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biostatistics

Introductory Biostatistics For The Health Sciences 1st Edition Michael R. Chernick, Robert H. Friis - Solutions

Suppose you are planning another experiment like the one in Exercise 9.20.Based on that data: (1) you are willing to assume that the standard deviation of the difference in means is 1.5°F, and (2) you anticipate that the average temperature in New York tends to be 3°F lower than the corresponding
Consider the paired t test that was used with the data in Table 9.1, what would the power of the test be if the alternative is that the mean temperature differs by 3 degrees between the cities? What is the power at a difference of 5 degrees? Why does the power depend on the assumed true difference
Find the critical values for t that correspond to the following:a. n = 12, = 0.05 one-tailed (right)b. n = 12, = 0.01 one-tailed (right)c. n = 19, = 0.05 one-tailed (left)d. n = 19, = 0.05 two-tailede. n = 28, = 0.05 one-tailed (left)f. n = 41, = 0.05 two-tailed g. n = 8, = 0.10
Find the area under the t-distribution between zero and the following values:a. 2.62 with 14 degrees of freedomb. –2.85 with 20 degrees of freedomc. 3.36 with 8 degrees of freedomd. 2.04 with 30 degrees of freedome. –2.90 with 17 degrees of freedomf. 2.58 with 1000 degrees of freedom
Recent advances in DNA testing have helped to confirm guilt or innocence in many well-publicized criminal cases. Let us consider the DNA test results to be the gold standard of guilt or innocence and a jury trial to be the test of a hypothesis. What types of errors are committed in the following
Redo Exercise 9.14 but use a one-tailed (left-tail) test.
Describe the differences between a one-tailed and a two-tailed test. Give examples of when it would be appropriate to use a two-tailed test and when it would be appropriate to use a one-tailed test.
Test the hypothesis that a normally distributed population has a mean blood glucose level of 100 (2 = 100). Suppose we select a random sample of 30 individuals from this population (X = 98.1, S2 = 126).a. What is the hypothesis set (null and alternative) for a two-tailed test?b. Find the
In the previous exercise there were two possible outcomes; reject the null hypothesis or fail to reject the null hypothesis. Explain in your own words what is meant by these outcomes.
We suspect that the average fasting blood sugar level of Mexican Americans is 108. A random sample of 225 clinic patients (all Mexican American) yields a mean blood sugar level of 119 (S2 = 100). Test the hypothesis that = 108.a. What is the hypothesis set for a two-tailed test?b. Find the
Again use the test in Exercise 9.9 to determine the power when the mean is 1.5 under the alternative hypothesis and the variance is again 5.
Use the test in Exercise 9.9 (i.e., critical values) to determine the power of the test when the mean is 1.0 under the alternative hypothesis, the variance is 5, and the sample size is 5.
Consider a sample of size 5 from a normal population with a variance of 5 and a mean of zero under the null hypothesis. Find the critical values for a 0.05 two-sided significance test of the mean equals zero versus the mean differs from zero.
Suppose we would like to test the hypothesis that mean cholesterol levels of residents of Kalamazoo and Ann Arbor, Michigan, are the same. We know that both populations have the same variance. State the appropriate hypothesis set (null and alternative). What test statistic should be used?
The Orange County Public Health Department was concerned that the mean daily fecal coliform level in a particular month at Huntington Beach, California, exceeded a safe level. Let us call this level “a.” State the appropriate hypothesis set (null and alternative) for testing whether the mean
Using the data from Exercise 9.5, state the hypothesis set (null and alternative hypotheses) for testing whether the mean blood lead level of smelter workers exceeds that of clerical workers.
In the example cited in Exercise 9.3, the physician measures the blood lead levels of smelter workers in the same factory and finds their mean blood lead level to be 15.3. State the hypothesis set (null and alternative hypotheses) for testing whether the mean blood lead level of clerical workers
Using the data from Exercise 9.3, state the hypothesis set (null and alternative hypotheses) for testing whether the population mean blood lead level exceeds 11.2. What is the name for this type of hypothesis test?
