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essentials of statistics
Essentials Of Statistics For The Behavioral Sciences 5th Edition Frederick J Gravetter, Larry B. Wallnau - Solutions
The following sample was obtained from a population with unknown parameters.Scores: 9, 1, 13, 1a. Compute the sample mean and standard deviation.(Note that these are descriptive values that summarize the sample data.)b. Compute the estimated standard error for M. (Note that this is an inferential
9. The following sample was obtained from a population with unknown parameters.Scores: 8, 0, 8, 8 39a. Compute the sample mean and standard deviation.(Note that these are descriptive values that summarize the sample data.)b. Compute the estimated standard error for M. (Note that this is an
8. A sample of n = 25 individuals is randomly selected from a population with a mean of p. = 65, and a treatment is administered to the individuals in the sample.After treatment, the sample mean is found to be M = 69.a. If the sample variance is s 2 = 100, are the data sufficient to reject HQ and
7. A sample is randomly selected from a population with a mean of p. = 50, and a treatment is administered to the individuals in the sample. After treatment, the sample is found to have a mean of M = 54 with a standard deviation of s = 6.a. If there are n = 9 individuals in the sample, are the data
6. A sample of n = 16 individuals is selected from a population with a mean of |x = 74. A treatment is administered to the individuals in the sample and, after treatment, the sample variance is found to be s = 64.a. If the treatment has a 3-point effect and produces a sample mean of M = 77, is this
5. What is the relationship between the value for degrees of freedom and the shape of the t distribution? What happens to the critical value of t for a particular alpha level when d/ increases in value?
4. Why is a t distribution generally more variable than a normal distribution?
Several factors influence the value obtained for a/ statistic. Some factors affect the numerator of the /statistic and others influence the size of the estimated standard error in the denominator. For each of the following, indicate whether the factor influences the numerator or denominator of the
A sample of n = 25 scores lias a mean of M a sample variance of s 2 = 100.42 and Compute the sample standard deviation (s) and the esli mated standard error for the sample mean (s u ).Briefly describe what is measured by the sample standard deviation and by the estimated standard error.
1. What factor determines whether you should use a.--score or a / statistic for a hypothesis test?
25. A psychologist develops a new inventory to measure described as depressed by a therapist is selected and depression. Using a very large standardization group given the test. Presumably, the higher the score on the of normal" individuals, the mean score on this test is inventory, the more
24. Researchers have often noted increases in violent crimes when it is very hot. In fact, Reifman. Larrick, and Fein (1991) noted that this relationship even extends to baseball. That is, there is a much greater chance of a batter being hit by a pitch when the temperature increases. Consider the
23. On a standardized anagram task (anagrams are sets of scrambled letters that must be arranged to form words), people successfully complete an average of u. = 26 anagrams with ct = 4. This distribution is normal. A researcher would like to demonstrate that the arousal from anxiety is distracting
22. A sample of n = 9 individuals was selected from a normally distributed population with a mean of (x =25 and a standard deviation of ct = 6. A treatment was administered to the sample and, after treatment, the nine individuals produced the following scores: 36, 27.30. 33, 22, 30, 30, 24, 38.a.
