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theory of statistics
Statistics For The Behavioral Sciences 5th Edition Susan A Nolan, Thomas Heinzen - Solutions
If the cutoffs for a z test are −1.96 and 1.96, determine whether you would reject or fail to reject the null hypothesis in each of the following cases:a. z = 1.06b. z =−2.06c. A z score beyond which 7% of the data fall in each tail
You are conducting a z test on a sample of 132 people for whom you observed a mean verbal score on the SAT, a university admissions test used in the United States and several other countries, of 490. The population mean is 500, and the standard deviation is 100. Calculate the mean and the spread of
You are conducting a z test on a sample of 50 people with an average verbal score on the SAT, a university admissions test used in the United States and several other countries, of 542(assume we know the population mean to be 500 and the standard deviation to be 100). Calculate the mean and the
State the percentage of scores in a one-tailed critical region for each of the following alpha levels:a. 0.05b. 0.10c. 0.01
For each of the following alpha levels, what percentage of the data will be in each critical region for a two-tailed test?a. 0.05b. 0.10c. 0.01
If the critical values for a hypothesis test occur where 2.5%of the distribution is in each tail, what are the cutoffs for z?
Rewrite each of the following probabilities, or alpha levels, as percentages:a. 0.19b. 0.04c. 0.92
Rewrite each of the following percentages as probabilities, or alpha levels:a. 5%b. 83%c. 51%
Using the z table in Appendix B, calculate the following percentages for a z score of 1.71:a. Above this z scoreb. Below this z scorec. At least as extreme as this z score
Using the z table in Appendix B, calculate the following percentages for a z score of −0.08:a. Above this z scoreb. Below this z scorec. At least as extreme as this z score
Calculate the following percentages for a z score of 0.74, with a tail of 22.96%:a. What percentage of scores falls below this z score?b. What percentage of scores falls between the mean and this z score?c. What proportion of scores falls below a z score of −0.74?
Calculate the following percentages for a z score of −1.61, with a tail of 5.37%:a. What percentage of scores falls above this z score?b. What percentage of scores falls between the mean and this z score?c. What proportion of scores falls above a z score of 1.61?
What is p-hacking and what are some examples of research behaviors that would constitute p-hacking?
What is HARKing and why can it be harmful?
Write the symbols for the null hypothesis and research hypothesis for a one-tailed test.
Why do researchers typically use a two-tailed test rather than a one-tailed test?
What is the difference between a one-tailed hypothesis test and a two-tailed hypothesis test in terms of critical regions?
Using everyday language rather than statistical language, explain why the word cuto might have been chosen to dene the point beyond which we reject the null hypothesis.
Using everyday language rather than statistical language, explain why the words critical region might have been chosen to dene the area in which a z statistic must fall for a researcher to reject the null hypothesis.
What do these symbolic expressions mean: H0: μ1 = μ2 and H1: μ1 ≠ μ2?
What does statistically signicant mean to statisticians?
What is the standard size of the critical region used by most statisticians?
What are critical values and the critical region?
What are the six steps of hypothesis testing?
What is the difference between parametric tests and nonparametric tests?
What sample size is recommended to meet the assumption of a normal distribution of means, even when the underlying population of scores is not normal?
In statistics, what do we mean by assumptions?
How is calculating a percentile for a mean from a distribution of means different from doing so for a score from a distribution of scores?
How do we calculate the percentage of scores below a particular positive z score?
When we look up a z score on the z table, what information can we report?
What is a percentile?
