Exam 12: Sampling Distributions

As many as 20% of all Americans contract influenza each year. A sample of 400 randomly selected Americans is chosen and the number with influenza is recorded. Let X represent the number with influenza in the sample. If X = 75 individuals are found to have influenza, what is the sample proportion?

Cats live for 14 years, on average, with a standard deviation of 2 years. A simple random sample of 78 recently deceased cats is selected, and the sample mean age at death x̄ of these cats is computed. We know the random variable x̄ has an approximately Normal distribution, but why?

The distribution of total body protein in healthy adult men is approximately Normal, with mean 12.3 kg and standard deviation 0.1 kg. If you take a random sample of 16 healthy adult men, what is the probability that their average total body protein is between 12.25 and 12.35 kg?

The distribution of vitamin C amount in the vitamin drops produced by a given factory is approximately Normal, with a mean of 60.0 mg and a standard deviation of 0.5 mg. If you take a random sample of 25 vitamin drops, what is, approximately, the probability that the average vitamin content x̄ is less than 59.9 mg?

Suppose a single random sample of size n is drawn from any population with mean μ and standard deviation σ. What does the central limit theorem say when n is sufficiently large?

The pH measurements of water specimens from various locations along a given river basin are Normally distributed, with mean 8 and standard deviation 0.3. You take water specimens from four randomly selected locations on this river basin. What is the probability that the mean pH measurement of these four specimens is greater than 8.2?

The distribution of total body protein in adult men with liver cirrhosis is approximately Normal, with mean 9.8 kg and standard deviation 0.1 kg. If you take a random sample of 16 adult men with liver cirrhosis, which of the following statements best describes the sampling distribution of the average total body protein?

The World Health Organization estimates that 5% of all adults in sub-Saharan Africa are living with HIV/AIDS. A survey takes a random sample of 1600 adults from all over sub-Saharan Africa and finds that 84 have HIV/AIDS. What is the approximate sampling distribution of the proportion who live with HIV/AIDS?

What does the law of large numbers state about the mean,x̄ , of observed values, when the number of observations drawn at random from a population-with finite mean μ and variance σ-increases?

The average age of trees in a large local park is 60 years, with a standard deviation of 2.2 years. A simple random sample of 400 trees is selected, and the sample mean age x̄ of these trees is computed. What is the probability that the average age x̄ of the 400 trees is more than 60.1 years?

The level of nitrogen oxides (NO_x) in the exhaust of cars of a particular model varies Normally, with mean 0.25 gram per mile (g/mi) and standard deviation 0.05 g/mi. Government regulations call for NO_x emissions no higher than 0.3 g/mi. A company has four cars of this model in its fleet. What is the probability that the average NO_x level of these cars is above the 0.3 g/mi limit?

You plan to randomly select 10 students from your campus and ask them how many minutes they exercised in the past seven days. What is the distribution of values taken by the average exercising time in all possible samples of size 10?

As many as 20% of all Americans contract influenza each year. A sample of 400 randomly selected Americans is chosen and the number with influenza is recorded. Let X represent the number with influenza in the sample. What is the probability that between 15% and 25% of the sample is observed to have influenza?

As many as 20% of all Americans contract influenza each year. A sample of 400 randomly selected Americans is chosen and the number with influenza is recorded. Let X represent the number with influenza in the sample. What is the probability that, at most, 25% of the sample is observed to have influenza?

The state of California reported a total of 1904 cases of pertussis (whooping cough) for all of the year 2013, 12% of which involved infants younger than 6 months. Infants this young cannot be vaccinated yet and must rely on herd immunity for protection from the virus. If we took many random samples of 100 California residents diagnosed with pertussis in 2013 and computed the proportion in each sample who are infants, what would the distribution of these values be

As many as 20% of all Americans contract influenza each year. A sample of 400 randomly selected Americans is chosen and the number with influenza is recorded. Let X represent the number with influenza in the sample. If X = 75 individuals are found to have influenza, then the sample proportion has approximately an

The sample proportion p̂ is a common statistic used when dealing with categorical data.

You plan to randomly select 10 students from your campus and ask them how many minutes they exercised in the past seven days. What is the distribution for all students of the times spent exercising in a week?

What is the sampling distribution of a statistic?

A large school district in southern California asked all of its eighth-graders to measure the length of their right foot at the beginning of the school year, as part of a science project. The data show that foot length is approximately Normally distributed, with a mean of 23.4 cm and a standard deviation of 1.7 cm. Suppose that 25 eighth-graders from this population are randomly selected. The distribution of the mean foot length in samples of 25 students is approximately Normal; what are the mean and standard deviation?