21. Reading achievement test scores for fifth-grade students form a normal distribution with a mean of u. = 60 and a standard deviation of ct = 10. A researcher would like to evaluate a new reading program by administering the program to a sample of fifthgrade students and then measuring their
20. A sample of n = 16 individuals is selected from a population that forms a normal distribution with u. = 40. A treatment is administered to the sample and, after treatment, the sample is measured to evaluate the effect of the treatment.a. Assuming that the population standard deviation is ct =
19. A psychologist has developed a standardized test for measuring the vocabulary skills of 4-year-old children.The scores on the test form a normal distribution with p. = 60 and ct = 10. A researcher would like to use this test to investigate the hypothesis that children who grow up as single
18. A sample of n = 9 scores is obtained from a normal population distribution with 0" = 12. The sample mean is M = 60.a. With a two-tailed test and a = .05. use the sample data to test the hypothesis that the population mean is p. = 65.b. With a two-tailed test and a = .05. use the sample data to
17. A sample of n = 4 individuals is selected from a normal population distribution with p = 70 and ct = 10.A treatment is administered to the individuals in the sample, and after the treatment, the sample mean is found to be M = 75.a. On the basis of the sample data, can you conclude that the
16. Under some circumstances a 6-point treatment effect can be very large, and in some circumstances it can be very small. Assume that a sample of n = 16 individuals is selected from a population with a mean of p, = 70. A treatment is administered to the sample and, after treatment, the sample mean
15. Assume that a treatment really does have an effect and that the treatment effect is being evaluated with a hypothesis test. If all other factors are held constant, how is the outcome of the hypothesis test influenced by the variability of the scores? To answer this question, do the following
14. Assume that a treatment really does have an effect and that the treatment effect is being evaluated with a hypothesis test. If all other factors are held constant, how is the outcome of the hypothesis test influenced by sample size? To answer this question, do the following two tests and
13. Suppose that scores on the Scholastic Achievement Test (SAT) form a normal distribution with p, = 500 and ct = 100. A high school counselor has developed a special course designed to boost SAT scores. A random sample of n = 16 students is selected to take the course and then the SAT. The sample
12. State College is evaluating a new English composition course for freshmen. A random sample of n = 25 freshmen is obtained and the students are placed in the course during their first semester. One year later, a writing sample is obtained for each student and the writing samples are graded using
11. A researcher is evaluating the effectiveness of a new physical education program for elementary school children. The program is designed to reduce competition and increase individual self-esteem. A sample of n = 16 children is selected and the children are placed in the new program. After 3
10. Numerous studies have demonstrated that listening to music while studying can improve memory (Hallam, Price, & Katsarou, 2002). To demonstrate this phenomenon, a researcher obtains a sample of college students and gives them a standardized memory test while they are listening to background
9. The distribution of scores from a self-esteem test for third-grade children forms a normal distribution with a mean of u. = 40 and a standard deviation of o = 12.A researcher is evaluating the effectiveness of a new program designed to increase self-esteem for children.A sample of n = 36
8. A researcher would like to test the effectiveness of a newly developed growth hormone. The researcher knows that under normal circumstances laboratory rats reach an average weight of jjl = 950 grams at 10 weeks of age. The distribution of weights is normal with a = 30. A random sample of n = 25
7. There is some evidence to suggest that certain herbs can have an effect on human memory. A researcher plans to use a standardized memory test to evaluate the effect of the herbs. Scores on the standardized test form a normal-shaped distribution with a mean of u, = 70 and a standard deviation of
6. The term error is used two different ways in the context of a hypothesis test. First, there is the concept of standard error, and. second, there is the concept of a Type I error.PROBLEMS 215a. What factor can a researcher control that will reduce the risk of a Type I error?b. What factor can a
5. Discuss the errors that can be made in hypothesis testing.a. What is a Type I error? Why might it occur?b. What is a Type II error? How does it happen?
4. Briefly explain the advantage of using an alpha level of .01 versus a level of .05. In general, what is the disadvantage of using a smaller alpha level?
3. What happens to the boundaries for the critical region when the alpha level is lowered—for example, from.05 to .01? Also, what happens to the probability of a Type I error when the alpha level is lowered?
2. The value of the z-score that is obtained for a hypothesis test is influenced by several factors. Some factors influence the size of the numerator of the z-score and other factors influence the size of the standard error in the denominator. For each of the following, indicate whether the factor
1. In the z-score formula as it is used in a hypothesis test,a. Explain what is measured by M — u, in the numerator.b. Explain what is measured by the standard error in the denominator.
5. What research situation is likely to lead to a Type II error?
4. Define a Type II error.
3. If the alpha level is changed from a = .05 to a = .01, the probability of a Type I error increases. (True or false?)
2. Why is the consequence of a Type I error considered serious?
1. What is a Type I error?
5. A decision to reject the null hypothesis means you have demonstrated that the treatment has no effect. (True or false?)
4. A small value (near zero) for the z-score statistic is evidence that the null hypothesis should be rejected. (True or false?)
