample of 25 observations.
What happens with the confidence interval when the sample size increases? Explain your answer.
What happens with the confidence interval when the confidence level decreases to 90%? Explain your answer.
Please answer ALL questions.
Question 5. [100 marks]
In 2016 the UK voted to leave the European Union. A group of 50 MPs is considering requesting a second referendum. The MPs are interested in whether citizens would change their voting behavior in a second referendum and hired a survey firm to conduct a poll of 1,000 voters. This poll found the following:
Of the 1,000 individuals surveyed, 530 say they would vote to remain part of the EU in a second referendum.
30.5 percent of those surveyed had voted to leave the EU in 2016 and were of ABC1 social status (middle class), while 19.0 percent of those surveyed voted to leave the EU and were of C2DE social status (working class/unemployed). 21.0 percent of those surveyed had voted to remain in 2016 and were C2DE.
188 people who voted to leave the EU in 2016 and who are ABC1 said they would vote “Leave” again; 168 who had voted remain in 2016 and are C2DE would not change their vote, and 143 people who are C2DE status and who voted to leave the EU in 2016 also would not change their vote.
97 respondents of ABC1 status who voted to remain in 2016 said that they would vote for independence in a second referendum.
Based on the above findings:
Construct a frequency table and the relative frequency table (with associated probabilities) that represents fully this data.
What is the probability of randomly selecting an individual who intends to vote to leave the EU and who voted for remain the first time around and who is C2DE social status?
Are social status and voting intention independent? Our past and future voting behaviors independent? Please explain your answer in both cases.
Describe stratified and cluster sampling. If you had to conduct this poll, which method would you use, and why?
Question 6. [100 marks]
A group of social scientists wants to examine whether, during the second half of the 19th century, districts in Ireland that were served by the new railway network were more likely to see severe population decline as compared with other Irish districts. They define it as severe a population fall of at least one-third between 1841 and 1901. Based on Census records, they know that 23% of Ireland’s 273 districts saw a population decline of at least one third between 1841 and 1901. Using maps of the rail network, they find that 132 districts got access to the rail in the period concerned, and of these, 39 saw severe population falls.
What is the estimate of the population proportion?
Determine whether the sample is large enough to use regular test statistics.
State the null and alternative hypotheses. Explain your choice.
Compute the test statistic.
Calculate the p-value, assuming that the standard is to use 95% confidence intervals.
Basing your view on this sample information, can you conclude that districts that got access to rail experienced worse than average population loss? Explain your answer. For more information on Standard Deviation check on this: https://en.wikipedia.org/wiki/Standard_deviation
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