## There is this disease which is extremely dangerous. Only 0.1% of the population has this disease. A test has been developed to detect a disease.

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Q.1.     There is this disease which is extremely dangerous.  Only 0.1% of the population has this disease.  A test has been developed to detect a disease. Among the diseased population, the test gives a positive result in 99% of the cases.  Among the healthy population, the test gives a negative result in 99.5 % of cases. Therefore, this test is very effective among the diseased population and healthy population.

Kumar is concerned since many of his family members have contracted the disease and died.  He decided to take the test.  He has decided that if the probability that he has the disease is at least 20%, he will resign his job, cash all his savings and enjoy while it lasts!  Otherwise he will sit tight and wait for further developments.  He takes the test and the result is positive.  Use what you have learnt in this course (till mid-term) and advise Kumar.                                                                                                   [5]

Q.2.     The following contingency table gives the relationship between a man’s age and his job grade in a large industrial area.  Each job has a grade reflecting the value of that job to the firm and the table has been generated from a sample of industrial workers.

 Job Grade Age Young Middle-Age Old 1 60 880 60 2 240 3530 230

An employee is selected at random.

(a)             Compute the probability that he is young.

(b)             Compute the probability he is young given that he is in Job Grade 1.

(c)             What can you infer from the computations of the marginal and conditional probabilities?     [1 + 1 + 2 = 4]

Q.3.        The number of patients consulting the village doctor on any day is a random variable with μ = 18 and σ = 2.5.  Use Chebyshev’s Theorem to estimate the probability that on a random day, the number of visitors will be between 8 and 28.                                                                                                       [2]

Q.4.        The population has Q1 = 42, Q2 = 44 and Q3 = 50.  From this limited information, do you think the distribution of the data is left skewed, right skewed or symmetric?  Explain your inference.            [2]

Q.5.        The life expectancy of an electronic item is normally distributed with a mean of 4 years and a standard deviation of 1 year.  How many years should this item last at a minimum, so that it is in the top 5% of the population?                                                                                                                         [3]

Q.6.        A primary health center health center reported that in a sample of 400 patients 80 had severe fever.  The director of health has asked you to compute a 90% confidence interval for the population proportion with severe fever.

(a)           Compute the required confidence interval.  Your answer should follow the logical sequence as done in class.

Incidentally the 95% confidence interval is (0.1608, 0.2392).

(b)          Explain why the 95% confidence interval will be larger than the 90% confidence interval.

(c)           The Director went and told the minister that the probability that the disease rate is between 16.08% and 23.92% is 0.95.  Explain (briefly) why this is nonsensical!

[5 + 1 + 2 = 8]

Q.7.        The furniture company promises to fulfill your order within 15 days of purchase. A sample of 49 past customers is taken. The average delivery time in the sample was 16.2 days with a samplestandard deviation of 5.6 days.  Test the promise at 1% significance level.  Use the Critical Value approach.

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