3. As the alpha level gets smaller, the size of the critical region also gets smaller.(True or false?)
2. Define the critical region for a hypothesis test.
1. What does the null hypothesis predict about a population or a treatment effect?
\ivy2f: Research examining sleep behavior demonstrates a S relationship between age and average amount of sleep.The following data show the mean amount of sleep and the standard error for five samples representing five different ages. Sketch a line graph showing these data including the standard
26. Welsh, Davis, Burke, and Williams (2002) conducted a study to evaluate the effectiveness of a carbohydrateelectrolyte drink on sports performance and endurance.Experienced athletes were given either a carbohydrateelectrolyte drink or a placebo while they were tested on a series of
25. People are selected to serve on juries by randomly picking names from the list of registered voters. The average age for registered voters in the county is p. = 39.7 years with ct = 1 1.8. The distribution of ages is approximately normal. During a recent jury trial in the county courthouse, a
24. The average age for licensed drivers in the county is p. = 42.6 years with a standard deviation of a = 12.The distribution is approximately normal.a. A researcher obtained a sample of n = 25 drivers who received parking tickets. The average age for these drivers was M = 40.5. Is this a
23. A normally distributed population has p. = 80 and ct = 20.a. Sketch the distribution of sample means for samples of n = 25 selected from this population. Show the expected value of M and the standard error in your sketch.b. What is the probability of obtaining a sample mean greater than M = 85
22. A normal distribution has p = 35 and cr = 8.a. Sketch the distribution of sample means for samples of n = 4 selected from this population. Show the expected value of M and the standard error in your sketch.b. For a sample of n = 4, what is the probability of selecting a random sample with a
21. A normal population has p, = 55 and ct = 8.a. Sketch the distribution of sample means for samples of size n — 16 selected from this population.b. What proportion of samples based on n = 16 will have sample means greater than 59?c. What proportion of samples with // = 16 will have sample means
20. A normal population has p. = 80 and ct = 12.a. Using this population, sketch the distribution of sample means for n = 4. Mark the location of the mean (expected value), and show the standard error in your sketch.b. Using your sketch from part (a), what proportion of samples of n = 4 will have
19. For a normal population with p. = 70 and ct = 20, what is the probability of obtaining a sample mean greater than 75a. For a random sample of n = 4 scores?b. For a random sample of n = 16 scores?c. For a random sample of n = 100 scores?
18. A normally distributed population has p. = 70 and a = 15.a. For this population, what proportion of the scores have values between 65 and 75?b. If random samples of size n = 9 are selected from this population, what proportion of the sample means will have values between 65 and 75?
17. A normal population has p. = 70 and a = 12.a. Sketch the population distribution. What proportion of the scores have values greater than X =73?b. Sketch the distribution of sample means for samples of size n = 16. What proportion of the sample means have values greater than 73?
16. The distribution of SAT scores is normal with a mean of p. = 500 and a standard deviation of a = 100. If a sample of n = 25 SAT scores is obtained, the sample mean should be around p = 500.a. What range of values should contain the sample mean 90% of the time? (That is. find the middle 90% of
15. A normal distribution has u. = 100 and cr = 20.a. Sketch the distribution of sample means for random samples of n = 25.b. Using ^-scores, find the boundaries that separate the middle 95% of the sample means from the extreme 5% in the tails of the distribution.'C. A sample mean of M = 106 is
14. A random sample of n = 16 scores is selected from a normal distribution with a mean of p. = 50 and a standard deviation of a = 10.a. What is the probability that the sample mean will have a value between 45 and 55?b. What is the probability that the sample mean will have a value between 48 and
13. If you are taking a random sample from a normally distributed population with p = 100 and ct = 12, which of the following outcomes is more likely?PROBLEMS 175 Explain your answer. (Hint: Calculate the c-score for each sample mean.)a. A sample mean greater than 106 for a sample of n = 4b. A
12. A normally distributed population has a 1 mean of p =80 and a standard deviation of cr = 20. Calculate the z-score for each of the following samples. Based on the z-score value, determine whether the sample mean is a typical, representative value or an extreme value for samples of this size.a.
