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Questions and Answers of Statistics

Mutations accumulate at a rate of 0.3 per million nucleotides during the first 0.5 yr of a study, and at a rate of 3.2 per million nucleotides during the second 1.5 yr. The DNA is 4.7 million
Due to an increase in radiation levels, the mutation rate increases linearly from 1.0 Ã— 10-6/yr at t = 0 to 3.0 x 10-6/yr at t = 2 Ã— 106. Find the expected number of mutations over the course
Due to a decrease in radiation, the mutation rate decreases linearly from 4.0 Ã— 10-6/yr at t = 0 to 1.0 Ã— 10-6/yr at t = 2 Ã— 106. Find the expected number of mutations over the course of
There are 100,000 gnats, and each leaves with probability 0.0067. Suppose many gnats are flying around in a room. Each leaves independently with a probability that depends on the insect repellent
There are 50,000 gnats, and each leaves with probability 0.037. Suppose many gnats are flying around in a room. Each leaves independently with a probability that depends on the insect repellent
Write down a sum giving the probability that the organism is alive at time t. Suppose an organism would live forever if it weren't for predators that attack at rate λ per year. Fortunately, only a
Write down the first few probabilities in a Poisson distribution with parameter (1 - q)λ. What do they add up to? Suppose an organism would live forever if it weren't for predators that attack at
The simulation in Exercise 2 with λ = 2.0. Using the simulations from the previous set of problems, compare the observed distribution of the number of hits per second with the Poisson distribution
Use this last fact to come up with the probability that the organism is alive at time t. What is the average lifetime? Could you have guessed it without doing any calculations? Suppose an organism
The simulation in Exercise 3 with λ = 0.5. Using the simulations from the previous set of problems, compare the observed distribution of the number of hits per second with the Poisson distribution
The simulation in Exercise 4 with λ = 0.8. Using the simulations from the previous set of problems, compare the observed distribution of the number of hits per second with the Poisson distribution
The simulation in Exercise 3 with λ = 0.5. Regroup the results into ten intervals with length 2 s, and compare with the Poisson distribution with λ = 0.5 and t = 2. Using the simulations from the
Suppose that n = 16, and that Xi takes the value 0 with probability 0.5 and the value 1 with probability 0.5.Suppose the random variables X1, X2 ... Xn are i.i.d. (independent and identically
Find and sketch the p.d.f. of X + Y. Suppose that X and Y are independent normally distributed random variables with X ≈ (5.0, 16.0) and Y ≈ N(10.0, 9.0).
Find and sketch the p.d.f. of the sum of nine independent samples from X. Suppose that X and Y are independent normally distributed random variables with X ≈ (5.0, 16.0) and Y ≈ N(10.0, 9.0).
Find and sketch the p.d.f. of the mean of nine independent samples from Y. Suppose that X and Y are independent normally distributed random variables with X ≈ (5.0, 16.0) and Y ≈ N(10.0, 9.0).
The normal p.d.f. with μ = 0 and σ2 = 1 takes on its maximum at x = 0. Show the above facts about the normal distribution.
The normal p.d.f. takes on its maximum at x = μ for any values of μ and σ. Show the above facts about the normal distribution.
The normal p.d.f. with μ = 0 and σ2 = 1 has points of inflection at x = -1 and x = 1. Show the above facts about the normal distribution.
The normal p.d.f. has points of inflection at x = μ + σ and x = μ - a for any values of μ and σ. Show the above facts about the normal distribution.
Use the law of total probability to show thatSuppose T1, T2, T3... are i.i.d. exponentially distributed random variables with Î» = 1.0 and thatLet fn be the p.d.f. for Sn.
Evaluate the integral to find f2, the p.d.f. for S2.Suppose T1, T2, T3... are i.i.d. exponentially distributed random variables with Î» = 1.0 and thatLet fn be the p.d.f. for Sn.
Use the same trick to find f3, the p.d.f. for S3.Suppose T1, T2, T3... are i.i.d. exponentially distributed random variables with Î» = 1.0 and thatLet fn be the p.d.f. for Sn.
Suppose that n = 50, and that Xi takes the value 0 with probability 0.25, the value 1 with probability 0.5, and the value 2 with probability 0.25.Suppose the random variables X1, X2 ... Xn are i.i.d.