Cheating in online gaming: Researchers used the normal curve to investigate cheating among online gamers playing a car racing game (Christensen et al., 2013). The graph below shows the winning times for one version of the game.a. Referring to Chapter 3, what kind of graph is this and why?(Hint:
Which was better, the book or the movie: FiveThirtyEight is a popular blog that uses statistics in creative ways to better understand politics, sports, science and health, economics, and culture. In one article (Hickey, 2015), the author uses z scores to standardize book reviews from goodreads.com
Cheating on standardized tests: In their book Freakonomics, Levitt and Dubner (2009) describe alleged cheating among teachers in the Chicago public school system. Certain classrooms had suspiciously strong performances on standardized tests that oen mysteriously declined the following year when a
Rural friendships and the General Social Survey: Earlier, we considered data from the GSS on numbers of close friends people reported having. The mean for this variable is 7.44, with a standard deviation of 10.98. Let’s say that you decide to use the GSS data to test whether people who live in
Probability and medical treatments: The three most common treatments for blocked coronary arteries are medication, bypass surgery, and angioplasty, which is a medical procedure that involves clearing out arteries and that leads to higher prots for doctors than do the other two treatments. The
The z distribution and a “super recognizer”: According to a news article, “Friends call Constable [Gary] Collins Rain Man or Yoda or simply The Oracle. But to Scotland Yard, London’s metropolitan police force, he is known as a ‘super recognizer’” (Bennhold, 2015).Prosopagnosia, also
The z distribution and a rogue cardiologist: A cardiologist in Munster, Indiana, was accused of performing unnecessary heart surgeries (Cresswell, 2015). Investigators found that the rates for one heart procedure were in the top 10% in the country for the city where this doctor worked. Lawyers
Percentiles, raw scores, and credit card the: Credit card companies will oen call cardholders if the pattern of use indicates that the card might have been stolen. Let’s say that you charge an average of $280 a month on your credit card, with a standard deviation of $75. The credit card company
A distribution of means and the General Social Survey: Refer to Exercise 6.49. Again, pretend that the GSS sample is the entire population of interest.a. Imagine that you randomly selected 80 people from this population, and that they had a mean of 8.7. Would you compare this sample mean to a
A distribution of scores and the General Social Survey: Refer to the previous exercise. Again, pretend that the GSS sample is the entire population of interest.a. Imagine that you randomly selected one person from this population who reported that he had 18 close friends. Would you compare his
Distributions and the General Social Survey: The General Social Survey (GSS) is a survey of approximately 2000 U.S. adults conducted each year since 1972, for a total of more than 38,000 participants.During several years of the GSS, participants were asked how many close friends they have. The mean
Distributions, personality testing, and social introversion: See the description of the MMPI-2 in the previous exercise. The mean T score is always 50, and the standard deviation is always 10. Imagine that you administer the MMPI-2 to 50 respondents who do not use Instagram or any other social
Distributions, personality testing, and depression: The revised version of the Minnesota Multiphasic Personality Inventory (MMPI-2)is the most frequently administered self-report personality measure.Test-takers respond to more than 500 true/false statements, and their responses are scored,
Distributions and life expectancy: Researchers have reported that the projected life expectancy for South African men diagnosed with human immunodeciency virus (HIV) at age 20 who receive antiretroviral therapy (ART) is 27.6 years (Johnson et al., 2013).Imagine that the researchers determined this
Raw scores, z scores, percentiles, and sports teams: Let’s look at baseball and football again. We’ll look at data for all of the teams in Major League Baseball (MLB) and the National Football League (NFL), respectively.a. In 2012, the mean number of wins for MLB teams was 81.00, with a
The z distribution and comparing scores on two tests of English language learning: The Test of English as a Foreign Language(TOEFL), with scores ranging from 0 to 120, has traditionally been the most commonly used exam of reading comprehension, vocabulary, writing, and grammar for English-language
z scores and comparisons of sports teams: A common quandary faces sports fans who live in the same city but avidly follow different sports. How does one determine whose team did better with respect to its league division? In 2012, the Atlanta Braves baseball team and the Atlanta Falcons football
Percentiles and eating habits: As noted in How It Works 6.1, Georgiou and colleagues (1997) reported that college students had healthier eating habits, on average, than did those individuals who were neither college students nor college graduates. The 412 students in the study ate breakfast a mean
The normal curve in the media: Statistics geeks rejoiced when the New York Times published an article on the normal curve (Dunn, 2013)! Biologist Casey Dunn wrote that “Many real-world observations can be approximated by, and tested against, the same expected pattern: the normal distribution.”
