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HARVARD UNIVERSITY (HU) |

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Probability and Statistics |

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^{nd}Sep 2023# Probability and Statistics Home Work, HU, Singapore Personnel tests are designed to test a job applicant’s cognitive and/or physical abilities. The Wonderlic IQ test is an example of the former

- Personnel tests are designed to test a job applicant’s cognitive and/or physical abilities. The Wonderlic IQ test is an example of the former; the Purdue Pegboard speed test involving the arrangement of pegs on a pegboard is an example of the latter. A particular dexterity test is administered nationwide by a private testing service. It is known that for tests administered last year, the distribution of job applicants’ scores was approximately normal with a mean of 78.5 and a standard deviation of 6.25. The minimum acceptable score for a job applicant is 75.

- If we randomly selected one job applicant in the last year, what is the probability that this applicant had a score less than 88?
- Find the 90 percentile of job applicants’ score distribution.
- Given that a randomly selected applicant has more than the minimum acceptable score, what is the probability that this applicant had a score less than 88?
- If we randomly selected 6 job applicants in the last year, what is the probability

that at least two of them had more than the minimum acceptable score?

- Assume that the number of visitors to a local post office in one minute is distributed as a Poisson variable during the lunch hour 1:00-2:00 PM. The average number of visitors to the post office is 1.68 per minute.

- Find the probability that between 7 and 10 visitors (inclusive) will visit the post

office within a five-minute period. - Let Y be the time between two consecutive visitors. What is the

distribution of Y? - Find the mean and standard deviation of Y.
- Find the probability that the time between two visitors is longer than 1 minute

but less than 2.5 minutes. - We start to observe and record the visitor at 1:00 PM on a particular day. Given that eight visitors visit the post office in the first ten minutes, what is the probability that the next visitor will visit the post office after 1:12 PM?

**(i)**The time between two consecutive customers entering a shoe shop during the off-peak hours can be modeled by an exponential distribution, with an average time of 1.60 minutes. Find the probability that the time between two consecutive customers entering the shoe shop is more than 45 seconds but less than 90 seconds.

**(ii)**According to a marketing survey, a family spends between $800 and $1,200 per month on transportation in a particular city. Suppose the money spent on transportation is uniformly distributed between these amounts. If we randomly select 10 families, what is the probability that at most three families spend less than $900 per month on transportation expenditures?- Suppose that the length of time it takes one variety of plant seeds to germinate is normally distributed with a mean of 10.5 days and a standard deviation of 2.5 days.

- What proportion of plant seeds will germinate between 8 and 12 days?]
- Find the interquartile range of this plant seed germination distribution.
- In a sample of 8 seeds, what is the probability that less than 2 of them will germinate after 12 days?

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