Can you guess the pattern? Why does the answer look so much like the Poisson distribution?Suppose T1, T2, T3... are i.i.d. exponentially distributed random variables with Î» = 1.0 and thatLet fn be
Suppose that n = 16 and that Xi takes the value 0 with probability 0.5 and the value 1 with probability 0.5. Compare with the results in Exercise 1.The Central Limit Theorem does not work if random
Suppose that n = 50 and that Xi takes the value 0 with probability 0.25, the value 1 with probability 0.5, and the value 2 with probability 0.25. Compare with the results in Exercise 2.The Central
Find the probability associated with each way. Based on the probabilities in Figure 7.8.1b, we can find the probability that a plant gains 5 cm in height from ten genes, Pr(H = 5).
Use the result of the previous problem to approximate the probability that the height is in the interval between 4.5 and 5.5 cm. Compare with the result in Exercise 26. Based on the probabilities in
Suppose that n = 16, and that Xi takes the value 0 with probability 0.5 and the value 1 with probability 0.5.Suppose the random variables X1, X2 ... Xn are i.i.d. and consider the averageFind the
Find the normal approximation for IQ with both genetic and environmental effects. What is the maximum possible IQ with the model? Suppose that scientists develop a new model of human IQ that includes
Suppose immigrants arrive into and emigrate from population a for 20 yr.Suppose immigration and emigration change the sizes of four populations with the following probabilities. Find the p.d.f. of
Suppose immigrants arrive into and emigrate from population b for 20 yr.Suppose immigration and emigration change the sizes of four populations with the following probabilities. Find the p.d.f. of
Suppose immigrants arrive into and emigrate from population c for 10 yr. How accurate do you think the normal approximation is?Suppose immigration and emigration change the sizes of four populations
Suppose immigrants arrive into and emigrate from population d for 10 yr. How accurate do you think the normal approximation is?Suppose immigration and emigration change the sizes of four populations
Ri = 4 with probability 0.5, Ri = 0.25 with probability 0.5. Find the approximate normal distribution for ln(P50), the log of the population size after 50 time steps. Although the Central Limit
Ri = 4 with probability 0.25, Ri = 0.25 with probability 0.75. Find the approximate normal distribution for ln(P25). Although the Central Limit Theorem applies to the sums of independent and
Let R be a random variable giving the per capita production in a population with p.d.f. g(x) = 5.0 for 1.0 ‰¤ x ‰¤ 1.2 (the values used in Example 6.1.2). Are the simulations in Example 6.1.2
Suppose that n = 50, and that Xi takes the value 0 with probability 0.25, the value 1 with probability 0.5, and the value 2 with probability 0.25.Suppose the random variables X1, X2 ... Xn are i.i.d.
Let R be a random variable giving the per capita production in a population with p.d.f. g(x) = 1.25 for 0.7 ‰¤ x ‰¤ 1.5 (the values used in Example 6.1.3). Are the simulations in Example 6.1.3
Experiment a is repeated 20 times. Write an integral that estimates the probability that there are between 30 and 40 mutants, and shade the corresponding area on a sketch of the approximate normal
Experiment b is repeated 80 times. Write an integral that estimates the probability that there are between 30 and 40 mutants, and shade the corresponding area on a sketch of the approximate normal
Experiment c is repeated 100 times. Write an integral that estimates the probability that there is an average of less than 1.8 mutants per experiment and shade the corresponding area on a sketch of
Experiment d is repeated 35 times. Write an integral that estimates the probability that there is an average of less than 1.8 mutants per experiment and shade the corresponding area on a sketch of
Find and sketch the p.d.f. of 3X. Suppose that X and Y are independent normally distributed random variables with X ≈ (5.0, 16.0) and Y ≈ N(10.0, 9.0).