The normal curve and real-life variables, part II: For each of the following variables, state whether the distribution of scores would likely approximate a normal curve. Explain your answer.a. Number of minutes that students check social media sites each weekb. Volume of water that people drink
The normal curve and real-life variables, part I: For each of the following variables, state whether the distribution of scores would likely approximate a normal curve. Explain your answer.a. Number of movies that a college student watches in a yearb. Number of full-page advertisements in a
Converting z scores to raw CFC scores: A study using the Consideration of Future Consequences scale found a mean CFC score of 3.20, with a standard deviation of 0.70, for the 800 students in the sample (Adams, 2012).a. Imagine that your z score on the CFC score is −1.2. What is your raw score?
z statistics and CFC scores: We have already discussed summary parameters for CFC scores for the population of participants in a study by Adams (2012). The mean CFC score was 3.20, with a standard deviation of 0.70. (Remember that we treated the sample of 800 participants as the entire population.)
The z distribution applied to admiration ratings: A sample of 148 of our statistics students rated their level of admiration for Hillary Clinton on a scale of 1 to 7. The mean rating was 4.06, and the standard deviation was 1.70. (For this exercise, treat this sample as the entire population of
The z distribution and hours slept: A sample of 150 statistics students reported the typical number of hours that they sleep on a weeknight. The mean number of hours was 6.65, and the standard deviation was 1.24. (For this exercise, treat this sample as the entire population of interest.)a. What is
z scores and the GRE: By design, the verbal subtest of the GRE has a population mean of 500 and a population standard deviation of 100 (the quantitative subtest has the same mean and standard deviation).a. Use symbolic notation to state the mean and the standard deviation of the GRE verbal test.b.
minutes. Here are the data for 40 of these students:
Distributions and getting ready for a date: We asked 150 students in our statistics classes how long, in minutes, they typically spend getting ready for a date. The scores ranged from 1 minute to 120 minutes, and the mean was
Normal distributions in real life: Many variables are normally distributed, but not all are. (Fortunately, the central limit theorem saves us when we conduct research on samples from non-normal populations if the samples are larger than 30!) Which of the following are likely to be normally
A sample of 100 people had a mean depression score of 85; the population mean for this depression measure is 80, with a standard deviation of 20. A different sample of 100 people had a mean score of 17 on a different depression measure; the population mean for this measure is 15, with a standard
Compute a z statistic for each of the following, assuming the population has a mean of 100 and a standard deviation of 20:a. A sample of 43 scores has a mean of 101.b. A sample of 60 scores has a mean of 96.c. A sample of 29 scores has a mean of 100.
A population has a mean of 55 and a standard deviation of 8.Compute μM and σM for each of the following sample sizes:a. 30b. 300c. 3000
Compute the standard error (σM) for each of the following sample sizes, assuming a population mean of 100 and a standard deviation of 20:a. 45b. 100c. 4500
Assume a normal distribution when answering the following questions.a. What percentage of scores falls below the mean?b. What percentage of scores falls between 1 standard deviation below the mean and 2 standard deviations above the mean?c. What percentage of scores lies beyond 2 standard
Compare the following scores:a. A score of 811 when μ = 800 and σ = 29 against a score of 4524 when μ = 3127 and σ = 951b. A score of 17 when μ = 30 and σ = 12 against a score of 67 when μ = 88 and σ = 16
Using the instructions in Example 6.9, compare the following“apples and oranges”: a score of 45 when the population mean is 51 and the standard deviation is 4, and a score of 732 when the population mean is 765 and the standard deviation is 23.a. Convert these scores to standardized scores.b.
A study of the Consideration of Future Consequences (CFC) scale found a mean score of 3.20, with a standard deviation of 0.70, for the 800 students in the sample (Adams, 2012). (Treat this sample as the entire population of interest.)a. If the CFC score is 4.2, what is the z score? Use symbolic
By design, the verbal subtest of the Graduate Record Examination(GRE) has a population mean of 500 and a population standard deviation of 100. Convert the following z scores to raw scores using symbolic notation and the formula.a. 1.5b. −0.5c. −2.0
By design, the verbal subtest of the Graduate Record Examination(GRE) has a population mean of 500 and a population standard deviation of 100. Convert the following z scores to raw scores without using a formula.a. 1.5b. −0.5c. −2.0
For a population with a mean of 1179 and a standard deviation of 164, convert each of the following z scores to raw scores.a. −0.23b. 1.41c. 2.06d. 0.03
For a population with a mean of 250 and a standard deviation of 47, convert each of the following z scores to raw scores.a. 0.54b. −2.66c. −1.00d. 1.79
For a population with a mean of 250 and a standard deviation of 47, calculate the z scores for 203 and 297. Explain the meaning of these values.
