Psych 2016 Exam 2
What is the power in hypothesis testing?
The probability of correctly rejecting a value stated in the null hypothesis.
Can the mean of a sampling distribution be a negative value? Explain.
Yes, it can. The mean can be any value between positive infinity (+∞) and negative infinity (−∞).
The sampling distribution of the sample mean approximates the shape of what type of distribution?
normal curve
What three factors can be increased to increase power?
sample size, effect size, and alpha.
The one-tailed tests. In their book, Common Errors in Statistics (and How to Avoid Them), Good and Hardin (2003) wrote, "No one will know whether your [one-tailed] hypothesis was conceived before you started or only after you had examined the data" (p. 347). Why do the authors state this as a concern for one-tailed tests?
The point the authors are making is that it is possible with the same data to retain the null hypothesis for a two-tailed test and reject the null hypothesis for a one-tailed test.
Suppose a researcher has participants rate how much their mood improves following an exercise on a bipolar scale from −3 (much worse) to +3 (much improved), with the midpoint, 0, indicating no change in mood. If following the exercise the researcher reports a 95% CI = 0.5 to 1.3, did she observe a significant increase in mood? Explain.
(a) One-sample t test; two-independent-sample t test. (b) The smaller the degrees of freedom, the larger the value of the test statistic needed to reach a decision to reject the null hypothesis. Hence, for a given sample size, N − 2 will be associated with less power than n − 1 degrees of freedom.
How would the standard error change if (a) the population standard deviation increased and (b) the sample size increased?
(a) The standard error would increase. (b) The standard error would decrease.
Name the t test used in hypothesis testing to evaluate the mean observed in one sample.
A one-sample t test is used to compare a mean value measured in a sample to a known value in the population.
The importance of sampling distributions. Turner and Dabney (2015) stated, "Sampling distributions play a key role in the process of statistical inference" (p. 23). Explain what the researchers meant by this statement.
A sampling distribution informs us of the likelihood of sample outcomes. Thus, we can infer (i.e., make a statistical inference) the likelihood of selecting any one sample, if we know the sampling distribution for the population from which the sample was selected
In the following studies, state whether the one-sample t test is an appropriate test statistic to analyze data. If not, then explain why it is not appropriate. A study evaluating the body mass index (BMI) score of athletes compared to the general population
Appropriate
What test is used as an alternative to the z test when the population variance is unknown?
The alternative test to the z test when population standard variance is unknown is the t-test.
How are the rejection regions, the probability of a Type I error, the level of significance, and the alpha level related?
All four terms describe the same thing. The level of significance is represented by alpha, which defines the rejection region or the region associated with the probability of committing a Type I error.
In the following studies, state whether the one-sample t test is an appropriate test statistic to analyze data. If not, then explain why it is not appropriate. A study measuring differences in attitudes about morality among people who self-identify as liberal or conservative
Appropriate
How does our estimate of the population variance change as the sample size increases? Explain.
As sample size increases, the sample variance more closely estimates the population variance.
The sample variance is used in the formula for standard error when the population variance is not known. Why is it appropriate to substitute the sample variance for the population variance?
Because the sample variance is an unbiased estimator of the population variance—the sample variance will equal the value of the population variance on average.
Alpha (α) is used to measure the error for decisions concerning true null hypotheses. What is beta (β) error used to measure?
Beta (β) error is a measure of error for decisions concerning false null hypotheses.
What are the critical values for a one-sample nondirectional (two-tailed) z test at a .05 level of significance?
Critical values = ±1.96.
Explain how conditional probabilities are related to sampling without replacement.
Data describe a set of measurements (made up of raw scores); a raw score describes individual measurements.
What is sampling distribution?
Distribution of all sample means that could be obtained in samples of a given size from the same population.
Name three measures used to estimate effect size for the one-sample t test.
Estimated Cohen's d, eta-squared, and omega-squared
20. T/F The central limit theorem explains why the mean is always at the center of a distribution.
False. The central limit theorem 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 a normal distribution, as the number of samples in the sampling distribution increases.
The value of a p value. In a critical commentary on the use of significance testing, Charles Lambdin (2012) explained, "If a p < .05 result is 'significant,' then a p = .067 result is not 'marginally significant'" (p. 76). Explain what the author is referring to in terms of the two decisions that a researcher can make.
If the p-value is less than 0.05 we shall reject the null hypothesis and if p-value is greater than 0.05 we shall accept the null hypothesis and reject the alternative.
Distinguish between the significance and the effect size of a result.
In hypothesis testing, the significance of an effect determines whether or not an effect exists in some population. Effect size is used as a measure of how big the effect is in the population.
The standard error measure is the standard deviation for what type of distribution?
It is the standard error or distance that sample mean values deviate from the value of the population mean.
Effect size: 0.8-1.4
Large
Sample size and power. Davis and Loprinzi (2016) evaluated a hypothesis related to engaging children, adolescents, and adults in physical activity. As part of their study, they reported a sample size of 106 children, 128 adolescents, and 440 adults. Assuming equal effect sizes across these age groups, which age group is likely to be associated with greater power to detect effects of physical activity? Explain.
