Stats Final Exam
The p-value represents the proportion under the curve that corresponds to the obtained statistics, such as z-scores and t-scores. True or False
True
A researcher fails to reject a false null hypothesis. This is known as: power 1-alpha Type I error Type II error
Type II error
In ANOVA, how do you calculate the degrees of freedom?
df = N (total of all scores in each group) - 1 or DFtotal = DFmodel + DFerror
A researcher sets alpha=.05 and conducts a study to compare two groups. If there truly is no difference in the population means, what is the probability of a Type I error? .00 .01 .05 .20
.05
The null hypothesis states that there is no difference/no effect. True or False
True
The Central Limit Theorem tells us three very important characteristics of the sampling distribution of the mean. Which of the following statements are NOT one of these characteristics? a. The typical value of the sample mean b. The amount of dispersion of the sample means c. The shape of the sampling distribution of the mean d. The confidence interval that contains the population mean
d. The confidence interval that contains the population mean
After conducting an independent-means t-test, a researcher computed an effect size (d) of .50. Jacob Cohen would interpret this effect size to mean that the: difference between means is large enough to notice. sample size was not large enough to reject Ho. means are not significantly different from each other. means are more than five standard deviations apart.
difference between means is large enough to notice.
The p value obtained is 0.0785 and the alpha level is 0.05, do we reject or faily to reject the null?
fail to reject the null
dfbetween is calculated by: n-1 k-1 k-2
k-1
You have just completed a study and found a 95% confidence interval with a width of 8 points. A colleague suggests that they would like to see a 99% confidence interval. The width of the 99% CI would be: 0 less than 8 points 8 points more than 8 points
more than 8 points (To be more confident you need to make the interval wider so that it is more likely to contain the population mean.)
Which of the following is a statistical assumption to violations of which the independent means t-test is USUALLY robust? normality of the dependent variable homogeneity of variance independence of observations equality of sample sizes
normality of the dependent variable
If our sample value is close to the null/population value, we conclude that: we can neither accept or reject the null. nothing happened in the study; there is no effect. something happened in the study; there is a significant effect. something happened in the study, but the effect is very small.
nothing happened in the study; there is no effect.
The probability of correctly rejecting a false null hypothesis is the definition of: Alpha Beta Inference Power
Power
Please check the t-distribution table and find the critical t value when df is 15 and the alpha is .05.
2.132
A researcher computes SSb=600, dfb=3, SSw=5000, dfw=100. Compute the obtained F-statistic. 0 4 50 200
4
Statistical significance = Hypothesis testing Effect size Statistical power
Hypothesis testing
Why do you use the t test instead of the z test?
If the population mean and population standard deviation is not known, we use t-tests to estimate the standard error of the mean
Dr. Tripper incorrectly rejects a true null hypothesis. This is an example of: Alpha Beta Type I error Type II error
Type I error
The independent means t-test, rather than the dependent (correlated) means t-test, is more appropriate when a pre-test and post-test is administered to the same individuals. the population variance is not known and must be estimated from the sample. sample size is small. observations are not related to one another.
observations are not related to one another.
What is the proportion (percentage or probability) of the curve that falls between the z scores of 0 and -1.96?
47.50%
The omnibus hypothesis in ANOVA would look like H0: m1 = m2 H0: m1 - m2 H0: m2 - m1 H0= m1 = m2 = m3 = m4
H0= m1 = m2 = m3 = m4
A colleague suggests running your statistical test with an alpha of .01, as opposed to an alpha of .05, to reduce the chance of a Type I error. What effect will this have on the chance of a Type II error? It will increase It will decrease It will remain the same
It will increase
What happens to the size of the confidence interval as the size of samples increases? The confidence interval band becomes wider. The confidence interval band remains the same. The confidence interval band becomes narrower.
The confidence interval band becomes narrower. (At the same level of confidence, the narrower band represents the more precise estimates.)
What happens to the size of the confidence interval as the level of confidence increases? The confidence interval band becomes wider. The confidence interval band remains the same. The confidence interval band becomes narrower.
The confidence interval band becomes wider.
