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probability statistics

Probability And Statistics For Engineers And Scientists 9th Edition Ronald E. Walpole, Raymond H. Myers, Sharon L. Myers, Keying E. Ye - Solutions

Computing the sample variance by dividing SS by n or dividing SS by (n – 1) has been debated. Which of the following explains the rationale for choosing to compute the sample variance as SS divided by (n – 1)?a. It is more important that a sample variance varies minimally from the population
The shape of the sampling distribution of the mean is approximately ____, whereas the shape of the sampling distribution of the variance is approximately ____. [Fill in the blanks]a. normal; normalb. positively skewed; normalc. normal; positively skewedd. positively skewed; positively skewed
Which of the following statements is true regarding the sampling distribution of the mean?a. The sample mean will be an unbiased estimator of the population mean.b. The shape of the sampling distribution will tend toward a positively skewed distribution.c. If we select a sample at random, then on
Which of the following explains why the following statement is true: = M?a. Scores in a population are distributed exactly the same as sample means in a sampling distribution.b. A given sample mean is always equal to the mean of the sampling distribution of sample means.c. The mean of a sampling
A researcher wants to select a sample from a population of 12 people using theoretical sampling. If the total number of possible samples that could be drawn at random from this population is equal to 20,736 samples, then what was the sample size selected?Answer: ____ [enter whole number]
A researcher selects a sample of 5 participants from a population of 10 people. How many possible samples of size 5 can be drawn at random from this population of 10 people using theoretical sampling?a. 1,000b. 10,000c. 100,000
A researcher uses an experimental sampling strategy to select 3 participants from this population. If the sample included Persons B and D, then which of the following is not a possible outcome for selecting the third person?a. The third person selected is Person A.b. The third person selected is
Which of the following is the only sample possible using the experimental sampling strategy?a. Persons B, B, and A.b. Persons A, D, and D.c. Persons B, C, and D.
A researcher constructs an experimental game to evaluate decision making. In this game, participants can select either a red, white, or blue card on each of two draws.The likelihood on a given draw of selecting a red card is p = .40, selecting a white card is p = .30, and selecting a blue card is p
A researcher constructs an experimental game to evaluate how rewards affect behavior.In this game, participants complete a task on each trial that results in a small, medium, or large reward given at random. On each trial, the likelihood of receiving a small reward is p = .35, receiving a medium
A researcher asks the following question: What is the probability that a sample statistic will be smaller than, larger than, or exactly equal to the value of a population parameter?What type of distribution can be used to specifically answer this question that the researcher asked?a. A sampling
In a sample, ____ are distributed along the x-axis, whereas in a sampling distribution of the same mean, ____ are distributed along the x-axis. [Fill in the blanks]a. scores; scores.b. sample means; scores.c. scores; sample means.
Which answer below correctly describes the following statement: µM = µ?a. The mean of a normal distribution in a sample is equal to the value of the population mean.b. The mean of a normal distribution in a population is equal to the value of the population mean.c. The mean of the sampling
Use the z transformation to identify the z score, given the following measures for a data set: µ = 5, σM = 1.5, M = 8.a. z = 2.0b. z = −2.0c. z = 0.5d. z = −0.5
What formula is used to find the probability of obtaining a sample mean from a given population?a. The standard normal transformation.b. The z transformation.c. Both a and b are correct.
If data are reported as 12 ± 20 (M ± σM), then the standard error of the mean is equal to 20.a. True, σM = 20.b. False, σM = 12.
The APA recommends which of the following ways to report the standard error of the mean in a research journal?a. Report σM in the text.b. Report σM in a table.c. Report σM in a graph.d. All of these are correct.
How is the standard error of the mean typically reported in a graph?a. Using error barsb. By plotting sample meansc. By listing group names along the x-axis
The guidelines for reporting statistical information is provided by the APA. What does APA stand for?a. American Psychological Associationb. Association of Psychological Assessmentc. American Psychological Affiliates
The image below shows the SPSS output table for computing the estimate for the standard error of the mean for the following 25 scores: 1, 2, 2, 3, 3, 3, 4, 4, 4, 4, 5, 5, 5, 5, 5, 6, 6, 6, 6, 7, 7, 7, 8, 8, and 9. [Please feel free to enter these data in SPSS and follow the directions given in
When you compute the estimate for the standard error of the mean, the SPSS output table also gives the value of the sample standard deviation and sample size.a. Trueb. False
Using SPSS, which options in the menu bar do you select to compute the estimate for the standard error of the mean?a. Analyze, then Descriptive Statistics and Descriptivesb. Analyze, then Descriptive Statistics and Crosstabsc. Analyze, then Descriptive Statistics and Frequencies
Which of the following states that increasing the number of observations or samples in a study will decrease the standard error?a. Law of large numbersb. Law of central tendencyc. Law of numeric anomalies
The following samples were selected by two researchers: Researcher A: n = 36, µ = 12,σ = 9; Researcher B: n = 36, µ = 12, σ = 6. Which sample is associated with a smaller standard error of the mean?a. The sample from Researchers A.b. The sample from Researcher B.c. None, they both have the same
As sample size increases, the standard error of the meana. increases.b. decreases.c. can increase or decrease.d. does not change.