11. For each of the following, assume that the sample was selected from a normally distributed population with p = 65 and cr = 10.a. For a sample of n = 4, would a sample mean of M = 70 be considered an extreme value compared to other samples of n = 4 scores? Explain your answer.b. For a sample of
10. The following samples were obtained from a population with a mean of p = 40 and a standard deviation of ct = 8. Find the z-score corresponding to each sample mean.a. A sample of n = 4 scores with M = 38b. A sample of n = 16 scores with M = 38c. A sample of n = 64 scores with M = 38
9. A population has a mean of p = 100 and a standard deviation of ct = 20. Find the z-score corresponding to each of the following sample means obtained from this population.a. M = 102 for a sample of n = 4 scoresb. M = 102 for a sample of n = 100 scoresc. M = 95 for a sample of n = 16 scoresd. M =
8. For a population with a standard deviation of ct = 20:a. How large a sample would be needed to have a standard error less than 10 points?b. How large a sample would be needed to have a standard error less than 4 points?c. How large a sample would be needed to have a standard error less than 2
7. Standard error measures the standard distance between a sample mean and the population mean. For a population with ct = 30,a. How large a sample would be needed to have a standard error of 10 points or less?b. How large a sample would be needed to have a standard error of 5 points or less?c. Can
6. IQ scores form normal distributions with a = 15.However, the mean IQ varies from one population to another. For example, the mean IQ for registered voters is different from the mean for nonregistered voters. A researcher would like to use a sample to obtain information about the mean IQ for the
5. A population has a mean of p. = 80 and a standard deviation of ct = 10.a. If you selected a random sample of n = 4 scores from this population, how much error would you expect between the sample mean and the population mean?b. If you selected a random sample of n = 25 scores, how much error
4. The distribution of sample means is not always a normal distribution. Under what circumstances will the distribution of sample means not be normal?
3. Describe the distribution of sample means (shape, expected value, and standard error) for samples of size n = 36 selected from a population with p. = 65 and cr = 12.
2. For each of the following, assume that the sample was selected from a population with p = 75 and ct = 20.a. What is the expected value of M for a sample of n = 4 scores?b. What is the standard error of M for a sample of n = 4 scores?c. What is the expected value of M for a sample of n = 25
1. Briefly define each of the following:a. Distribution of sample meansb. Expected value of Mc. Standard error of M
27. Find each of the following values for a normal distribution with a mean of p = 500 and a standard deviation of (j = 100. (See Box 6.1 for help.)a. The percentile rank for X = 575b. The percentile rank for X = 350c. The 90th percentile 'd. The 25th percentile
26. For a normal distribution, find the percentile rank for each of the following c-score locations. (See Box 6.1.)a. z - -1.25b. z = -0.60c. z = 0.25d. z = 1.40
25. The distribution of SAT scores is normal with p = 500 and cr = 100. What SAT scores, X values, separatea. The middle 60% from the rest of the distribution?b. The middle 80% from the rest of the distribution?c. The middle 95% from the rest of the distribution?
24. A normal distribution has a mean of 80 and a standard deviation of 10. For this distribution, find each of the following probability values:a. p(X > 75) = ?b. p(X < 65) = ?c. p(X < 100) = ?d. P(65 < X < 95) = ?e. P(84 < X < 90) = ?
23. Drivers pay an average of p, = $690 per year for automobile insurance. The distribution of insurance payments is approximately normal with a standard deviation of a = 1 10.a. What proportion of drivers pay over $800 per year for insurance?b. What is the probability of randomly selecting a
The county government obtains juries by selecting people from the list of registered voters. The average age for registered voters in the county is p. =39.7 years with a standard deviation of a = 1 1.8.The distribution of ages is approximately normal.Given this distribution,a. What proportion of
22.->a. What proportion of the population spends more \than $150 per week on groceries? \b. What is the probability of randomly selecting a
21. A consumer survey indicates that the average household spends p. = $ 1 55 on groceries each week. The distribution of spending amounts is approximately normal with a standard deviation of ct = $25. Based on this distribution.
20. The distribution of scores on the SAT is approximately normal with p. = 500 and a = 1 00.a. What proportion of the population have SAT scores above 650?b. What proportion of the population have SAT scores below 540?c. What is the minimum SAT score needed to be in the highest 20% of the
The distribution of IQ scores is normal with \x = 100 and cr = 15. What proportion of the population have IQ scoresa. Above 130?b. Below 90?c. Above 1 10?