In a factory where he conducted a research study, an occupational medicine physician found that the mean blood lead level of clerical workers was 11.2.State the null and alternative hypotheses for testing that the population mean blood lead level is equal to 11.2. What is the name for this type of
Chapters 8 and 9 discussed methods for calculating confidence intervals and testing hypotheses, respectively. In what manner are parameter estimation and hypothesis testing similar to one another? In what manner are they different from one another?
The following terms were discussed in Chapter 9. Give definitions of them in your own words:a. Hypothesis testb. Null hypothesisc. Alternative hypothesisd. Type I errore. Type II errorf. p-value g. Critical region h. Power of a test i. Power function j. Test statistic k. Significance level
In Exercise 8.18, how many individuals must you select to obtain the halfwidth of a 99% confidence interval no larger than 0.5 mmHg?
Change exercise 8.18 to assume there are 400 randomly selected individuals with a mean of 75 and standard deviation of 12. Construct a 99% confidence interval for the mean.
The mean diastolic blood pressure for 225 randomly selected individuals is 75 mmHg with a standard deviation of 12.0 mmHg. Construct a 95% confidence interval for the mean.
Repeat Exercise 8.16 for 99% confidence intervals.
The standard hemoglobin reading for normal males of adult age is 15 g/100 ml. The standard deviation is about 2.5 g/100 ml. For a group of 36 male construction workers, the sample mean was 16 g/100 ml.a. Construct a 95% confidence interval for the male construction workers.What is your
The mean weight of 100 men in a particular heart study is 61 kg with a standard deviation of 7.9 kg. Construct a 95% confidence interval for the mean.
What would the number of experimental subjects have to be under the assumptions in Exercise 8.13 if you want to construct a 99% confidence interval with half-width no greater then 0.4? Under the same criteria we decide that n should be large enough so that a 95% confidence interval would have this
Suppose you want to construct a 95% confidence interval for mean aggression scores as in Exercise 8.11, and you can assume that the standard deviation of the estimate is 5. How many experimental subjects do you need for the half-width of the interval to be no larger than 0.4?
Suppose the sample size in exercise 8.11 is 169 and the mean score is 55 with a standard deviation of 5. Construct a 99% confidence interval for the population mean.
In a sample of 125 experimental subjects, the mean score on a postexperimental measure of aggression was 55 with a standard deviation of 5. Construct a 95% confidence interval for the population mean.
Suppose that a sample of pulse rates gives a mean of 71.3, as in Exercise 8.9, with a standard deviation that can be assumed to be 9.4 (close to the estimate observed in exercise 8.9). How many patients should be sampled to obtain a 95% confidence interval for the mean that has half-width 1.2 beats
Suppose we randomly select 20 students enrolled in an introductory course in biostatistics and measure their resting heart rates. We obtain a mean of 66.9(S = 9.02). Calculate a 95% confidence interval for the population mean and give an interpretation of the interval you obtain.
How can bootstrap confidence intervals be generated? Name the simplest form of a bootstrap confidence interval. Are bootstrap confidence intervals exact?
Explain the bootstrap principle. How can it be used to make statistical inferences?
Two situations affect the choice of a calculation of a confidence interval: (1)the population is known; (2) the population variance is unknown. How would you calculate a confidence interval given these two different circumstances?
State the advantages and disadvantages of using confidence intervals for statistical inference.
What are the desirable properties of a confidence interval? How do sample size and the level of confidence (e.g., 90%, 95%, 99%) affect the width of a confidence interval?
What are the advantages and disadvantages of using point estimates for statistical inference?
What are the desirable properties of an estimator of a population parameter?