Which sample will be associated with a larger standard error of the mean?a. Sample 1b. Sample 2c. None, both samples will have the same value for standard error.
A researcher selects two samples of 36 participants each. In the first sample, the population mean was 32 and the variance was 9. In the second sample, the population mean was 4 and the variance was
A researcher selects a sample of 64 participants from a population with a mean of 10 and a variance of 16. What is the standard error of the mean?a. 0.5b. 2.0c. 4.0
The ______ is a numeric measure of sampling error, with larger values indicating greater sampling error or greater differences that can exist from one sample to the next. [Fill in the blank]a. standard unitb. standard normalc. standard error
From the statistician’s perspective, which of the following is most important for sample variance?a. It more important for the sample variance to be an unbiased estimator of the population variance.b. It more important for the sample variance to vary minimally from the value of the population
What is the skewed distribution rule for sample variance?a. It explains that sample variances will vary minimally from the population variance.b. It explains that regardless of the distribution of scores in a population, the sampling distribution of sample variances selected at random from that
If a random sample is selected from a population with a variance equal to 12, then what can we expect the value for the sample variance to equal on average when we divide SS by df?a. 12b. Any positive value.c. Any value selected at random.
What is the central limit theorem?a. It explains that sample means will vary minimally from the population mean.b. It explains that regardless of the distribution of scores in a population, the sampling distribution of sample means selected at random from that population will approach the shape of
The standard error of the mean (σM) representsa. the minimal “error” or deviations from the population mean.b. the standard “error” or distance that sample mean values deviate from the value of the population mean.c. Both a and b are correct.
Any sample statistic obtained from a randomly selected sample that equals the value of its respective population parameter, on average, is calleda. a biased estimator.b. an unbiased estimator.
If a random sample is selected from a population with a mean equal to 15, then what can we expect the value of the sample mean to be, on average?a. 15b. Any positive value.c. Any value selected at random.
A researcher draws a sample of size 2 (n = 2) from a population of size 5 (N = 5). How many possible samples of this size can the researcher draw using experimental sampling?a. 10 possible samplesb. 25 possible samplesc. 50 possible samples
Experimental research is typically conducted using a sampling strategy in whicha. the order of selecting participants matters.b. the same participant can never be selected twice in the same sample.c. participants are replaced after being sampled.d. all of the above.
A researcher draws a sample of size 4 (n = 4) from a population of size 50 (N = 50). How many possible samples of this size can the researcher draw using theoretical sampling?a. 200 possible samplesb. 230,300 possible samplesc. 6,250,000 possible samples
In terms of characterizing samples, statistical theory was developed using a sampling strategy in whicha. the order of selecting participants matters.b. the same participant can be selected twice in the same sample.c. participants are replaced after being sampled.d. all of the above.
Which sampling method is the most commonly used in behavioral research?a. Sampling with replacementb. Sampling without replacement
The number of left-handed people in a hypothetical population of 50 students is equal to 5 (p = .10). What is the probability of selecting 2 left-handed people from this population when using sampling with replacement?a. p = .05b. p = .01c. p = .10
Which of the following correctly describes random sampling?a. A sampling procedure in which participants have an equal chance of being selected to participate in a study.b. A sampling procedure in which the probability of selecting each participant is the same.c. Both a and b are correct.
A researcher uses a sample ________ to make inferences about the value of a population ________ of interest. [Fill in the blanks]a. statistic; parameterb. parameter; statistic
A researcher wants to know the probability of selecting a nontraditional student from a school population where 8% of students are nontraditional. If the researcher selects a sample of 100 students, why is it inappropriate to use the normal approximation of the binomial distribution to estimate
In a local college, 10% of students are athletes. For this college, what is the probability that exactly 25 students are athletes in a random sample of 200 students selected from this school? [Hint: This is a binomial distribution, where p = .10 and q = .90.]a. Less than 5%.b. Greater than 95%.c.
