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fundamentals of statistics
Questions and Answers of
Fundamentals Of Statistics
What are Cohen’s (1988) guidelines for interpreting different effect sizes?
Why are measures of effect size comparable across different research studies?
A researcher conducts an analysis in which the null hypothesis is not rejected, yet a large measure of effect size is calculated. What advice might you give the researcher, and why?
What is the difference between the r2 statistic and Cohen’s d measure of effect size?
For each of the situations below, calculate the r2 and Cohen’s d measures of effect size and specify whether each is a “large,” “medium,” or “small” effect as outlined in the chapter.a.
For each of the following hypotheses, create a chart that shows the four possible outcomes that could occur as a result of hypothesis testing. There should be two correct decisions and two incorrect
If we reject the null hypothesis and we were wrong to do so, what type of error have we made? Give an example of a research situation in which this could occur.
If we fail to reject the null hypothesis but should have rejected it, what type of error have we made? Provide an example of a research situation in which this could occur.
For each of the situations below, describe what would be considered a Type I and Type II error.a. A consumer group wants to see if people can tell whether they are drinking tap water or bottled
For each of the situations below, describe what would be considered a Type I and Type II error.a. A political scientist compares older and younger adults' attitudes toward the death penalty.b. A
Below are two hypothetical situations:Situation 1: \(N_{i}=30, \alpha=.05\) (two-tailed test)Situation 2: \(N_{i}=30, \alpha=.01\) (two-tailed test)a. Find the critical values for each of the two
Below are two hypothetical situations:Study A: \(N_{1}=20, \bar{X}_{1}=14.50, s_{1}=2.50\)\(N_{2}=20, \bar{X}_{2}=12.75, s_{2}=3.25\)Study B: \(N_{1}=65, \bar{X}_{1}=14.50, s_{1}=2.50\)\(N_{2}=65,
Imagine two studies examining the same research question find the same means and standard deviations for two groups: \(\bar{X}_{1}=10.00, s_{1}=3.00\), and \(\bar{X}_{2}=12.00, s_{2}=2.00\). However,
Two researchers separately testing the same research hypothesis end up with the same sample size ( \(N_{i}=12\) ) and standard deviations for the two groups ( \(s_{1}=.50\) and \(s_{2}=.75\) ).
A study compares the effectiveness of a new (Drug A) and an old medicine (Drug B) and finds no difference in effectiveness: \(t(68)=1.34, p>.05\). The researchers run the study a second time: Both
Two research studies compare two groups on the same dependent variable. Both studies are based on the same sample size \(\left(N_{i}=17\right)\), and somehow both studies end up with the same means
A researcher wishes to test a drug that claims to improve memory. The researcher sets up an experiment where a control group receives a placebo (i.e., a sugar pill) and an experimental group receives
The researcher in Exercise 12 notices the large amount of variability within each of the two groups. Investigating, he finds the research assistants counting the number of correctly remembered
For each of the situations below, calculate both the \(r^{2}\) and Cohen's \(d\) measure of effect size and specify whether each is a "small," "medium," or "large" effect as outlined in the
For each of the situations below, calculate both the \(r^{2}\) and Cohen's \(d\) measure of effect size and specify whether each is a "small," "medium," or "large" effect as outlined in the
A researcher is investigating the birth-order theory that middle siblings are greater risk takers than are first-born siblings. The following statistics are obtained for the study as a measure of
Below are sets of data for two studies: The data for the second study double the data from the first study.Original data: Group A: \(20,14,18,12,24,14,22\)Group B: \(15,18,16,13,10,17,8\)Doubled
Conceptually, what are between-group and within-group variability? What do they consist of or represent?
In testing the difference between group means, how are between-group and within-group variability related to each other? What has to exist for the null hypothesis to be rejected?
In testing the difference between group means, what is the implication of a large amount of within-group variability (error) on the amount of between-group variability (effect) needed to reject the
What does ANOVA stand for, and why?
In testing the difference between means, why is the inferential statistic called the F-ratio?
What is the shape of the theoretical distribution of F-ratios? Why is it this shape?
What is the modality of the theoretical distribution of F-ratios?
If you were conducting a one-way ANOVA involving four groups (Group 1 to Group 4), what are the two ways you could state the null hypothesis (H0)?
Which of these is the correct way to state the alternative hypothesis: H1: all μs are not equal or H1: not allμs are equal? Why is one way correct and the other way incorrect?
Why does hypothesis testing in the one-way ANOVA involve two degrees of freedom?
Why is dfBG = # groups = 1 and dfBG = Σ(Ni − 1)?
How is stating the critical value and decision rule for the F-ratio different from the t-test?
In measuring between-group and within-group variance, what does the symbol MS represent?