For a population with a mean of 250 and a standard deviation of 47, calculate the z score for 250. Explain the meaning of the value you obtain.
A population has a mean of 1179 and a standard deviation of 164.Calculate z scores for each of the following raw scores:a. 1000b. 721c. 1531d. 1184
A population has a mean of 250 and a standard deviation of 47.Calculate z scores for each of the following raw scores:a. 391b. 273c. 199d. 160
Create a histogram for these three sets of scores. Each set of scores represents a sample taken from the same population.
Each of the following equations has an error. Identify, x, and explain the error in each of the following equations.a. σM =μ√Nb. z =(μ − μM)σM(for a distribution of means)c. z =(M − μM)σ(for a distribution of means)d. z =(X − μ)σM(for a distribution of scores)
What does a z statistic—a z score based on a distribution of means—tell us about a sample mean?
Why does the standard error become smaller simply by increasing the sample size?
What is the difference between standard deviation and standard error?
What does the symbol σM stand for?
What does the symbol μM stand for?
Why is the central limit theorem such an important idea for dealing with a population that is not normally distributed?
What are the mean and the standard deviation of the z distribution?
Give three reasons why z scores are useful.
What is a z score?
Explain how the word standardize is used in everyday conversation, then explain how statisticians use it.
How does the size of a sample of scores affect the shape of the distribution of data?
What point on the normal curve represents the most commonly occurring observation?
Explain how the word normal is used in everyday conversation, then explain how statisticians use it.
Preregistration, crowdsourcing, and fake news: Ethical researchers are increasingly using the Internet to modernize their research and conduct it in a more ethical way. In one study, not yet peer-reviewed, Yale researchers found that
Treatment for depression: Researchers conducted a study of 18 patients whose depression had not responded to treatment(Zarate, 2006). Half received one intravenous dose of ketamine, a hypothesized quick x for depression; half received one intravenous dose of placebo.
Alcohol abuse interventions: Sixty-four male students were ordered, aer they had violated university alcohol rules, to meet with a school counselor. Borsari and Carey (2005)randomly assigned these students to one of two conditions.Those in the rst condition were assigned to undergo a newly
Horoscopes and predictions: People remember when their horoscopes had an uncanny prediction—say, the prediction of a problem in love on the exact day of the breakup of a romantic relationship—and decide that horoscopes are accurate. Munro and Munro (2000) are among those who have challenged
Testimonials and Harry Potter: Amazon and other online bookstores offer readers the opportunity to write their own book reviews, and many potential readers scour these reviews to decide which books to buy. Harry Potter books attract a great deal of these reader reviews. One Amazon reviewer,“bel
Probability and sumo wrestling: In their book Freakonomics, Levitt and Dubner (2005) describe a study conducted by Duggan and Levitt (2002) that broached the question: Do sumo wrestlers cheat? Sumo wrestlers garner enormous respect in Japan, where sumo wrestling is considered the national sport.
Rejecting versus failing to reject an invitation: Imagine you have found a new study partner in your statistics class. One day, your study partner asks you to go on a date. This invitation takes you completely by surprise, and you have no idea what to say. You are not attracted to the person in a
Type I versus Type II errors: Examine the statements from the previous exercise repeated here. For each, if this conclusion were incorrect, what type of error would the researcher have made? Explain your answer.
Decision about null hypotheses: For each of the following ctional conclusions, state whether the researcher seems to have rejected or failed to reject the null hypothesis (contingent, of course, on inferential statistics having backed up the statement). Explain the rationale for your decision.
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