# Statistics

Name:

ID Number:

Assignment

**You should include Excel output to support your answers or reasoning as needed.**

**Q1** A sample of 200 executives is asked the question: “If you were given an offer by another company equal to or slightly better than your present position, would you remain with the company or take the other position?” The responses of the 200 executives in the survey were cross-classified with their length of service with the company, giving the results below:

Length of Service |
||||

Loyalty |
< 1 | 1 – 5 | 6 – 10 | > 10 |

Remain | 10 | 30 | 5 | 75 |

Leave | 25 | 15 | 10 | 30 |

(a) If an executive is picked at random, what is the probability that they are loyal (ie would remain with the company)? [2]

(b) If an executive is picked at random, what is the probability that they have been with the company for more than 10 years? [2]

(c) If an executive who has been with the company for more than 10 years is picked at random, what is the probability that they are loyal? [2]

(d) Which two of the above answers would you compare to judge whether Loyalty and Time of Service are independent? What does this comparison suggest? [2]

** Q2** A consumer survey has derived the following probability distribution for the number of coffee cups drunk per day by adult New Zealanders:

no of cups | 0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | >7 |

male | 0.40 | 0.34 | 0.12 | 0.05 | 0.04 | 0.03 | 0.01 | 0.01 | 0.00 |

female | 0.47 | 0.35 | 0.08 | 0.06 | 0.02 | 0.01 | 0.01 | 0.00 | 0.00 |

(a) Separately for each gender, what is the probability of drinking more than two cups per day? [2]

Males:

Females:

(b) Calculate for each gender the average number of cups of coffee drunk per day. [3]

Males:

Females:

**Q3** Internal auditors sometimes check random samples of transactions within a database. Suppose that in a particular set of transactions, 2% contain an error of some kind. The auditor takes a random sample of 20 transactions for checking. Let *X* denote the number of transactions found to be in error in the sample.

(a) State the probability distribution of *X* (including the values of all parameters) and find the probability that 2 transactions are found to be in error. [3]

(b) If three or more transactions are found to be in error then a larger sample is taken for checking. How often will this happen? (Use the appropriate template). [3]

(c) What assumption is required for the validity of the above answers? [1]

**Q4** An orange juice producer sources his oranges from a large grove. The amount of juice extracted from a single orange is normally distributed with mean 141 ml and standard deviation 12 ml.

(a) What is the probability that a randomly-chosen orange will contain less than 150 ml of juice? [3]

(b) What amount of juice would only 1 in 100 oranges exceed? [3]

(c) Describe the probability distribution of the average amount of juice from a sample of 20 oranges? What is the probability that this average is less than 150ml?

[3]

**Q5** The owner of a petrol station wants to study fuel-purchasing habits of motorists at his station. A random sample of 60 motorists during a particular week gave the following results:

(a) Set up a 95% confidence interval estimate for the proportion of motorists who buy diesel. Write a sentence to explain the meaning of your answer. [3]

**Confidence interval: **

**Explanation:**

(b) Set up a 95% confidence interval estimate for the mean amount of fuel purchased by a customer. Write a sentence to explain the meaning of your answer. [3]

**Confidence interval: **

**Explanation:**

(c) Before carrying out the survey, the owner thought that 10% of his customers used diesel, and that the average amount of fuel was no more than 40 litres per customer. Do the results of the survey suggest that he was wrong? Explain. [2]

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(d) Do we need to assume for the analysis in (b) that the amounts purchased follow a normal distribution? Discuss. [1]

**Q6** **Extension Question** – this is not compulsory, but is an opportunity to earn some bonus marks if you want to attempt this more challenging question. You can get up to 4 extra marks by doing so.

Consider the situation of Question 2 again. Calculate the probability that, in a group comprising three men and two women, exactly two are coffee-drinkers. State any assumptions you are making.

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