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**Task:**

**1. A committee of three is to be chosen from a group of four men and five women. Find the probability that the committee contains: **

- all women [1 marks]
- at least one woman [2 marks]
- exactly two men [1 marks]
- exactly one men, given that at least one man is chosen.

**2. Leaking from underground gasoline tanks at service stations can damage the environment. It is estimated that 22% pf these tanks leak. Suppose you examine 15 tanks chosen at random, independent of each other. **

- What is the mean number of the leaking tanks in such sample of 15? [1 mark]
- What is the probability that 10 or more of the 15 tanks leak? [ 1 mark]
- What is the probability that at least 5 of the 15 tanks leak? [1 mark]
- What is the probability that at most 5 of the 15 tanks leak? [1 mark]
- Now you examine a random sample of 1000 tanks nationally, what is the probability at least 275 of these tanks are leaking?

**3. The number of vehicles pass a bridge in one minute interval were recorded over 1000 minutes given in the dataset “no_vechicles.csv” **

- Read the dataset in R and obtain the frequency distribution. [3 marks]
- Compute the mean number of vehicles that pass the bridge per minute. [2 marks]
- Using part (b) as the lambda value, use the parameter of a Poisson distribution to compute the probabilities for each of the observed values in the frequency distribution. [2 marks]
- Compute the expected frequencies for each of the values in the frequency distribution. [1 mark]
- Compare the expected frequencies to those actually observed values from the “no_vechile.txt”, dataset and comment on your findings (i.e the actual and expected values agree)

**4. A radar unit is used to measure speeds of cars on a motorway. The speeds are normally distributed with a mean of 80 km/hr and a standard deviation of 10 km/hr. **

- What is the probability that a car picked at random is travelling at more than 100 km/hr?
- What is the probability that a car picked at random is travelling at less than 82 km/hr?
- What is the probability that a car picked at random is travelling between 78 km/hr and 95 km/hr?

**5. It is often felt that female account holders have different account balances on average than male account holders. **

- Using the “bank.csv” dataset, check the validity of the statement. Use 5% level of significance. [5 marks]
- Check all assumptions

**6. Using the “bank.csv” dataset, obtain 95% confidence interval for the average age of the account holders and the average total transaction and interpret them.**

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