Testing Hypotheses for Means
Assignment: Testing Hypotheses for Means. This week you have explored three different approaches to t tests. By this point, you know that each test has assumptions about the data and type of research questions it can answer. For this Assignment, you will be provided with three scenarios.
As you read the scenarios, be sure and think about aligning the appropriate t test with the question. Consider whether the data are independent samples and if two samples are being compared.
To prepare for this Assignment:
· Review the Learning Resources and the media programs related to t tests.
· For additional support, review the Skill Builder: Research Design and Statistical Design and the Skill Builder: Hypothesis Testing for Independent Samples t-test, which you can find by navigating back to your Blackboard Course Home Page. From there, locate the Skill Builder link in the left navigation pane.
· Also, review the t test scenarios found in this week’s Learning Resources and consider the three different approaches of t tests:
· Independent sample t test
· Paired sample t test
· One sample t test
· Based on each of the three research scenarios provided, open the High School Longitudinal Study dataset or the Afrobarometer dataset from this week’s Learning Resources using SPSS software, then choose and run the appropriate t test.
· Once you perform your t test analyses, review Chapter 11 of the Wagner text to understand how to copy and paste your output into your Word document.
For this Assignment:
Write a 2 to 3-paragraph analysis of your t test results for each research scenario and include the SPSS syntax and output. If you are using the Afrobarometer Dataset, report the mean of Q1 (Age). If you are using the HS Long Survey Dataset, report the mean of X1SES. Do not forget to evaluate if the t test assumptions are met, justify the selection of type of t test, and report the effect size. Based on your results, provide an explanation of what the implications of social change might be.
Use proper APA format, citations, and referencing for your analysis, research questions, and output.
REFERENCES
Frankfort-Nachmias, C., Leon-Guerrero, A., & Davis, G. (2020). Social statistics for a diverse society (9th ed.). Thousand Oaks, CA: Sage Publications.
· Chapter 8, “Testing Hypothesis” (pp. 243-279)
Wagner, III, W. E. (2020). Using IBM® SPSS® statistics for research methods and social science statistics (7th ed.). Thousand Oaks, CA: Sage Publications.
· Chapter 6, “Testing Hypotheses Using Means and Cross-Tabulation” (previously read in Week 5)
· Chapter 11, “Editing Output” (previously read in Week 2, 3, and 4)
Walden University, LLC. (Producer). (2016l). The t test for independent samples [Video file]. Baltimore, MD: Author.
Note: The approximate length of this media piece is 5 minutes.
In this media program, Dr. Matt Jones, demonstrates the t Test for independent samples in SPSS.
Research Design and Statistical Analysis
As a student, you have research questions you want to answer. For example, an agriculture student may want to help farmers select an effective fertilizer for their corn crop. Perhaps farmers in an area of the country have traditionally used cow manure, but turkey dung has now become available at a good price.
The student may want to examine the two types of fertilizers to see if there is a difference in their effectiveness and whether farmers will be willing to change from cow manure to turkey dung. To address this research question, the student will need to think about both research design and statistical analysis.
Research Design
The research design for a study is the overall plan for how a researcher will collect data. Research design focuses on obtaining the right data to answer a research question. For example, there are many different ways to design a research study that examines the effectiveness of different types of fertilizer.
Will the agricultural student survey farmers who have used each type of fertilizer and get their reports on how their corn crops have been growing? Or will the student raise corn and use one type of fertilizer on some corn crops and another type on the second set of corn crops and then compare how the two sets of corn crops are growing? In deciding on research design, the researcher will want to think about which of these designs will give the farmers the best answer about whether the fertilizers differ in effectiveness.
Determining Cause and Effect
· bullet
During the research process, the researcher will ideally measure each variable of interest in a study to determine cause and effect. One of the key elements for determining cause and effect in a research study is control. Good research design requires the researcher to be aware of factors besides the independent variable that may have an effect on the dependent variable.
These factors are considered threats to validity. The researcher may try to control these factors in some manner in order to make a better judgment about whether the independent variable is, in fact, creating change in the dependent variable. For example, weather and moisture can also affect how corn crops grow, so the researcher needs to control for those factors. Realistically, some factors are more important than others, and the researcher needs to have an understanding of the factors most likely to be in play.
· bullet
Another key element for determining cause and effect in a research study is random assignment. The researcher investigating cow manure and turkey dung as fertilizers will probably want to test the fertilizers on essentially identical plots of ground. Researchers use random assignment to help them ensure that experimental units (e.g., plots) do not differ in some systematic way at the start of the study.
For example, the student would not want the plots receiving cow dung to receive greater sunlight, on average, than the plots receiving turkey dung. It will be important, then, to randomly assign the plots to receive either cow manure or turkey dung.
The following three broad categories of research design are covered in this Skill Builder.
· 1
General Research Design Categories
1
Experiential
· 2
2
Quasi-experimental
· 3
3
Correlational
Statistical Analysis
After measuring each variable of interest in their study (e.g., fertilizer type and crop yield), researchers use the data they have collected to conduct statistical analyses to answer their research questions. Statistical analyses involve the use of probabilistic models to analyze the data. Statistical analysis, broadly stated, is evaluating models to determine whether variables are associated with one another.
