# Calculate the final probabilities of the four possible junk–keyword outcome combinations, i.e. the probability of an email being ”junk” and containing the ”offer” word, ”non-junk” and containing the ”offer” word etc.

STATISTICS

Paper, Order, or Assignment Requirements

Use correct probability notation in your responses to all questions. Marks are awarded for presentation including correct notation.

Correct probability notation implies statements such as:

P(X<5) = 0.1, P(5<X<10) = 0.65 etc,

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Questions

1 Junk email

A systems administrator did a quick analysis and found that 35% of the emails in his inbox can be classified as junk. In addition, he noted that 80% of junk emails contain the word offer in their subject line, whereas, only 15% of the non-junk emails have a similar subject line.

(a) Calculate the final probabilities of the four possible junk–keyword outcome combinations, i.e. the probability of an email being ”junk” and containing the ”offer” word, ”non-junk” and containing the ”offer” word etc. You might find it useful to create a tree diagram illustrating the above information but it is not necessary to include it in your submitted manuscript.

(b) Show that the overall probability of an email containing the keyword ”offer”, regardless of its ”junk” or ”non-junk” status, is 0.345.

(c) Calculate the conditional probability that an email is junk given that it contains the keyword ”offer”.

2 Cell cultures

Many lab-based experiments rely on cultured cells. Culturing cells can be a difficult process, and many culturing attemps fail. In one particular lab, numerous cultures are begun, and only those cultures that thrive are used in subsequent experiments.

Suppose that the probability that any one culture will succeed is 0.2, the success or failure of any given culture does not depend on or influence the success or failure of the other cultures, and 10 cultures are attempted.

(a) Let X be the number of successful cultures out of the 10 attempted cultures. Explain why X is a binomial random variable.

Using Rcmdr, plot:

(b) the probability function for this example

(c) the cumulative probability function (distribution function) for this example.

Using Rcmdr, calculate:

(d) At most 3 (i.e. less or equal to 3) cultures are successful.

(e) At least 3 (i.e. greater or equal to 3) cultures are successful

(f) Exactly 3 cultures are successful.

3 Biochemical reaction

In a stable biochemical system a reaction occurs, on average, 3.5 times per minute. Assume that the number of reaction occurrences per minute is a random variable (say X) that follows a Poisson distribution. Using Rcmdr calculate the probability of:

(a) no reaction occurring during a minute.

(b) 1 to 5 reactions occurring during a minute.

(c) more than 5 reactions occurring during a minute.

Use the probability notation in your answers.

4 Intelligence Quotient

One of the assumptions of intelligence tests is that the scores follow approximately a Normal Distri- bution. For example, the Intelligence Quotient (IQ) score assumes that the mean score is 100 and the standard deviation is 15.

(a) Use Rcmdr to estimate the proportion of the population that has an IQ score between 90 − 110.

(b) Calculate the z-score for an IQ score of 115 and use it to estimate the proportion of the population that scores below 115. Show your working.

(c) A high-IQ society admits members who score in the top 5% or above. Using Rcmdr, find the minimum IQ score that is needed in order to be admitted.

(d) It has been (falsely) reported that listening to Mozart before you take an IQ test boosts your score by 8.5 IQ points on average. Propose a new distribution for the distribution of scores of Mozart listeners. You can assume that the Mozart Effect follows a Normal distribution with mean 8.5 and standard deviation of 1.75.