Statistics and Probability Answers

A biologist is studying the levels of arsenic that are naturally produced in some ground water sources. A mean level of 8.0 parts per billion (ppb) and less is considered safe for agricultural use. A random sample of 60 tests of the ground water at different locations in the town of Howick yields a sample mean of 8.46 ppb with a standard deviation of 1.3 ppb. Is there enough evidence to conclude that the arsenic levels are too dangerous to use for agricultural purposes at the 98% confidence level? Conduct a complete hypothesis test.

The organizer of the Montreal International Art Exhibit is trying to determine its optimal operating hours for its next one-day exhibition. Studies have shown that the arrival times at any given exhibition form a normal distribution with the average time that visitors arrive being 2 hours and 56 minutes after doors open, with a standard deviation of 48 minutes.
a) If the organizer sets the opening of the exhibition at 10:00 a.m., at what time would they expect 95% of the visitors to have arrived?
b) If the organizer sets the opening of the exhibition at 9:00 a.m., at what time after the doors open will only 15% of the visitors have arrived?
c) At what time should the organizer open the exhibition if they would like 70% of the visitors to have arrived by 1:00 p.m. so that they can award the first door prize?

Crispy Chips is a potato chip company that is quite popular for its low-fat, low-calorie bags of potato chips. The procedure used at its production plant allows for 65 chips to be inserted into each bag for distribution to consumers. However, given that chip-making is not an exact science, there is a standard deviation of 5 chips per individual bag. If we can assume that the amount of chips in each bag forms a normal distribution, calculate the following:
a) Calculate the z-score if there are 75 chips in a bag.
b) What is the probability that less than 61 potato chips will be in a bag?
c) Determine the probability that more than 79 potato chips will be in a bag.
d) Find the probability that there will be between 60 and 80 potato chips in a bag.

TubeView, a digital television distribution company based in Gatineau (QC), hires a marketing firm to gauge the satisfaction levels of their customers. The CEO of TubeView has claimed that past polls have suggested that the customer satisfaction level is at 90%.
If this claim was true, what is the probability that in a random sample of 12 customers:
a) Exactly 8 customers are satisfied with TubeView’s service?
b) At least 8 customers are satisfied with TubeView’s service?
c) All of the customers they contact are satisfied with TubeView’s service?
d) At most 2 customers are not satisfied with TubeView’s service?
e) P(5 ≤ X < 9) where X = satisfied customer?

Determine the direction of the hypothesis test (one-sided left, one-sided right, bidirectional)  Determine the test statistic (z* or t*) and the p-value for each of the following situations and  Determine if they would cause the rejection of the null hypothesis if the confidence level was set at 95% in each case. (Hint: be wary of the sample size)
a) Ho: μ = 50 mL, Ha: μ ≠ 50 mL, sample mean = 48.1 mL, sample standard deviation = 5, n = 40 b) Ho: μ ≤ 8.4 m3, Ha: μ > 8.4 m3, sample mean = 10 m3, s = 3.5, n = 25 c) Ho: μ ≥ 20oC, Ha: μ < 20oC , sample mean = 17.1oC, s =4.6oC, n = 12 d) Ho: μ = 357 s, Ha: μ ≠ 380 s, sample mean = 410 s, s = 75, n = 40 e) Ho: μ ≤ 46 units, Ha: μ > 46 units, sample mean = 50 units, s = 9.5, n = 41

Assume that you have a bag of 10 billiard balls. You know that there may be either black or white balls in the bag, but you do not know how many black and white balls there are. Using the Bayesian formula answer the following:

1. What is the initial probability p(nB) that the bag contains nB black balls?

during summer vacation Tanuja wants to visit three cities, kolkata, bhubaneswar and chennai randomly. find the probability that she will visit (i) bhubaneswar before chennai (ii) bhubaneswar just before kolkata.

Carol lives in the east end of Montreal. To get to school for her morning classes Carol has the option of taking the bus, going underground with the metro, or driving her car downtown to get to Concordia. Carol prefers the metro, especially in the winter, and she will opt for it 35% of the time. The bus is perhaps a more convenient option since it stops near her house, so she will choose that option 55% of the time. The car is the most expensive option, but it is the most reliable, as demonstrated by the fact that she is only late for class 5% of the time when she drives. The bus is the least reliable of the options as it gets Carol to class on time 65% of the time, whereas this value increases by 10% with the metro.

c) What is the probability that Carol drove to school and is on time for class?
d) What is the probability that Carol opted for the bus given that she was late to class?
e) What is the probability that Carol is on-time for class given that she did not drive?

Carol lives in the east end of Montreal. To get to school for her morning classes Carol has the option of taking the bus, going underground with the metro, or driving her car downtown to get to Concordia. Carol prefers the metro, especially in the winter, and she will opt for it 35% of the time. The bus is perhaps a more convenient option since it stops near her house, so she will choose that option 55% of the time. The car is the most expensive option, but it is the most reliable, as demonstrated by the fact that she is only late for class 5% of the time when she drives. The bus is the least reliable of the options as it gets Carol to class on time 65% of the time, whereas this value increases by 10% with the metro.

a) What is the probability that Carol is late to class on a given morning?
b) What is the probability that Carol took the metro and is on time for class?

The batteries to the remote control for your television have just run out. You find your collection of miscellaneous "AA" batteries and grab 2 of them to replace the used ones. the box you used to fish out the replacements contained 14 batteries, but you were unaware that 5 of them were faulty and did not work.

Given that the remote control is now working what us the probability that the nest two batteries you select from your remaining stash will also work?