A colleague has calculated a 90% confidence interval about the sample mean as: 105 The probability is 0.90 that the population mean is located between 105 and 125. The probability is 0.10 that the sample mean is larger than 125 or smaller than 105. The probability is 0.90 that the sample mean is located between 125 and 125. Without an estimate of variability, there is not enough information available to make an interpretation.
The probability is 0.90 that the population mean is located between 105 and 125.
The p value obtained is 0.0285 and the alpha level is 0.05, do we reject or fail to reject the null?
reject the null
What would a small effect be for two group means? .20 .50 .80
.20
Based on the normal curve below, what is the total proportion (or probability) under the curve that falls below the mean (0)?
.50
Use the F table in your text to find the critical value with dfb=5 and dfw=29 (alpha=.05).
2.54
Use the F table in your text to find the critical value with dfb=2 and dfw=100 (Alpha=.05).
3.09
What is the proportion (percentage or probability) of the curve that falls between the z scores of 0 and 1.96?
47.50%
Please check the t-distribution table and find the critical t value when df is 3 and the alpha is .01.
5.841
What is the proportion (percentage or probability) of the area that falls below the z score of -1.5 like the graph shown below?
6.68%
A researcher finds a mean of 80, and a standard deviation of 6, based on a sample of 100 observations. What is the 95% confidence interval? 68.1, 91.9 75.2, 84.8 78.8, 81.2 79.5, 80.5
78.8, 81.2 (80 +- 1.96*(6/sqrt(100)) 80 +- 1.96*.6 80 +- 1.176)
What is the proportion (percentage or probability) of the curve that falls between the z scores of -1.96 and 1.96?
95%
Based on question 11, if a student has a z score of 1.65, what is the student's percentile in the group?
95.05
What is the proportion (percentage or probability) of the area that falls below the z score of 1.65?
95.05%
If the proportion under the curve that corresponds to the obtained z score is smaller than rejection areas, what decision can we make about rejecting the null hypothesis? Reject the null hypothesis Fail to reject the null hypothesis Can not be determined based on given information
Reject the null hypothesis
Describe a study that can utilize ANOVA testing.
Suppose a researcher wanted to examine the effects of three different pets on anxiety reduction in a stressful situation (cat, dog, hamster). 15 of the subjects were randomly assigned perform a stressful task with each of the animals present. Each of the subject's mean heart rate during the task was recorded. Test the appropriate hypotheses at the alpha = 0.05 level to decide if the mean heart rate differs between the groups.
What is the SSwithin in the ANOVA table?
Under Error Sum of Squares
What is SSbetween in the ANOVA table?
Under Model Sum of Squares
What research scenario will you choose the dependent t test?
Used when observations are not independent of each other (ex. pre- and post-test design)
A Type I error occurs when we: correctly reject a false null hypothesis. incorrectly reject a true null hypothesis. incorrectly reject a false null hypothesis. correctly fail to reject a false null hypothesis.
incorrectly reject a true null hypothesis.
One of the three ways to increase statistical power in our research is to: increase effect size. decrease effect size. decrease sample size. increase the β (type-II error) level.
increase effect size. 1. increase sample size 2. increase effect size 3. increase alpha (type I error) level
A colleague contacts you regarding the creation of a confidence interval around a mean that is too wide to provide much information. Which of the following options would lead to a smaller interval width? decreasing the sample size increasing the confidence level from 95% to 99% increasing the sample size none of these option would decrease the interval width
increasing the sample size (Increasing the sample size decreases sampling variation in the mean, which in turn decreases the width of the confidence interval.)
The MSwithin group is based on deviations between the first and second group mean. group means and the grand mean. individual scores and the grand mean. individual scores and their respective group mean.
individual scores and their respective group mean.
In one study the sample mean is 5 points from the hypothesized value. In a second study the sample mean is 15 points from the hypothesized value. The t-value in the second study: will be the same size will definitely be bigger will definitely be smaller may be smaller, larger, or the same size
may be smaller, larger, or the same size
A researcher conducting an ANOVA obtains a p-value of .03. This means that if the null is true there are 3 chances out of 100 of obtaining the computed F-statistic. obtaining a statistically significant F-statistic. obtaining an F-statistic smaller than the one computed. obtaining an F-statistic as large or larger than the one computed.
obtaining an F-statistic as large or larger than the one computed.