In your own words define the following terms:a. Descriptive statisticsb. Inferential statisticsc. Point estimate of a population parameterd. Interval (confidence interval) estimate of a population parametere. Type I errorf. Biased estimator of a population parameter g. Mean square error
Assume that we have normally distributed data. From the standard normal table, find the probability area bounded by ±1 standard deviation units about a population mean and by ±1 standard errors about the mean for any distribution of sample means of a fixed size. How do the areas compare?
Based on a sample of six cases, the mean incubation period for a gastrointestinal disease is 26.0 days with a standard deviation of 2.83 days. The population standard deviation () is unknown, but = 28.0 days. Assume the data are normally distributed and normalize the sample mean. What is the
The following questions relate to comparisons between the standard normal distribution and the t distribution:a. What is the difference between the standard normal distribution (used to determine Z scores) and the t distribution?b. When are the values for t and Z almost identical?c. Assume that a
Here are some questions about sampling distributions in comparison to the parent populations from which samples are selected:a. Describe the difference between the distribution of the observed sample values from a population and the distribution of means calculated from samples of size n.b. What is
The following questions pertain to the central limit theorem:a. Describe the three main consequences of the central limit theorem for the relationship between a sampling distribution and a parent population.b. What conditions must be met for the central limit theorem to apply?c. Why is the central
Using the data from Exercise 7.8, for a sample of 25 female students, calculate the standard error of the mean, draw the sampling distribution about , and find:a. P(200 < X < 220)b. P(X < 196)c. P(X > 224)
A health researcher collected blood samples from a population of female medical students. The following cholesterol measurements were obtained: = 211, = 44. If we select any student at random, what is the probability that her cholesterol value (X) will be:a. P(150 < X < 250)b. P(X < 140)c. P(X
The average height of male physicians employed by a Veterans Affairs medical center is 180.18 cm with a standard deviation of 4.75 cm. Find the probability of obtaining a mean height of 184.93 cm or greater for a sample size of:a. 5b. 10c. 20
Based on the findings obtained from Exercises 7.4 and 7.5, what general statement can be made regarding the effect of sample size on the probabilities for the sample means?
Repeat Exercise 7.4 with a sample size of 36.
The population mean () blood levels of lead of children who live in a city is 11.93 with a standard deviation of 3. For a sample size of 9, what is the probability that a mean blood level will be:a. Between 8.93 and 14.93b. Below 7.53c. Above 16.43
The average fasting cholesterol level of an entire community in Michigan is = 200 ( = 20). A sample (n = 25) is selected from this population. Based on the information provided, sketch the sampling distribution of .
Calculate the standard error of the mean for the following sample sizes ( =100, = 10). Describe how the standard error of the mean changes as n increases.a. n = 4b. n = 9c. n = 16d. n = 25e. n = 36
Define in your own words the following terms:a. Central limit theoremb. Standard error of the meand. Student’s t statistic
It is suspected that a random variable has a normal distribution with a mean of 6 and a standard deviation of 0.5. After observing several hundred values, we find that the mean is approximately equal to 6 and the standard deviation is close to 0.5. However, we find that 53% percent of the
The population of 25-year-old American women has a remaining life expectancy that is also normally distributed and differs from that of the males in Exercise 6.17 only in that the women’s average remaining life expectancy is 5 years longer than for the males.a. What proportion of these
Suppose that the population of 25-year-old American males has an average remaining life expectancy of 50 years with a standard deviation of 5 years and that life expectancy is normally distributed.a. What proportion of these 25-year-old males will live past 75?b. What proportion of these
Assume the weights of women between 16 and 30 years of age are normally distributed with a mean of 120 pounds and a standard deviation of 18 pounds.If 100 women are selected at random from this population, how many would you expect to have the following weights (round off to the nearest integer):a.
A community epidemiology study conducted fasting blood tests on a large community and obtained the following results for triglyceride levels (which were normally distributed): males— = 100, = 30; females— = 85, =25. If we decide that persons who fall within two standard deviations of the
Repeat Exercise 6.12 again, but this time with a mean of 110 and a standard deviation of 15.