For each of the following, calculate the degrees of freedom (df) and identify the critical value of F (assumeα = .05).a. # groups = 4, Ni = 11b. # groups = 5, Ni = 6c. # groups = 6, Ni = 8
For each of the following, calculate between-group variance (MSBG). a. N; 9,X 4.00, X2 = 9.00, X3 = 2.00 = b. N=10, X = 27.00, X2 = 39.00, X3 = 33.00, X4 = 37.00 c. N=7,X =3.75, X2 = 4.25, X3 = 3.85,
For each of the following, calculate within-group variance (MSWG). a. N; 18, 51 2.00, 52=4.00, 53-5.00 = b. N; = 5, 5 = 12.00, s2 = 9.00, 53 = 11.00, s = 8.00 c. N; = 11, 5 = .27, 52 = .51, 53= .46,
What does R2 in a one-way ANOVA represent?
What is the most specific conclusion that can be drawn when the null hypothesis for a one-way ANOVA has been rejected?
Does rejecting the null hypothesis for a one-way ANOVA support or not support a research hypothesis?
Although research has examined factors women take into consideration when making birth control decisions, relatively few studies have looked at men. As male oral contraceptives were starting to
What is the purpose of conducting analytical comparisons?
What are two types of analytical comparisons? In what research situations would you use one versus the other?
What is between-group and within-group variance in an analytical comparison?
What are concerns researchers have about unplanned comparisons?
What is familywise error? How can it be controlled?
Earlier in this chapter, a study was described examining males’ concerns regarding oral contraceptives(Jaccard et al., 1981). The study hypothesized that males would rate risks to their health as
Imagine that a researcher conducts a study consisting of five groups, each of which consists of seven participants (Ni = 7). The researcher conducts a one-way ANOVA and rejects the null hypothesis.
Researchers Latané and Rodin (1969) were interested in studying helping behavior. In one of their experiments, they set up a situation in which the participant is brought into the waiting room
For each of the following, calculate the degrees of freedom ( \(d f\) ) and identify the critical value of \(F\) (assume \(\alpha=.05\) ).a. \(\#\) groups \(=3, N_{i}=10\)b. \(\#\) groups \(=5,
For each of the following, calculate the degrees of freedom ( \(d f\) ) and identify the critical value of \(F\) (assume \(\alpha=.05\) ).a. \(\#\) groups \(=3, N_{i}=8\)b. \(\#\) groups \(=6,
For each of the following, calculate between-group variance \(\left(M S_{\mathrm{BG}}\right)\).a. \(\quad N_{i}=13, \bar{X}_{1}=3.00, \bar{X}_{2}=5.00, \bar{X}_{3}=10.00\)b. \(\quad N_{i}=11,
For each of the following, calculate between-group variance \(\left(M S_{\mathrm{BG}}\right)\).a. \(N_{i}=20, \bar{X}_{1}=5.50, \bar{X}_{2}=3.75, \bar{X}_{3}=4.25\)b. \(N_{i}=8, \bar{X}_{1}=14.85,
For each of the following, calculate within-group variance \(\left(M S_{\mathrm{WG}}\right)\).a. \(N_{i}=7, s_{1}=1.00, s_{2}=2.00, s_{3}=6.00\)b. \(N_{i}=6, s_{1}=13.00, s_{2}=16.00, s_{3}=12.00,
For each of the following, calculate within-group variance \(\left(M S_{\mathrm{WG}}\right)\).a. \(N_{i}=15, s_{1}=.75, s_{2}=.48, s_{3}=.63\)b. \(N_{i}=19, s_{1}=4.52, s_{2}=4.86, s_{3}=4.28,
For each of the following, calculate the \(F\)-ratio \((F)\) and create an ANOVA summary table.a. \(N_{i}=20, \bar{X}_{1}=20.00, s_{1}=8.00, \bar{X}_{2}=25.00, s_{2}=10.00, \bar{X}_{3}=15.00,
For each of the following, calculate the \(F\)-ratio \((F)\) and create an ANOVA summary table.a. \(\quad N_{i}=10, \bar{X}_{1}=14.00, s_{1}=2.50, \bar{X}_{2}=12.50, s_{2}=2.10, \bar{X}_{3}=17.00,
Using the means and standard deviations calculated in Exercise 1 for the helping behavior study, determine if the number of seconds until the participant helps the person in distress is the same for
To evaluate the effectiveness of four different teaching methods, algebra students are randomly assigned to one of four methods (A, B, C, or D) and then given a test. The descriptive statistics for
Although men and women have been found to perform differently on tests of mental ability, less is known about possible reasons for these differences. A researcher hypothesizes that one's beliefs play
Past research has examined relationships between people's backgrounds and their personality. For example, Eysenck (1982) found a relationship between blood type and introversion, and Pellegrini
Employers seek ways to improve the performance of their employees. Gardner, Van Dyne, and Pierce (2004) hypothesized that performance is influenced by organizational self-esteem, defined as an
A researcher studying the effects of different learning techniques upon memory hypothesizes people learn better when training is distributed over time rather than massed all at once. She conducts an
What is the relationship between the \(t\)-test and an ANOVA? In Chapter 9, we looked at two different types of instruction in reading comprehension (imagery and repetition). A \(t\)-test for
An important issue in court cases is the accuracy of eyewitness testimony. Behavioral scientists have suggested eyewitnesses can be influenced by how a question is phrased. A researcher conducts a
Looking back at Exercise 17, the most specific conclusion that could be drawn from rejecting the null hypothesis was that the estimates of speed for the three types of wording were not all equal to
The hypothetical study in Exercise 15 examined the effects of different learning techniques upon memory. Given the hypothesis is that people learn better when training is distributed over time rather
The chess study discussed in this chapter (de Bruin et al., 2007) hypothesized that the number of checkmates should be greater for students asked to make predictions while learning to play chess than
An experimenter ran a study that consisted of four groups, each with 16 participants. This research study was exploratory research, and the experimenter did not have any specific hypotheses or
Imagine the \(F\)-ratio for a one-way ANOVA is rejected for a study involving six groups, each with nine participants. A researcher plans on comparing each group with every other group, controlling
For what types of situations might one calculate the confidence interval for the mean rather than the t-test for one mean?