A researcher is testing the null hypothesis that the population mean is 50, at the .05 significance level. If the p-value associated with the t-test is .073, the researcher should conclude the population mean equals 50 the population mean may be 50 the population mean is different from 50 the sample mean is different from the population mean
the population mean may be 50 (Even though we fail to reject the null hypothesis, we do not know the true null hypothesis.)
Independence assumption is: the variable is normally distributed in each of the two populations two group variances are equal the scores on the variable are independent of each other
the scores on the variable are independent of each other
Normality assumption is: the variable is normally distributed in each of the two populations two group variances are equal the scores on the variable are independent of each other
the variable is normally distributed in each of the two populations
Homogeneity assumption is: the variable is normally distributed in each of the two populations two group variances are equal the scores on the variable are independent of each other
two group variances are equal
A Type II error occurs when: we reject a false null hypothesis. we reject a true null hypothesis. we fail to reject a true null hypothesis. we fail to reject a false null hypothesis.
we fail to reject a false null hypothesis.
A researcher wishes to test the null hypothesis that the population mean equals 100, at the .05 significance level. The following statistical information is obtained, t=4.21, df=85. Should the researcher reject the null hypothesis? yes no can not be determined from the information given
yes
A researcher sets alpha=.05 and conducts a study to compare two groups. If there truly is a difference in the population means, what is the probability of a Type I error? .00 .01 .05 .20
.00
Based on the normal curve below, where can you find the z scores in the table? Column A Column B Column C
Column A
where can you find the value (proportion or probability) for area between mean (0) and z (1) under the curve in the Z table? Column A Column B Column C
Column B
Which of the following is the first step in hypothesis testing? Constructing the sampling distribution for the null hypothesis Developing a null and alternative hypothesis. Setting the cutoff value for rejecting the null hypothesis. It does not matter where you begin when you test hypotheses.
Developing a null and alternative hypothesis
Precision of inference is related to the size of the sample and is also related to width of confidence band. True or False
True
A researcher comparing two groups obtains a p-value of .19. The probability of getting: the obtained test statistic is .19 if the null hypothesis is true the obtained test statistic is .19 if the alternative hypothesis is true a test statistic as extreme or more extreme than the obtained test statistic is .19 if the null hypothesis is true a test statistic as extreme or more extreme than the obtained test statistic is .19 if the alternative hypothesis is true
a test statistic as extreme or more extreme than the obtained test statistic is .19 if the null hypothesis is true
The representativeness of a sample affects the ___________________ of an inference to a population. reliability accuracy precision estimate
accuracy
A research comparing 4 groups fails to reject the null hypothesis of no mean difference using an F-test. The next step should be to: conduct a Tukey test conduct a chi-square test conduct no more statistical tests conduct a t-test for each pair of means
conduct no more statistical tests
A researcher conducts an F-test and obtains a F-value of 2.36. The critical value with an alpha=.05 is 3.56. The researcher should reject the null hypothesis. conduct a chi-square test. fail to reject the null hypothesis. conduct a t-test for each pair of means.
fail to reject the null hypothesis.
The equation of the Standard error of the mean to compute confidence intervals and statistical inference: population sample xbar = mew population mean sample (o xbar) = population mean / square root of n S xbar = sample mean / square root of n S xbar = mew
population mean sample (o xbar) = population mean / square root of n
Based on the normal curve below, what is the total proportion (or probability) under the curve that falls beyond the mean (0)?
.50
Based on the normal distribution curve, what is the total proportion (or probability) under the curve?
1
What is the intervals that capture a 95% confidence interval? 1.282 1.65 1.96 1
1.96
If the standard deviation of FCAT Math scores is 50 in the population of examinees, the standard deviation of the sampling distribution of means (standard error of the mean) computed from samples of size 25 will be 2 10 20 100
10
A friend calls and indicates they are trying to decide what size sample to collect. Which of the following sample sizes would lead to the confidence interval with the shortest width? 15 40 80 200
200 (As sample size increases the variation in sample means decreases (e.g., sample means tend to be closer to the population mean), and thus the sample interval width decreases.)
