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Normal and binomial probabilities and percentiles

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A company that sells annutities must base the annual payout on the probability distribution of the length of life of the participants in the plan. Suppose the probability distribution of the lifetimes of the participants is approximately a normal distribution with a mean of 68 years and a standard deviation of 3.5 years. Find the age at which payments have ceased for approximately 86% of the plan participants.

A company that sells annuitites must base the annual payout on the probability distribution of the length of life of the participants in the plan. Suppose the probability distribution of the lifetimes of the participants is approximately a normal distribution with a mean of 68 years and and a standard deviation of 3.5 years. What proportion of the plan recipients would recieve payments beyond age 75?

A national trend predicts that women will account for half of all business travelers in the next 3 years. To attract these women business travelers, hotels are providing more amenities that women particularly like. A recent survey of american hotels found that 70% offer hairdryers in the bathrooms. Consider a random and independent sample of 20 hotels. Find the probability of all the hotels in the sample that offer hairdryers in the bathrooms.

A national trend predicts that women will account for half of all business travelers in the next 3 years. To attract these women business travelers, hotels are providing more amenities that women particularly like. A recent survey of american hotels found that 70% offer hairdryers in the bathrooms. Consider a random and independent sample of 20 hotels. Find the probability that at least 9 of the hotels in the sample do not offer hairdryers in the bathrooms.

The amount of time necessary for assembly line workers to complete a product is a normal random variable with a mean of 15 minutes and a standard deviation of 2 minutes. The probability is __________ that a product is assembled between 16 and 21 minutes.

A campus program evenly enrolls undergraduate and graduate students. If a random sample of 4 students is selected from the program to be interviewed about the introduction of a new fast food outlet on the ground floor of the campus building, what is the probability that all 4 students selected are undergraduate students?
0.0256
0.0625
0.16
1.00

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This problem shows how to calculate probabilities and percentiles from normal and binomial probability distributions.

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Normal/approximation to Binomial

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1. If you determine that for the graph of a standard normal distribution the cumulative area to the left of a z core is 0.4, what is the cumulative area to the right of that z score?
2. Assume that the readings on the thermometers are normally distributed with a mean of 0 degree and a standard deviation of 1.00 degree C. A thermometer is randomly selected and tested. In each case, draw a sketch, and find the probability of each reading in Celsius degrees.
a. Less than -2.50
b. Greater than 1.96
c. Between 1.05 and 2.05
d. Between -3.90 and 1.50

3. Assume that the readings on the thermometers are normally distributed with a mean of 0 degree and a standard deviation of 1.00 degree Celsius. Find the indicated probability, where z is the reading in degrees.
a. P(z>1.645)
b. P(z<-1.96 or z>1.96)
c.
4. Assume that the readings on the thermometers are normally distributed with a mean of 0 degree and a standard deviation of 1.00 degree Celsius. Find the temperature reading corresponding to the given information.
a. If 1.0% of the thermometers are rejected because they have readings that are too high and another 1.0% are rejected because they have readings that are too low, find the two readings that are cutoff values separating the rejected thermometers from the others.

5. What is the difference between a z score and an area under the graph of a normal probability distribution? Can a z score be negative? Can an area be negative?
6. A common design requirement is that an item must fit the range of people who fall between the 5th percentile for women and the 95th percentile for men. If this requirement is adopted, what is the minimum sitting distance and what is the maximum sitting distance? For the sitting distance, use the buttock to knee length. Men have buttock to knee lengths that are normally distributed with a mean of 23.5 in. and a standard deviation of 1.1 in. Women have buttock to knee lengths that are normally distributed with a mean of 22.7 in. and a standard deviation of 1.0 in.
7. Here are the numbers of sales per day that were made by Kim Ryan, a telemarketer who worked four days before being fired: 1, 11, 9, 3. Assume that samples of size 2 are randomly selected with replacement from this population of four values.
a. List the 16 different possible sample and find the mean of each of them.
b. Identify the probability of each sample, then describe the sampling distribution of sample means.
c. Find the mean of sampling distribution
d. Mean of the population

8. Because a statistics student waited until the last minute to do a project, she has only enough time to collect heights from female friends and female relatives. She then calculates the mean height of the females in her sample. Assuming that females have heights that are normally distributed with a mean of 63.6 in. and a standard deviation of 2.5 in., can she use the central limit theorem when analyzing the mean height of her sample?
9. The new Lucky Lady Casino wants to increase revenue by providing buses that can transport gamblers from other cities. Research show that thes gamblers tend to be older, they tend to play slot machines only, and they have losses with a mean of $182 and a standard deviation of $105. The buses carry 35 gamblers per trip. The casino gives each bus passenger $50 worth of vouchers that can be converted to cash, so the casino needs to recover that cost in order to make a profit. Find the probability that if a bus is filed with 35 passengers, the mean amount lost by a passenger will exceed $50. Based on the result, does the casino gable when it provides such buses?
10. Engineers must consider the breadths of male heads when designing motorcycles helmets. Men have head breadths that are normally distributed with a mean of 6.0 in. and a standard deviation of 1.0 in.
a. If one male is randomly selected, find the probability that his head breadth is less than 6.2 in.
b. The Safeguard Helmet company plans an initial production run of 100 helmets. Find the probability that 100 randomly selected men have a mean head breadth less than 6.2 in.
c. The production manager sees the result from part (b) and reasons that all helmets should be made for men with head breadths less than 6.2 in., because they would fit all but a few men. What is wrong with that reasoning?

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