Repeat Exercise 6.12 but with a standard deviation of 9 instead of 12.
In a health examination survey of a prefecture in Japan, the population was found to have an average fasting blood glucose level of 99.0 with a standard deviation of 12. Determine the probability that an individual selected at random will have a blood sugar reading:a. Greater than 120 (let the
The mean height of a population of girls aged 15 to 19 years in a northern province in Sweden was found to be 165 cm with a standard deviation of 15 cm. Assuming that the heights are normally distributed, find the heights in centimeters that correspond to the following percentiles:a. Between the
A first year medical school class (n = 200) took a first midterm examination in human physiology. The results were as follows (X = 65, S = 7). Explain how you would standardize any particular score from this distribution, and then solve the following problems:a. What Z score corresponds to a test
The inverse of Exercise 6.8 is to be able to find a Z score when you know a probability. Assuming a standard normal distribution, identify the Z score indicated by a # sign that is associated with each of the following probabilities:a. P(Z < #) = 0.9920b. P(Z > #) = 0.0005c. P(Z < #) = 0.0250d. P(Z
Another way to express probabilities associated with Z scores (assuming a standard normal distribution) is to use parentheses according to the format:P(Z > 0) = 0.5000, for the case when Z = 0. Calculate the following probabilities:a. P(Z < –2.90) =b. P(Z > –1.11) =c. P(Z < 0.66) =d. P(Z >
The areas under a standard normal curve also may be considered to be probabilities.Find probabilities associated with the area:a. Above Z = 2.33b. Below Z = –2.58c. Above Z = 1.65 and below Z = –1.65d. Above Z = 1.96 and below Z = –1.96e. Above Z = 2.33 and below Z = –2.33
Determine the areas under the standard normal curve that fall between the following values of Z:a. 0 and 1.00b. 0 and 1.28c. 0 and –1.65d. 1.00 and 2.33e. –1.00 and –2.58
Referring to the properties shown in Table 6.3, find the standard normal score (Z score) associated with the following percentiles: (a) 5th, (b) 10th, (c)20th, (d) 25th, (e) 50th, (f) 75th, (g) 80th, (h) 90th, and (i) 95th.
If you were a clinical laboratory technician in a hospital, how would you apply the principles of the standard normal distribution to define normal and abnormal blood test results (e.g., for low-density lipoprotein)?
The following questions pertain to the standard normal distribution:a. How is the standard normal distribution defined?b. How does a standard normal distribution compare to a normal distribution?c. What is the procedure for finding an area under the standard normal curve?d. How would the typical
The following questions pertain to some important facts to know about a normal distribution:a. What are three important properties of a normal distribution?b. What percentage of the values are:i. within 1 standard deviation of the mean?ii. 2 standard deviations or more above the mean?iii. 1.96
Define the following terms in your own words:Continuous distribution Normal distribution Standard normal distribution Probability density function Standardization Standard score Z score Percentile
In the example in Section 5.9, consider the probability that three items have mismatched labels and one of these items is found.a. Calculate the probability that all three items would pass inspection and, therefore, there would be two additional ones out of the 84 remaining in the field.b.
a. Define the probability density and cumulative probability function for an absolutely continuous random variable.b. Which of these functions is analogous to the probability mass function of a discrete random variable?c. Determine the probability density function and the cumulative distribution
Compute the mean and variance of the binomial distribution Bi(n, p). Find the arithmetic values for the special case in which both n = 10 and p = 1/2.