Which of the following interpretations for a 95% confidence interval is(are) accurate? The population mean will fall in a given confidence interval 95% of the time The sample mean will fall in the confidence interval 95% of the time. 95% of the confidence intervals created around sample means will contain the population mean All three statements are accurate
95% of the confidence intervals created around sample means will contain the population mean (Population means don't move from sample to sample so the first choice isn't OK. The sample mean is always in the interval so the second choice isn't OK. The third choice is accurate.)
Why do the researchers say "failing to" reject the null instead of "accepting" the null hypothesis?
Because we never know the null hypothesis
Practical significance = Hypothesis testing Effect size Statistical power
Effect size
Dr. Goodwill is concerned that students who are not native English speakers are at a disadvantage when taking a mathematics state standardized test. He also believes that this disadvantage diminishes if students are taught in English-based classes rather than those that offer students instruction in their native language. To test his hypothesis, he examines the 8th grade math test scores of students in regular 8th grade math. Students fall into one of the following categories: (1) Native English speakers, (2) Non-Native English speakers in a class taught in English only, and (3) Non-Native English speakers in a class taught in their native language. The best statistical test to use in this situation would be a/an independent-means t-test. dependent-means t-test. F-test (ANOVA). one mean z-test.
F-test (ANOVA).
In ANOVA, the equation for F is: F=Xbarb / Xbarw F= MSb / MSw F= Population mean / sample mean F= n-k
F= MSb / MSw
If the critical t value is 2.3 and the obtained t value is 2.0, what decision can we make about rejecting the null hypothesis? Reject the null hypothesis Fail to reject the null hypothesis Can not be determined based on given information
Fail to reject the null hypothesis
If the p-value we obtained is greater thant the alpha level we predetermined, what decision can we make about rejecting the null hypothesis? Reject the null hypothesis Fail to reject the null hypothesis Can not be determined based on given information
Fail to reject the null hypothesis
The alpha level represents the rate of Type II error. True or False
False
The independent t test robust to violation of the independence assumption. True or False
False
The smaller the value of the degree of freedom (df) for a sample, the better the sample variance ( s2) represents the population variance (σ2). True or False
False (The greater the value of the degree of freedom (df) for a sample, the better the sample variance ( s2) represents the population variance (σ2))
Statistical power tells us our level of Type I error. True or False
False (Type II error rate)
How can a researcher reduce the width of a confidence interval without reducing her level of confidence? Use a smaller critical value of t or Z Increase her sample size Reduce her sample size Increase her power
Increase her sample size
What happens if we have a large effect size but no statistical significance?
Lack of power so we have to collect more data
If obtained F value is greater than the critical value, do we reject the null or fail to reject the null? Fail to reject the null Reject the null
Reject the null
If the p value is less than the alpha level, we reject the null or fail to reject the null? Reject the null Fail to reject the null
Reject the null
If the critical z value is -4.5 and the obtained z score is -4.9, what decision can we make about reject the null hypothesis? Reject the null hypothesis Fail to reject the null hypothesis Can not be determined based on given information
Reject the null hypothesis
MSb is the Mean Square Between Groups and is calculated by: SSbetween / dfbetween SSwithin/dfwithin
SSbetween / dfbetween
MSw is the Mean Square Within Groups and is calculated by: SSbetween / dfbetween SSwithin/dfwithin
SSwithin/dfwithin
Which of the following statements about degree of freedom (df) is FALSE? There is a different t-distribution for every different degrees of freedom. The shape of the sampling distribution of t depends depends on the degree of freedom. The actual amount of variability in the sampling distribution of t depends on the degree of freedom. The degree of freedom is a measure of the number of independent pieces of information that can be used to estimate a population parameter.