Based on the z scores computed in SPSS, which raw score is farthest from the mean?Answer: __ [enter whole number]
Based on the z scores computed in SPSS, which of the following is true?a. A score of 8 falls closest to the mean.b. A score of 10 is farther from the mean than a score of 4.c. Th ere are more scores below the mean than above the mean.
Based on the data summarized in the output, which of the following statements is true?a. Th ere are more scores below the mean than above the mean.b. No other score occurs as often as the value equal to the mean.c. Th e score that is farthest from the mean is located below the mean.
Based on the data summarized in the output, which of the following scores falls at least 1 standard deviation from the mean? [Hint: Th e answer is directly in the output; no calculations are required]a. 8b. 2c. 2 and 8
Based on the data summarized in the output, what is the mean of this distribution? [Hint:Th e answer is directly in the output; no calculations are required]Answer: __ [enter whole number]
A normal distribution has a standard deviation equal to 10. What is the mean of this normal distribution if the probability of scoring below x = 10 is .5000?Answer: ____ [enter whole number]
A professor records the following final exam scores: 78 ± 7 (M ± SD). Students scoring in the top 14% get an A. What is the cutoff score for the top 14% in this example?a. 78b. 70.44c. 85.56
A researcher records the following times (in seconds) that it took a group of mice to complete a maze: 23 ± 4.1 (M ± SD). The top 10% of mice finishing with the fastest times will be placed in a “maze bright” group. What is the cutoff time to be placed in this group?a. 17.75 secondsb. 28.25
If a score in the bottom 10% qualifies for placement in a remedial class, how many incoming students will be placed in the remedial class?Answer: ____ [enter whole number]
If a score in the top 10% qualifies for placement in an advanced class, how many incoming students will be placed in the advanced class?Answer: ____ [enter whole number]
A therapist aims to reduce the mean number of symptoms expressed by their patients to two symptoms per day. Following a series of therapy sessions, they determine that the z score for two symptoms per day is 0. Assuming the data are normally distributed, was their goal reached?a. No, because the z
A researcher finds that the average preschool-aged child drinks two cups of milk per day.Assuming these data are normally distributed, the z score for the average cups per day in this example is____ Answer: ____ [enter whole number]
The standard normal transformationa. converts any normal distribution with any mean and any variance to a standard normal distribution with a mean of 1 and variance of 0.b. converts any normal distribution with any mean and any variance to a standard normal distribution with a mean of 0 and
The unit of measurement distributed along the x-axis of a standard normal distribution is referred to as a _ score. [Fill in the blank]Answer: __ [enter the missing letter]
The standard normal distribution is normally distributed with a mean of ____ and a variance of ____. [Fill in the blanks]a. 0; 1b. 1; 0c. any value; any positive valued. any positive value; any value
One characteristic of a normal distribution is that the mean, median, and mode are the same value.
One characteristic of a normal distribution is that the total area under the curve is equal to 1.0.
The mean can take on ____ and the standard deviation can take on ____. [Fill in the blanks]a. any value; any valueb. any positive value; any valuec. any value; any positive value
What is the implication for the tails of a normal distribution being asymptotic?a. That scores above the mean are distributed the same as scores below the mean.b. That extreme scores are possible in the tails of a normal distribution.c. That there are an infinite number of possible normal
The normal distribution is symmetrical, which means thata. as you travel away from the mean the tails of the distribution are always approaching the x-axis but never touch it.b. the distribution of data above the mean is the same as the distribution of data below the mean.c. the mean of a normal
The following distribution most closely approximates a ____. [Fill in the blank]a. normal distributionb. positively skewed distributionc. negatively skewed distribution
A binomial distribution is distributed with a mean equal to 8 and a standard deviation equal to 2. If the real limits of a binomial outcome are 9.5 and 10.5, then which of the following is the normal approximation for the probability, p, of obtaining this binomial outcome?a. p = .121b. p = .101c. p
What are the real limits for an outcome of 10 in a binomial variable?a. 9.5 and 10.5b. 9 and 11c. +10 and −10
To use the normal approximation for the binomial distribution, np and nq must be equal toa. any positive valueb. at least 10c. at least 20
The larger the sample size, the more closely a binomial distribution approximates a normal distribution.a. Trueb. False
A probability distribution for binomial outcomes approximates the shape of what type of distribution?a. bimodal distributionb. normal distributionc. skewed distribution
The image below shows nine raw scores (x) and their corresponding standard z scores.[The image gives enough information for you to reproduce it in SPSS. Please feel free to enter these data in SPSS and follow the directions given in Section 6.6 to reproduce the z scores.] Based on the data
When you convert raw scores to standard z scores in SPSS, the z scores appear in an SPSS output table.a. True, the z scores appear in an SPSS output table.b. False, the z scores appear in the Data View tab of SPSS.