Larger the sample size,lesser the standard error of mean,so test statistic value increases,p-value decreases,hence,we will be more likely to reject the null hypothesis,so type II error decreases,hence power increases.
Effect size: 0.5
Medium
A researcher conducts a one-sample z test and makes the decision to reject the null hypothesis. Another researcher selects a larger sample from the same population, obtains the same sample mean, and makes the decision to retain the null hypothesis using the same hypothesis test. Is this possible? Explain.
No, the larger the test statistic, the smaller the p value of the test is. The smaller p value will have more of a chance of rejecting the null hypothesis.
In the following studies, state whether the one-sample t test is an appropriate test statistic to analyze data. If not, then explain why it is not appropriate. A study testing whether night-shift workers sleep the recommended 8 hours per day
Not appropriate, z test because we are testing for a single μ
Name two measures of proportion of variance for the one-sample t test. Which measure is the most conservative?
Omega-Squared (ω2)
Is a one-tailed test associated with greater power than a two-tailed test? Explain.
One-tailed tests are associated with greater power, assuming the value stated in the null hypothesis is false.
Define point estimation and interval estimation.
Point estimation is a statistical procedure that involves the use of a sample statistic (e.g., a sample mean) to estimate a population parameter (e.g., a population mean). Interval estimation is a statistical procedure in which a sample of data is used to find the interval or range of possible values within which a population parameter is likely to be contained.
What three factors can be decreased to increase power?
Population standard deviation, beta error, and standard error.
What are two decisions that a researcher makes in hypothesis testing?
Reject the null hypothesis and retain the null hypothesis.
Making decisions in hypothesis testing. Toll, Kroesbergen, and Van Luit (2016) tested their hypothesis regarding real math difficulties among children. In their study, the authors concluded: "Our hypothesis [regarding math difficulties] was confirmed" (p. 429). In this example, what decision did the authors make: Retain or reject the null hypothesis?
Reject the null hypothesis.
Distinguish between sampling with replacement and sampling without replacement.
Sampling with replacement will result in the same individual being chosen again whereas sampling without replacement will not.
Effect size: 0.2
Small
What are the three steps to compute an estimation formula?
Step 1: Compute the sample mean and standard error. Step 2: Choose the level of confidence and find the critical values at that level of confidence. Step 3: Compute the estimation formula to find the confidence limits.
Four Steps of Hypothesis Testing
Step 1: State the hypotheses Step 2: Set the criteria for a decision Step 3: Compute the test statistics Step 4: Make a decision
The usefulness of confidence intervals. To clarify the kind of information conveyed by a level of confidence, Thompson (2007) stated, "CIs are extremely useful because they convey not only our point estimate, but also, via the width of the intervals, something about the precision of our estimates" (p. 427). What is the name of the estimate that conveys the width of the intervals?
The assumption of normality, because if the data are skewed, this would directly violate this assumption.
Define the central limit theorem.
The central limit theorem states that regardless of the distribution of scores in a population, the sampling distribution of sample means selected from that population will be approximately normal.
A researcher tests the null hypothesis that the mean intelligence score in the population of adult learners using the standard IQ test is µ = 100. In his sample, he identifies a 95% CI = 99.1 to 102.3. What would the decision have been for this test using the one-sample t test? Explain your answer.
The decision would have been to retain the null hypothesis because the value of the null hypothesis, µ = 100, is contained within the identified confidence interval.
Will each of the following increase, decrease, or have no effect on the value of the test statistic in a one-sample t test? The difference between the sample and population mean is increased.
The difference between the sample and the population mean is increased increase the value of the test statistic in a one-sample t Test.
Will each of the following increase, decrease, or have no effect on the value of the test statistic in a one-sample t test? The sample variance is doubled.
The difference between the sample and the population mean is increased increase the value of the test statistic in a one-sample t Test.
25. A researcher computes the following 95% CI = 12 to 24. What is the point estimate for this 95% CI?
The point estimate for the population proportion is the sample proportion, and the margin of error is the product of the Z value for the desired confidence level (e.g., Z=1.96 for 95% confidence) and the standard error of the point estimate.
Explain why the following statement is true: The population standard deviation is always larger than the standard error when the sample size is greater than one (n > 1).
The population standard deviation is always larger than the standard error when n > 1 because the population standard deviation is divided by the square root of the sample size to calculate the standard error.
27. As α increases, so does the power to detect an effect. Why, then, do we restrict α from being larger than .05?
The probability of a type 1 error, given the null hypothesis, is true is called the significance level of the test α. And the probability of type 2 error is β. The power of a test is rejecting null hypothesis giving it is falsely represented as 1-β If we choose small α , then β increases, making it difficult to reject the null hypothesis. (Type 2 error will be common). If we choose largely α, type 2 error will be less, then it will easier to reject the null hypothesis and increases the chance of getting type 1 error. Thus, to balance Type 1 error and Type 2 error, we use a significance level around 0.05.
What is a Type I error (α)?
The probability of rejecting a null hypothesis that is actually true.
What is a Type II error (β)?
The probability of retaining a null hypothesis that is actually false. Researchers do not directly control for this type of error.
