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![13. Suppose that a local appliances shop has found from experience that
the demand for tubelights is roughly distributed as Poisson with a
mean of 4 tubelights per week. If the shop keeps 6 tubelights during a
particular week, what is the probability that the demand will exceed
the supply during that week?
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- Suppose that we have disjoint normal populations A and B with equal population variances. Suppose we plan a sample of size 4 from from population A and a sample of size 9 from population B which we will pool to form the pooled variance. If you know the population variances are both in fact equal to 6, what is the probability that the pooled variance of the two samples will turn out to be less than 8 given your information?9.An automobile insurer has found that repair claims are Normally distributed with a mean of $530 and a standard deviation of $480. (a) Find the probability that a single claim, chosen at random, will be less than $490. (b) Now suppose that the next 60 claims can be regarded as a random sample from the long-run claims process. Find the probability that the average X¯ of the 60 claims is smaller than $490. (c) If a sample larger than 60 claims is considered, there would be __________ chance of getting a sample with an average smaller then $490. (NOTE: Write ''LESS'', ''MORE'' or ''AN EQUAL''.)2. An automobile battery manufacturer claims that its midgrade battery has a mean life of 50 months with a standard deviation of 6 months. Suppose the distribution of battery lives of this particular brand is approximately normal. On the assumption that the claims are true, find the probability that a randomly selected battery of this type will last less than 48 months. (Use the software link for every question) (a) Let X = number of months a battery will last. Write the question above in terms of this variable X (b) Find the probability that a single battery of this type will last less than 48 months. (c) Find the probability that the mean of a random sample of 36 batteries will be less than 48 months. (d) Why do you think the values from part (b) and part (c) are different? Explain.
- An elevator has a placard stating that the maximum capacity is 1344lb long dash—8 passengers. So, 8 adult male passengers can have a mean weight of up to 1344 divided by 8 equals 168 pounds.1344/8=168 pounds. If the elevator is loaded with 8 adult male passengers, find the probability that it is overloaded because they have a mean weight greater than 168lb. (Assume that weights of males are normally distributed with a mean of 171 lb and a standard deviation of 27 lb) Does this elevator appear to be safe? The probability the elevator is overloaded is____Suppose that the average height of men in the United States is approximately normallydistributed with a mean of approximately 70 inches and a standard deviation ofapproximately 4 inches. (a) If you take a sample of 4 men, find Pr {68 ≤ Y̅ ≤ 73}, the probability that the sample mean Y̅ will be between 68 and 73 inches. (b) If you take a sample of 16 men, find Pr {68 ≤ Y̅ ≤ 73}, the probability that the sample mean Y̅ will be between 68 and 73 inches. (c) If you take a sample of n men where n > 16, which one of the following is true? (circle one of the three statements.)Pr{68 ≤ Y̅ ≤ 73} > the probability that you found in (b)Pr{68 ≤ Y̅ ≤ 73} = the probability that you found in (b)Pr{68 ≤ Y̅ ≤ 73} < the probability that you found in (b)Some previous studies have shown a relationship between emergency-room admissions per day and level of pollution on a given day. A small local hospital finds that the number of admissions to the emergency ward on a single day ordinarily (unless there is unusually high pollution) follows a Poisson distribution with mean = 2.0 admissions per day. Suppose each admitted person to the emergency ward stays there for exactly 1 day and is then discharged. The hospital is planning a new emergency-room facility. It wants enough beds in the emergency ward so that for at least 95% of normal-pollution days it will not need to turn anyone away. What is the smallest number of beds it should have to satisfy this criterion? Answer the previous question for a random day during the year.
- 4. Suppose that 60% of college students own a smart phone manufactured by Apple. If a random sample of 200 college students is selected, use the normal distribution to approximate the probability that the number of students in the sample owning a smartphone made by Apple will be less than 105. Is it reasonable to use the normal approximation here?A dean in the business school claims that GMAT scores of applicants to the school's MBA program have increased during the past 5 years. Five years ago, the mean and standard deviation of GMAT scores of MBA applicants were 550 and 60, respectively. 30 applications for this year's program were randomly selected and the GMAT scores recorded. If we assume that the distribution of GMAT scores of this year's applicants is the same as that of 5 years ago, find the probability of erroneously concluding that there is not enough evidence to supports the claim when, in fact, the true mean GMAT score is 590. Assume αα is 0.02.5. Suppose the time spent by a randomly selected student at a campus computer laboratory has a gamma distribution with mean 20 min and variance 80 min². (a) What are the values of a and B? (b) What is the probability that a student spends between 20 and 40 min at the laboratory? (c) Only 1% of students will spend more than how many minutes at the laboratory?
- Chains "A" and "B" are made of the same steel. "A" consists of three links and "B" consists of six links, each link having a normal distribution function of the resisting strength with a mean of 60,000 psi and standard deviation of 5,000 psi. Determine which chain is generally weaker (from a probabilistic point of view) by plotting the probability distribution functions of their resisting strengths on the same diagram.The number of forklift topples in an area follows a poisson-process with rate 3 accidents per week. Suppose that we observed one forklift topple in the first week. What is the probability that we will observe at most one forklift topples in the next 2 days?Suppose the horses in a large stable have a mean weight of 1354 lbs, and a variance of 9604. What is the probability that the mean weight of the sample of horses would be less than 1364 lbs if 43 horses are sampled at random from the stable?