Using SPSS, which options in the menu bar do you select to convert raw scores to standard z scores?a. Analyze, then Descriptive Statistics and Descriptives.b. Analyze, then Descriptive Statistics and Crosstabs.c. Analyze, then Descriptive Statistics and Frequencies.
Identify the z score that most closely approximates the bottom 2.5% of scores in a distributiona. z = −1.96b. z = −1.645c. z = −0.675
Identify the z score associated with a probability of p = .4013 toward the tail in a normal distribution.a. z = 0.25b. z = 0.40c. z = 1.29
Which of the following is a step for locating the cutoff score for a given proportion of data?a. Locate the z score associated with a given proportion in the unit normal table.b. Transform the z score into a raw score (x).c. Both a and b are correct.
Identify the probability of a score between a z score of 0.60 and 1.20.a. p = .3849b. p = .6106c. p = .1592
Identify the probability of a score at or below a z score of −1.00.a. p = .0918b. p = .1587c. p = .3413
Identify the probability of a score at or above a z score of 0.50.a. p = .3085b. p = .1915c. p = .1230
Which of the following is a step for locating the proportion of scores in any normal distribution?a. Transform a raw score (x) into a z score.b. Locate the corresponding probability for that z score in the unit normal table.c. Both a and b are correct.
Which of the following identifies the correct probability toward the tail for a z score of−1.25? [Hint: use the unit normal table]a. .1056b. .1788c. .3944
The normal distribution is used to determine the probability of a certain outcome _____ all other outcomes. [Fill in the blank]a. timesb. divided byc. in relation to
In which column in the unit normal table are the z scores located?a. Column Ab. Column Bc. Column C
A data set is normally distributed with a mean of 10 and a standard deviation of 3. Which of the following is the z transformation for x = 16 in this normal distribution?a. z = (16-3)/10 = +1.3b. z = (10-16)/3 = −2.0c. z = (16-10)/3 = +2.0
A set of data is normally distributed with a given mean and standard deviation. If you compute a z transformation for the mean, what will the z score be equal to?a. 0b. 1c. Not enough information
A z score is a score that is on the x-axis of what type of distribution?a. Any normal distributionb. A standard normal distributionc. A bimodal distribution
What term is often used to describe behavior that falls within 2 SD of the mean in a normal distribution?a. statistical normb. outliersc. symmetry
The area under a normal curve ranges between 0 and 1 and can never be negative. What type of statistic also has these same characteristics?a. varianceb. probabilityc. meand. frequency
A normal distribution has a mean equal to 5. Which of the following statements is true?a. The mode will be larger than the mean.b. The median will be larger than the mode.c. Both the median and mode will be equal to the mean.d. The mean will be larger than the median and the mode.
Which of the following is NOT a characteristic of the normal distribution?a. The mean can be any positive or negative number.b. The variance can be any positive number.c. The shape of the normal distribution is symmetrical.d. The tails of a normal distribution touch the x-axis at 3 standard
Project: Project: This project can be done in groups or as individuals. Each group or person must find a set of data, preferably but not restricted to their field of study. The data needs to fit the regression framework with a regression 2variable, x, and a response variable, y. Carefully make the
Show the necessary steps in converting the equation to the equivalent form .
Suppose that an experimenter postulates a model of the type, Y = β + β x + ϵ , i = 1, 2, …, n, when, in fact, an additional variable (say, x independent of x ) also contributes linearly to the response. The true model is then given by: Y = β + β x +β x + ϵ , i = 1, 2, …, n. Establish
In Review Exercise 11.62, the student was required to show that for a standard simple linear regression model. Does the same hold for a model with zero intercept?Show why or why not.
Consider the situation presented in Exercise 11.62 but suppose that n = 2 (that is, only two data points are available).Prove that the coefficient of correlation is +1 or –1 as the regression line increases or decreases.
Show, in the case of a least squares fit to the simple linear regression model that
For a simple linear regression model where the ϵ are independent and normally distributed with zero means and equal variances σ , show that and 12 2i 20 0 Y|x 0 1 01 2have zero covariance.
Assuming that the ϵ are independent and normally distributed with zero means and common variance σ , show that B , the least squares estimator of β in μ = β + β x, is normally distributed with mean β and variance

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