Assumptions for the one-sample t test. In a critical evaluation of the one-sample t test, Rochon and Kieser (2011) explained that "a one-sample t test is used when inference about the population mean µ is made for a sample of n independent observations" (p. 411). Identify the assumption for the one-sample t test that the authors refer to. Explain your answer.
The reason that researchers would remove outliers in a data set that are nonnormal is to satisfy the assumption for the two-independent-sample t test that the data are normally distributed. Removing outliers would make the data less skewed and thus presumably more normal in shape.
Who determines the level of confidence for an interval estimate?
The researcher
The sample mean is an unbiased estimator of the population mean.
The sample is taken from the population randomly. This means that the sample mean on average is equal to the population mean.
Explain why the following statement is true: µ = µM.
The sample mean is an unbiased estimator of the population mean.
A researcher selects a sample of 30 participants and makes the decision to retain the null hypothesis. She conducts the same study testing the same hypothesis with a sample of 300 participants and makes the decision to reject the null hypothesis. Give a likely explanation for why the two samples led to different decisions.
The sample size in the second sample was larger. Therefore, the second sample had more power to detect the effect, which is likely why the decisions were different.
Will each of the following increase, decrease, or have no effect on the value of the test statistic in a one-sample t test? The sample size is increased.
The sample size is increased increase the value of the test statistic in a one-sample t Test.
Will each of the following increase, decrease, or have no effect on the value of the test statistic in a one-sample t test? The level of significance is reduced from .05 to .01.
The sample variance is doubled decrease the value of the test statistic in a one-sample t Test.
Why are the degrees of freedom for the t distribution and the degrees of freedom for sample variance the same?
The t distribution is a sampling distribution with a standard error that is computed using the sample variance to estimate the population variance. Hence, the degrees of freedom for sample variance (n − 1) are also associated with each t distribution.
28. Estimating effect size. Yuan and Maxwell (2005) investigated how the power of a study influences the decisions researchers make in an experiment. In their introduction concerning effect size, they stated that "the exact true effect size is generally unknown even after the experiment. But one can estimate the effect size . . . [and] when the sample size is large, the estimated effect size is near the true effect size" (Yuan & Maxwell, 2005, p. 141). Why is the "exact true effect size" generally unknown even after the experiment? Explain. How does increasing the sample size improve our estimate of effect size when we use estimated Cohen's d?
The two-independent-sample t test, because the two groups being compared had different participants assigned (children who did and did not participate in sports).
Effect on the value of Cohen's d: The population variance is increased.
This change will decrease the value of Cohen's d.
Effect for one-sample z test d. The difference between the sample mean and population mean is decreased.
This change will decrease the value of the test statistic.
Effect on the value of Cohen's d: The sample size is decreased.
This change will have no effect on the value of Cohen's d.
Effect for one-sample z test b. The sample variance is doubled.
This change will have no effect on the value of the test statistic
Effect on the value of Cohen's d: The difference between the sample and population mean is increased.
This change will increase the value of Cohen's d.
Effect on the value of Cohen's d: The sample variance is reduced.
This change will increase the value of Cohen's d.
Effect for one-sample z test a. The sample size is increased.
This change will increase the value of the test statistic.
Effect for one-sample z test c. The population variance is decreased.
This change will increase the value of the test statistic.
A researcher conducts a hypothesis test and concludes that his hypothesis is correct. Explain why this conclusion is never an appropriate decision in hypothesis testing.
This conclusion is never an appropriate decision in hypothesis testing because we only reject the null hypothesis or fail to reject the null hypothesis.
28. Will increasing sample size (n) increase or decrease the value of standard error? Will this increase or decrease power?
This will decrease standard error, thereby increasing power.
What values do you need to know to compute the formula for standard error?
To compute standard error, you need to know the population standard deviation and the sample size.
T/F The mean of a sampling distribution is equal to the population mean from which samples are selected.
True
T/F The sampling distribution of sample means is approximately normally distributed only if sample size(n) is sufficiently large enough(greater than 30).
True
T/F The standard error of the mean is the standard deviation of a sampling distribution of sample means.
True
T/F The value of the sample mean equals the population mean on average.
True
T/F There is a 5% probability of selecting a sample mean that is farther than 2 SEM.
True
The value of the sample mean can vary from sample to sample.
True
Distinguish between sampling where order matters and sampling where order does not matter.
When order matters, each time we select participants in a different order, it is counted as a different possible sample. When order does not matter, selecting participants in a different order is counted as the same sample.
Sample standard error
When you increase the value of an existing score, the standard error will decrease When you decrease the value of an existing score, the standard error will increase
Population standard error
When you increase the value of an existing score, the standard error will increase When you decrease the value of an existing score, the standard error will decrease
Describing the z test. In an article describing hypothesis testing with small sample sizes, Collins and Morris (2008) provided the following description for a z test: "Z is considered significant if the difference is more than roughly two standard deviations above or below zero (or more precisely, |Z| > 1.96)" (p. 464). Based on this description, a. Are the authors referring to critical values for a one-tailed z test or a two-tailed z test? b.What alpha level are the authors referring to?
a. two-tailed z test b.α = .05