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39% probability he will receive at least one upgrade during the next two weeks. And a standard deviation A measure of the variability of proportions computed from samples of the same size. An airline claims that there is a 0. The probability of receiving an upgrade in a flight is independent of any other flight, hence, the binomial distribution is used to solve this question. Find the mean and standard deviation of the sample proportion obtained from random samples of size 125. Which lies wholly within the interval, so it is safe to assume that is approximately normally distributed. Assuming that a product actually meets this requirement, find the probability that in a random sample of 150 such packages the proportion weighing less than 490 grams is at least 3%. Sam is a frequent flier who always purchases coach-class. A sample is large if the interval lies wholly within the interval. This outcome is independent from flight. The information given is that p = 0.

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Because it is appropriate to use the normal distribution to compute probabilities related to the sample proportion. After the low-cost clinic had been in operation for three years, that figure had risen to 86%. An airline claims that 72% of all its flights to a certain region arrive on time. If Sam receives 18 or more upgrades to first class during the next. B. Sam will make 4 flights in the next two weeks. Suppose that one requirement is that at most 4% of all packages marked 500 grams can weigh less than 490 grams. Often sampling is done in order to estimate the proportion of a population that has a specific characteristic, such as the proportion of all items coming off an assembly line that are defective or the proportion of all people entering a retail store who make a purchase before leaving.

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In one study it was found that 86% of all homes have a functional smoke detector. Using the binomial distribution, it is found that there is a: a) 0. Suppose random samples of size n are drawn from a population in which the proportion with a characteristic of interest is p. The mean and standard deviation of the sample proportion satisfy. 43; if in a sample of 200 people entering the store, 78 make a purchase, The sample proportion is a random variable: it varies from sample to sample in a way that cannot be predicted with certainty. Assuming this proportion to be accurate, find the probability that a random sample of 700 documents will contain at least 30 with some sort of error.

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Binomial probability distribution. Be upgraded exactly 2 times? Using the value of from part (a) and the computation in part (b), The proportion of a population with a characteristic of interest is p = 0. Be upgraded 3 times or fewer? 5 a sample of size 15 is acceptable. Thus the population proportion p is the same as the mean μ of the corresponding population of zeros and ones. D. Sam will take 104 flights next year. An ordinary die is "fair" or "balanced" if each face has an equal chance of landing on top when the die is rolled.

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The probability is: In which: Then: 0. An online retailer claims that 90% of all orders are shipped within 12 hours of being received. Suppose this proportion is valid. Here are formulas for their values. The proportion of a population with a characteristic of interest is p = 0. A state public health department wishes to investigate the effectiveness of a campaign against smoking. P is the probability of a success on a single trial.

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To learn more about the binomial distribution, you can take a look at. A humane society reports that 19% of all pet dogs were adopted from an animal shelter. Suppose that in a population of voters in a certain region 38% are in favor of particular bond issue. Suppose that in 20% of all traffic accidents involving an injury, driver distraction in some form (for example, changing a radio station or texting) is a factor. Samples of size n produced sample proportions as shown. A state insurance commission estimates that 13% of all motorists in its state are uninsured. An economist wishes to investigate whether people are keeping cars longer now than in the past. First verify that the sample is sufficiently large to use the normal distribution. The Central Limit Theorem has an analogue for the population proportion To see how, imagine that every element of the population that has the characteristic of interest is labeled with a 1, and that every element that does not is labeled with a 0. 71% probability that in a set of 20 flights, Sam will be upgraded 3 times or fewer.

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Item b: 20 flights, hence. Of them, 132 are ten years old or older. 10 probability that a coach-class ticket holder who flies frequently will be upgraded to first class on any flight, hence. Find the probability that in a random sample of 600 homes, between 80% and 90% will have a functional smoke detector. For each flight, there are only two possible outcomes, either he receives an upgrade, or he dos not. The sample proportion is the number x of orders that are shipped within 12 hours divided by the number n of orders in the sample: Since p = 0. Lies wholly within the interval This is illustrated in the examples. Thus the proportion of times a three is observed in a large number of tosses is expected to be close to 1/6 or Suppose a die is rolled 240 times and shows three on top 36 times, for a sample proportion of 0. At the inception of the clinic a survey of pet owners indicated that 78% of all pet dogs and cats in the community were spayed or neutered. An outside financial auditor has observed that about 4% of all documents he examines contain an error of some sort. You may assume that the normal distribution applies. Historically 22% of all adults in the state regularly smoked cigars or cigarettes. 38, hence First we use the formulas to compute the mean and standard deviation of: Then so.

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The parameters are: - x is the number of successes. Find the probability that in a random sample of 250 men at least 10% will suffer some form of color blindness. To be within 5 percentage points of the true population proportion 0. In the same way the sample proportion is the same as the sample mean Thus the Central Limit Theorem applies to However, the condition that the sample be large is a little more complicated than just being of size at least 30. Clearly the proportion of the population with the special characteristic is the proportion of the numerical population that are ones; in symbols, But of course the sum of all the zeros and ones is simply the number of ones, so the mean μ of the numerical population is. He knows that five years ago, 38% of all passenger vehicles in operation were at least ten years old.

A random sample of size 1, 100 is taken from a population in which the proportion with the characteristic of interest is p = 0. Find the probability that in a random sample of 450 households, between 25 and 35 will have no home telephone. Nine hundred randomly selected voters are asked if they favor the bond issue. For large samples, the sample proportion is approximately normally distributed, with mean and standard deviation. Some countries allow individual packages of prepackaged goods to weigh less than what is stated on the package, subject to certain conditions, such as the average of all packages being the stated weight or greater. A consumer group placed 121 orders of different sizes and at different times of day; 102 orders were shipped within 12 hours. Item a: He takes 4 flights, hence. This gives a numerical population consisting entirely of zeros and ones. In actual practice p is not known, hence neither is In that case in order to check that the sample is sufficiently large we substitute the known quantity for p. This means checking that the interval. Find the indicated probabilities. Suppose that 2% of all cell phone connections by a certain provider are dropped. Assuming the truth of this assertion, find the probability that in a random sample of 80 pet dogs, between 15% and 20% were adopted from a shelter. 6 Distribution of Sample Proportions for p = 0. In an effort to reduce the population of unwanted cats and dogs, a group of veterinarians set up a low-cost spay/neuter clinic.

First class on any flight. 1 a sample of size 15 is too small but a sample of size 100 is acceptable. C. What is the probability that in a set of 20 flights, Sam will. Show supporting work. Suppose that 8% of all males suffer some form of color blindness. Would you be surprised. 90,, and n = 121, hence. N is the number of trials.

Find the probability that in a random sample of 50 motorists, at least 5 will be uninsured. Find the probability that in a random sample of 275 such accidents between 15% and 25% involve driver distraction in some form. He commissions a study in which 325 automobiles are randomly sampled. In a survey commissioned by the public health department, 279 of 1, 500 randomly selected adults stated that they smoke regularly. Viewed as a random variable it will be written It has a mean The number about which proportions computed from samples of the same size center. In a random sample of 30 recent arrivals, 19 were on time.

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