Find the z-score (the number of standard deviations from the mean) for the following measurements. 1. A value of 11.0 drawn from a normal distribution with mean 13.0 and standard deviation 1.2. 2. A,
The probability of a value between 7.0 and 13.0 drawn from a normal distribution with mean 10.0 and variance 4.0. Use the cumulative distribution function for the standard normal, Φ(z), to find the
The masses of a type of insect are normally distributed with a mean of 0.38 g and a standard deviation of 0.09 g. What is the probability that a given insect has mass less than 0.40 g? Using a table
Scores on a test are normally distributed with mean 70 and standard deviation 10. What is the probability that a student scores more than 85? Using a table or computer program that can calculate the
Measurement errors are normally distributed with a mean of 0 mm and a standard deviation of 0.01 mm. Find the probability that a given measurement is within 0.012 mm of the true value. Using a table
The number of insects captured in a trap on different nights is normally distributed with mean 2950 and standard deviation 550. What is the probability of capturing between 2500 and 3500
43% of trees are infested by a certain insect. What is the chance of randomly choosing 40 trees, fewer than 10 of which are infested? Use the normal approximation to the binomial with and without the
30% of the cells in a small organism are not functioning. What is the probability that an organ consisting of 250 cells is functioning if it requires 170 cells to work? Use the normal approximation
In a certain species of wasp, 75% of individuals are female. Find the probability that between 70 and 80 wasps (inclusive) are female in a sample of 100 wasps. Use the normal approximation to the
10% of people are known to carry a certain gene. In a sample of 200, what is the probability that between 5 and 15 people carry the gene? Use the normal approximation to the binomial with and without
Seeds have fallen into a region with an average density of 20 seeds per square meter. What is the probability that a particular square meter contains fewer than 15 seeds? Use the normal approximation
Mutations occur along a piece of DNA at a rate of 0.023 per thousand codons. What is the probability of 30 or more mutations in a piece of DNA 1 million codons long? Use the normal approximation to
Insects are caught in a trap at a rate of 0.21 per minute. What is the probability of catching between 10 and 15 insects in an hour? Use the normal approximation to the Poisson distribution with and
Molecules enter a cell at a rate of 0.045 per second. What is the probability that between 150 and 200 molecules enter during an hour? Use the normal approximation to the Poisson distribution with
A fair coin is flipped eight times. Find the probability of no more than one head. The normal approximation cannot be used in the above case. Why not? Find another way to compute the probabilities
5% of people are infected with a disease. Find the probability of choosing 50 people and finding none who are infected. The normal approximation cannot be used in the above case. Why not? Find
Under the circumstances in Exercise 21, find the probability of catching more than one insect in a minute. The normal approximation cannot be used in the above case. Why not? Find another way to
Under the circumstances in Exercise 20, find the probability of no mutation in a piece of DNA ten thousand codons long. The normal approximation cannot be used in the above case. Why not? Find
Show that E(Z) = 0 without writing down any integrals. Suppose that if X ≈ N(μ, σ2) and Z = X - μ / σ.
Show that Var(Z) = 1 without writing down any integrals. Suppose that if X ≈ N(μ, σ2) and Z = X - μ / σ.
Experiment a is repeated 20 times. Find the probability that there is a total of between 30 and 40 mutants, and compare with the sketch in that Exercise 5. Recall the data describing the
Experiment b is repeated 80 times. Find the probability that there are between 30 and 40 mutants, and compare with the sketch in that Exercise 6. Recall the data describing the probabilities of the
Experiment c is repeated 100 times. Find the probability that there is an average of less than 1.8 mutants, and compare with the sketch in that Exercise 7. Recall the data describing the
Experiment d is repeated 25 times. Find the probability that there is an average of less than 1.8 mutants, and compare with the sketch in that Exercise 8. Recall the data describing the probabilities
Immigrants arrive into and emigrate from population a for 20 years. What is the probability that the population grows if immigrants arrive over a period of 40 years? Why does it become
Immigrants arrive into and emigrate from population b for 20 years. What is the probability that the population grows if immigrants arrive over a period of 10 years? Why does it become
Consider again the situation in Section 7.8, Exercises 29-32, describing the putative genetic basis of IQ.1. What is the probability of an IQ above 100 with just the genes of large effect?2. What is
Consider again the situation in Section 7.8, Exercises 29-32, describing the putative genetic basis of IQ.1. What is the probability of an IQ above 100 with just environmental factors?2. What is the