Under the conditions given for Exercise 5.18, calculate the probability that the child will have two dominant genes if:a. One of the parents is a carrier and the other parent has two dominant genesb. Both of the parents are carriers
Sickle cell anemia is a genetic disease that occurs only if a child inherits two recessive genes. Each child receives one gene from the father and one from the mother. A person can be characterized as follows: The person can have:(a) two dominant genes (cannot transmit the disease to a child), (b)
For the binomial distribution, do the following:a. Give the conditions necessary for the binomial distribution to apply to a random variable.b. Give the general formula for the probability of r successes in n trials.c. Give the probability mass function for Bi(10, 0.40).d. For the distribution inc,
Consider the following 2 × 2 table that shows incidence of myocardial infarction(denoted MI) for women who had used oral contraceptives and women who had never used oral contraceptives. The data in the table are fictitious and are used just for illustrative purposes.Assume that the proportions in
Based on the following table of hemoglobin levels for miners, compute the probabilities described below. Assume that the proportion in each category for this set of 90 miners is the true proportion for the population of miners.a. Compute the probability that a miner selected at random from the
Give a definition or description of the following:a. C(4, 2)b. P(5, 3)c. The addition rule for mutually exclusive eventsd. The multiplication rule for independent events
Provide definitions for each of these terms:a. Elementary eventsb. Mutually exclusive eventsc. Equally likely eventsd. Independent eventse. Random variable
In how many ways can four different colored marbles be arranged in a row?
Use Formula 5.8, combinations of r objects taken out of n, to determine the following combinations:a. C(7, 4)b. C(6, 4)c. C(6, 2)d. C(5, 2)e. What is the relationship between 5.11 (d) and 5.9 (e)?f. What is the relationship between 5.11 (b) and 5.9 (d)?
Nine volunteers wish to participate in a clinical trial to test a new medication for depression. In how many ways can we select five of these individuals for assignment to the intervention trial?
Refer to Formula 5.7, permutations of r objects taken from n objects. Compute the following permutations:a. P(8, 3)b. P(7, 5)c. P(4, 2)d. P(6, 4)e. P(5, 2)
Repeat Exercise 5.4 but this time assume that the probability of having a male offspring is 0.514 and the probability of having a female offspring is 0.486. In this case, the elementary outcomes are not equally likely. However, the trials are Bernoulli and the binomial distribution applies. Use
In an ablation procedure, the probability of acute success (determined at completion of the procedure) is 0.95 when an image mapping system is used.Without the image mapping system, the probably of acute success is only 0.80. Suppose that Patient A is given the treatment with the mapping system and
A pharmacist has filled a box with six different kinds of antibiotic capsules.There are a total of 300 capsules, which are distributed as follows: tetracycline(15), penicillin (30), minocycline (45), Bactrim (60), streptomycin (70), and Zithromax (80). She asks her assistant to mix the pills
What is the expected distribution—numbers and proportions—of each of the six faces (i.e., 1 through 6) of a die when it is rolled 1000 times?
A certain laboratory animal used in preclinical evaluations of experimental catheters gives birth to only one offspring at a time. The probability of giving birth to a male or a female offspring is equally likely. In three consecutive pregnancies of a single animal, what is the probability of
In the science exhibit of a museum of natural history, a coin-flipping machine tosses a silver dollar into the air and tallies the outcome on a counting device.What are all of the respective possible outcomes in any three consecutive coin tosses? In any three consecutive coin tosses, what is the
In this exercise, we would like you to toss four coins at the same time into the air and record and observe the results obtained for various numbers of coin tosses. Count the frequencies of the following outcomes: 1) zero heads, 2)one head, 3) two heads, 4) three heads, 5) four heads.a. Toss the
By using a computer algorithm, an investigator can assign members of twin pairs at random to an intervention condition in a clinical trial. Assume that each twin pair consists of dizygotic twins (one male and one female). The probability of assigning one member of the pair to the intervention
Answer the following questions:a. Can a population have a zero variance?b. Can a population have a negative variance?c. Can a sample have a zero variance?d. Can a sample have a negative variance?
that it seems suspicious. Such extreme values are called outliers. Which estimate of location do you trust more when this observation is included, the mean or the median?
The eleventh observation of 931 is so different from all the others in Exercise
Which statistics varied the most from Exercise 4.19 to Exercise 4.20? Which statistics varied the least?

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