What is the difference between a point estimate and the confidence interval for the mean?
Why is a 100% confidence interval of relatively little use?
“There is a .95 probability that the population mean is between 3 and 6.” Why is this conclusion about a confidence interval inappropriate?
For each of the following sets of numbers, calculate a 95% confidence interval for the mean (σ not known);before calculating the confidence interval, the sample mean (X) and standard deviation (s)
A team of researchers interested in reducing alcohol-related problems in college fraternity members asked 159 members to report the number of drinks consumed on a typical occasion (Larimer et al.,
How does the calculation of the confidence interval for the mean change depending on whether the population standard deviation σ is known?
For each of the following sets of numbers, calculate a 95% confidence interval for the mean (σ known); before calculating the confidence interval, the sample mean (X̅) and standard deviation (s)
Although the link between obesity and one’s physical condition is well established, a research team stated that less was known about the relationship between obesity and one’s mental state of
What happens to the width of a confidence interval for the mean as sample size increases?
What are the two ways sample size affects the width of the confidence interval for the mean?
Why might a researcher want to estimate the sample size needed for a confidence interval of a desired width?
Counselors at two colleges (College Blue and College Gold) want to estimate the average grade point average (GPA) of students attending their respective colleges. The College Blue counselor collects
Earlier in this chapter, a 95% confidence interval for the mean (σ known) was calculated for the SAT example(N = 12). Assuming the sample mean (X = 550.00) and population standard deviation (σ =
In the SAT example, for N = 12, the 95% confidence interval had a width of about 115 points (the difference between the lower limit of 493.36 and the higher limit of 606.64). If we again use 100 as
What happens to the size of a confidence interval if the level of confidence is changed from 95% to 99%?From 95% to 90%?
What are relative strengths and weaknesses of confidence intervals of different levels of confidence?
Using our earlier example of SAT (N = 12, X̅ = 550.00, σ = 100)…a. Calculate the 90% and 99% confidence interval for the mean (σ known).b. Which confidence interval is wider, and why?
As part of an examination of baseballs used in Major League Baseball games (Rist, 2001), a research team weighs a sample of 80 baseballs manufactured in 2000. They find the mean weight to be 5.11
For each of the following situations, calculate a \(95 \%\) confidence interval for the mean ( \(\sigma\) not known), beginning with the step, "Calculate the degrees of freedom ( \(d f\) ) and
For each of the following situations, calculate a \(95 \%\) confidence interval for the mean ( \(\sigma\) not known), beginning with the step, "Calculate the degrees of freedom ( \(d f\) ) and
The exercises in Chapter 7 included a study designed to measure the number of chocolate chips in a bag of Chips Ahoy cookies (Warner \& Rutledge, 1999). From this example, for the sample of 18 bags
For each of the following situations, calculate a \(95 \%\) confidence interval for the mean ( \(\sigma\) known), beginning with the step, "Identify the critical value of \(z\). ."a. \(\bar{X}=7.00,
For each of the following situations, calculate a \(95 \%\) confidence interval for the mean ( \(\sigma\) known), beginning with the step, "Identify the critical value of \(z . "\)a. \(\bar{X}=50.00,
An exercise in Chapter 7 referred to a program designed to improve scores on the Graduate Record Examination (GRE). The class had 25 students and they scored a mean of 1075.00 on the GRE. Assuming
William and Meagan are working on their senior projects, both of which involve estimating the average alcohol consumption of students on their college campuses. Imagine they achieve the same mean and
In the Chips Ahoy cookie example (Exercise 6), a \(95 \%\) confidence interval of \((1204.30,1317.70)\) was constructed for \(N=18\) bags of cookies. Assuming the mean and standard deviation remained
In the ambulance response time example (Exercise 8), a \(95 \%\) confidence interval was constructed for \(N=67\) ambulance runs. Assuming the mean and standard deviation remained the same
In the Chips Ahoy cookie example (Exercise 6), a \(95 \%\) confidence interval with a width of approximately 100 (the difference between the lower limit of \(1,204.30\) and the higher limit of
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