Ri = 4 with probability 0.5, Ri = 0.25 with probability 0.5. Find the probability that P50 lies between 1 and 10000. In a population growing by reproduction, the logarithm of population size can be
Ri = 4 with probability 0.25, Ri = 0.25 with probability 0.75. Find the probability that P25 lies between 1 and 3. In a population growing by reproduction, the logarithm of population size can be
R has p.d.f. g(x) = 5.0 for 1.0 ≤ x ≤ 1.2. Find the probability that P50 lies between 50 and 150. In a population growing by reproduction, the logarithm of population size can be approximated
R has p.d.f. g(x) = 1.25 for 0.7 ≤ x ≤ 1.5. Find the probability that P50 lies between 50 and 150. In a population growing by reproduction, the logarithm of population size can be approximated
Find the binomial distribution and its normal approximation for the number of tall offspring. What is the probability of getting ten or fewer? Suppose that the alleles A and a for height are
Find the binomial distribution and its normal approximation for the number of intermediate offspring. What is the probability of getting eight or fewer? Suppose that the alleles A and a for height
1 min. Starting with 50 molecules, each leaving with probability 0.2/min, find the normal probability distribution approximating the number remaining at the above time. Use it to estimate the
2 min. Starting with 50 molecules, each leaving with probability 0.2/min, find the normal probability distribution approximating the number remaining at the above time. Use it to estimate the
Use the normal approximation to find the probability that the results lie between 96 and 106 people (inclusive). Figure 6 in Section 6.1, as part of Example 5, illustrating stochastic immigration,
The population size can be thought of as twice a binomial random variable, because we are counting pairs. Use this idea, and the continuity correction, to estimate the probability that the results
A gene has a mutation rate of 0.02 mutations per generation. Estimate the probability of more than 50 mutations in a period of 2000 generations. Genes in different organisms have different rates of
The probability of a value less than 0.7 drawn from a normal distribution with mean 0 and variance 1. Use the cumulative distribution function for the standard normal, Φ(z), to find the above
A gene has a mutation rate of 0.002 mutations per generation. Estimate the probability of exactly one mutation in a period of 2000 generations. How does this compare with the exact answer in Section
The probability of a value greater than -0.1 drawn from a normal distribution with mean 0 and variance 1. Use the cumulative distribution function for the standard normal, Φ(z), to find the above
The probability of a value greater than 11.0 drawn from a normal distribution with mean 13.0 and standard deviation 1.2. Use the cumulative distribution function for the standard normal, Φ(z), to
The probability of a value less than 0.9 drawn from a normal distribution with mean 0.5 and standard deviation 0.3. Use the cumulative distribution function for the standard normal, Φ(z), to find
The probability of a value between 10.0 and 12.0 drawn from a normal distribution with mean 10.0 and variance 25.0. Use the cumulative distribution function for the standard normal, Φ(z), to find
Identify the experimental units on which the following variables are measured: a. Gender of a student b. Number of errors on a midterm exam c. Age of a cancer patient d. Number of flowers on an
Fifty people are grouped into four categories- A, B, C, and D-and the number of people who fall into each category is shown in the table:a. What is the experimental unit? b. What is the variable
A manufacturer of jeans has plants in California, Arizona, and Texas. A group of 25 pairs of jeans is randomly selected from the computerized database, and the state in which each is produced is
During the spring of 2010, the news media were already conducting opinion polls that tracked the fortunes of the major candidates hoping to become the president of the United States. One such poll
Would you want to be the president of the United States? Although many teenagers think that they could grow up to be the president, most don't want the job. In an opinion poll conducted by ABC News,
The social networking site called Face book has grown quickly since its inception in 2004. In fact, Face book's United States user base grew from 42 million users to 103 million users between 2009
How long does it take you to adjust to your normal work routine after coming back from vacation? A bar graph with data from the Snapshots section of USA Today is shown below:a. Are all of the
Construct a stem and leaf plot for these 50 measurements:a. Describe the shape of the data distribution. Do you see any outliers? b. Use the stem and leaf plot to find the smallest observation. c.
a. Approximately how many class intervals should you use?b. Suppose you decide to use classes starting at 1.6 with a class width of .5 (i.e., 1.6 to < 2.1, 2.1 to < 2.6). Construct the relative
Consider this set of data:a. Construct a stem and leaf plot by using the leading digit as the stem. b. Construct a stem and leaf plot by using each leading digit twice. Does this technique improve

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