The actual amount of variability in the sampling distribution of t depends on the degree of freedom. (The actual amount of variability in the sampling distribution of t depends on the sample size n, but the shape depends on df. We say that the t statistic has (n-1) degrees of freedom.)
Which of the following statements is NOT part of the central limit theorem? The mean of the sampling distribution of the mean is equal to the population mean. The variance of the sampling distribution is inversely proportional to the sample size. The mean of the sampling distribution of the mean is equal to the population mean divided by the square root of the sample size. The variance of the sampling distribution of the mean is equal to the population variance divided by the sample size.
The mean of the sampling distribution of the mean is equal to the population mean divided by the square root of the sample size.
A researcher is comparing 2 groups of 35 achievement scores. For which of the following scenarios would you strongly advise the researcher NOT to conduct a pooled variance t-test? The scores are not independent within the groups The variances for group 1 is 50 and for group 2 is 100 The achievement variable is positively skewed for both groups
The scores are not independent within the groups
As n increases, the sampling distribution of the mean approaches the shape of a normal distribution regardless of the shape of the population distribution. True or False
True
The independent t test robust to violation of the normality assumption. True or False
True
The p-value is the exact probability that the statistic we calculated on our observed sample could actually occur in our null distribution by chance alone. True or False
True
We always test a null hypothesis against an alternative. True or False
True
A 95% confidence interval for a mean runs from .6 to .8. Which of the following statements about the confidence interval is worded correctly? The population mean falls between .6 to .8 95% of the time. The sample mean falls between .6 to .8 95% of the time We are 95% confident that the sample mean is between .6 and .8. We are 95% confident that the population mean is between .6 and .8.
We are 95% confident that the population mean is between .6 and .8. (The first option is incorrect because the population mean does not move. The second and third options are incorrect because the sample mean is always in the interval.)
We want to compare three methods of instructions. We want to determine if the means of achievement scores of student undergoing the three methods are the same or different. What would happen if we conduct more than one t-test to compare the three methods? We would fail to reject our null hypothesis. We would increase our Type I error rate each time we do a t-test. We would increase our Type II error rate each time we do a t-test. We would fail to reject the null hypothesis for one of the comparisons.
We would increase our Type I error rate each time we do a t-test. (If we conduct more than one t-test, we would increase our Type I error rate each time we do a t-test.)
What research scenario will you choose the independent t test?
When observations are independent of each other (ex. comparing two separate groups of observations such as mean scores for boys and girls on the FCAT)
What sample statistic provides the best interval estimate of the population mean (u) when the population variance is NOT known? Xbar +/- (Z)(Sxbar) Xbar +/- (t) (population mean xbar) Xbar +/- (Z) (population mean xbar) X bar +/- (t) (Sxbar)
X bar +/- (t) (Sxbar)
Twenty classrooms are randomly assigned to treatment A and twenty classrooms are randomly assigned to treatment B. Suppose there are 25 students in each class. Which is the most appropriate way to compare the effectiveness of the two treatments? z-test one-sample t-test independent samples t-test on class means independent samples t-test on individual scores
independent samples t-test on class means (The individual students are dependent within classrooms. So it is good to use classrooms as units of analysis.)
A researcher comparing two groups, calculates a p-value of .037. If alpha=.05, the statistical decision is to: reject the null hypothesis fail to reject the null hypothesis reject the alternative hypothesis fail to reject the alternative hypothesis
reject the null hypothesis
For the independent means t-test, the 95% confidence interval is between 1.547 and 4.587. The statistical decision is to: reject the null hypothesis fail to reject the null hypothesis reject the alternative hypothesis fail to reject the alternative hypothesis
reject the null hypothesis
A researcher, with an alpha=.05, conducts an F-test and obtains a p-value of .002. The researcher should conduct a Tukey test. reject the null hypothesis. fail to reject the null hypothesis. conduct a t-test for each pair of means.
reject the null hypothesis.
Statistical power is the probability of: making a Type I error. making a Type II error. rejecting the null hypothesis when it is true. rejecting the null hypothesis when it is false.
rejecting the null hypothesis when it is false.
What context can you use the z test?
when the standard deviation and the population mean is know, z-tests are used