11. For a normal distribution, identify the z-score locatiorj that separatesa. The top 10% of the distribution (right side) from the bottom 90% (left side)i ViccOA normal distribution has a mean of |x = 40 and a standard deviation of a = 8. For each of the following scores, indicate whether the
For a normal distribution, identify the z-score location that would separate the distribution into sections so that there isa. 60% in the body on the right-hand sideb. 85% in the body on the right-hand sidec. 90% in the body on the left-hand sided. 95% in the body on the left-hand side
A normal distribution has a mean of (x = 60 and a standard deviation of cr = 10. For each of the following scores, indicate whether the tail of the distribution is located to the right or the left of the score, and find the proportion of the distribution located in the tail. .a. X = 55b. X = 51c. X
Answer each of the following for a positively skewed distribution. (Caution: This is a trick question.)a. Find the proportion of the distribution located in the tail beyond z = 2.00.b. What z-score value separates the top 40% of the distribution from the rest?
Identify the z-score values that separate each of the following sections from the rest of the scores in a normal distribution.a. The middle 30%b. The middle 50%c. The middle 70%d. The middle 95%
For a normal distribution, find the z-score values that divide the distribution so that they separatea. The middle 80% from the extreme 20% in the tailsb. The middle 85% from the extreme 15% in the tailsc. The middle 90% from the extreme 1 0% in the tailsd. The middle 95% from the extreme 5% in the
For each of the following pairs of z-scores, find the proportion of a normal distribution that is located between the two z-score values.a. Between z = 1.50. and z = 2.00b. Between z = -1.00 and z = -0.50c. Between z = and z = 2.00d. Between z = -1.50 and z = 0.25
10b. The top 40% from the bottom 60%c. The top 80% from the bottom 20%For a normal distribution, find the probability of randomly selecting a z-scorea. Between z = +0.50 and z = -0.50b. Between z = + 1.00 and z = -1.00c. Between z = + 1 .50 and z = - 1 50d. Between z = +2.00 and z = -2.00
9. For a normal distribution, identify the z-score location that would separate the distribution into sections so that there isa. 20% in the tail on the right-hand sideb. 25% in the tail on the right-hand sidec. 15% in the tail on the left-hand sided. 30% in the tail on the left-hand side
8. Find each of the following probabilities for a normal distribution.a. p(z > -0.25)b. p(z> 1.75)c. p(z< 0.90)d. p(z< -1.25)
7. Find each of the following probabilities for a normal distribution that has been transformed into z-scores. Ia. p(z > 1.50)b. p(z > -2.00)J^>.Lc. p(z < 0.50)d. p(z < -0.75)
6. For each of the following z-score values, sketch a normal distribution and draw a vertical line at the location of the z-score. Then, determine whether the, body is to the right or the left of the z-score and find the proportion of the distribution located in the body.a. z = 0.25b. z = -1.50 c
5. For each of the following z-score values in a normal distribution, determine whether the tail is to the right or the left of the z-score and find the proportion of the distribution located in the tail.a. z = 2.00b. z = 0.50c. z = -1.00d. z = -0.75
4. What is sampling with replacement, and why is it used?
3. What requirements must be satisfied to have a random sample!
2. Ajar contains 10 black marbles and 40 white marbles.a. If you randomly select a marble from the jar, what is the probability that you will get a white marble?b. If you are selecting a random sample of n = 3 marbles and the first 2 marbles are both white, what is the probability that the third
In a psychology class of 90 students, there are 30 males and 60 females. Of the 30 men, 5 are freshmen.Of the 60 women, 10 are freshmen. If you randomly sample an individual from this class, what is the probability of obtaininga. A female?b. A freshman?c. A male freshman?PROBLEMS 15]
28. A population consists of the following N = 5 scores:0,6, 4, 3, and 12.a. Compute u, and ct for the population.b. Find the c-score for each score in the population.c. Transform the original population into a new population of N = 5 scores with a mean of u. = 60 and a standard deviation of o = 8.
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