Presuming the researcher has made valid measurements, the results of the statistical model can then be used to make inferences back to the real world. For example, the student will somehow measure the effectiveness of each type of fertilizer, perhaps by focusing on crop yield, and will use those measurements in a statistical model to answer their research question of interest.
In the ideal case, the statistical analysis will be useful in convincing a critic that only the experimental manipulation or chance (a type I error) can explain an experimental effect that is statistically significant.
There are several types of statistical analyses, including Pearson’s correlation, ANOVA, and regression. In this Skill Builder, we will briefly discuss two types of statistical analyses:
· 1
1
t-test for independent groups
· 2
2
matched pairs t-test
CO
Identifying the Research Design
There are many possible research designs, and we will only consider a few examples. While there are many ways to categorize research designs, we will use three general terms: true experiment, quasi-experiment, and correlational.
True Experiment
The gold standard in doing research is the true experiment, which involves the random assignment of experimental units to treatment conditions along with experimenter manipulation and control of factors that may affect the dependent variable. The random assignment involves, for example, the experimental units being assigned to treatment conditions or control conditions by the use of a random number generator.
Consider a study in which the researcher wants to study the effects of marijuana use on short term memory. Two groups will be created. One group will smoke a marijuana joint and then take a memory test, and the other will only take the memory test. How can participants be assigned to conditions?
I may flip a coin for each experimental unit and assign it to Condition 1 if a head appears and Condition 2 if a tail occurs. I will continue until one treatment has been filled and then assign the remainder to the other condition. There are also methods using random number generators that can be used to assign experimental units to treatments.
After the random assignment, the experimenter controls or manipulates all factors that may affect the dependent variable. In the simplest case, one factor is manipulated, the independent variable and the other potentially relevant factors are controlled by the experimenter by carefully following well-developed experimental procedures. If the effect is produced, the independent variable is said to cause an effect on the dependent variable.
For example, if statistically significant differences are found between the group using marijuana and the group not using marijuana, then we would reject the null hypothesis of no difference between the groups, and the results would suggest that marijuana use vs. no use has a causal link with memory.
Note that the fertilizer study described above would be another example of a true experiment if the student used random assignment and controlled factors (e.g. sunlight) that could impact the dependent variable of crop yield.
Quasi Experiment
A quasi-experimental study approximates an experiment, but experimental units are not randomly assigned to conditions. In this design, though, there is often still an experimental treatment and experimenter manipulation and control. Quasi-experimental designs are sometimes used when a random assignment of individuals to groups is not feasible.
For example, if a researcher wanted to compare the effects of two types of reading interventions, the researcher might not be able to randomly assign elementary school students to one of the two interventions. The researcher, instead, might have to employ each intervention in existing classrooms within the school.
As another example, consider a slightly different study of short term memory than the one we described above. In this case, the researcher gives the memory test to two groups: marijuana users are assigned to the group that consumes marijuana and non-users are assigned to a second group that does not smoke marijuana. This procedure may be used because it may not be ethical to randomly assign some individuals to smoke marijuana.
The researcher in this study, though, could still make the experiences of the two groups as similar as possible during the study to still maintain an element of control over extraneous factors that could affect the dependent variable.
It is notable that this plan is not an ideal research design for this study because the immediate marijuana consumption is confounded with the historical marijuana use – that is, any effects on memory could be due to either past marijuana use or the specific marijuana use during the course of the study.
It would be difficult to know which scenario is occurring. This design, therefore, illustrates the problem of uncontrolled sources of variance being confounded with the independent variable.
Although quasi-experiments are less desirable than true experiments, they are sometimes the best a researcher can do when practical constraints are taken into consideration.
Application of Statistical Analysis
Topic 3 of 5
Learning Objective:
Explain the difference between research design and statistical design.
Introduction
Research design is concerned with the overall plan for data collection and how the variables in the study will be measured. Statistical analysis is the next step in the research process and involves using the data the researcher has collected in a variety of probabilistic models in order to answer the research questions.
Based on the statistical analyses that the researcher conducts, the researcher can then draw real-world conclusions that pertain to their research questions.
For example, if we want to compare the effectiveness of two types of fertilizer, we will run statistical analyses to examine whether the two fertilizers show a statistically significant difference in crop yield. We will also examine practical significance by examining what our statistical results show about how different the crop yields are for each type of fertilizer.
Through examining both statistical and practical significance, we can make real-world conclusions about whether one fertilizer is more effective than the other and whether it’s a good idea to use one fertilizer instead of the other.
There are many types of statistical analysis, but in this lesson, we will focus on two specific types of statistical analyses: t-test for independent groups and matched pairs t-test.
t-Test for Independent Groups
One type of commonly used statistical analysis is the t -test for independent groups. In this statistical test, researchers compare two independent groups to see if they show a mean difference on a continuous dependent variable. In this test, the null hypothesis is:
HO : μ1 = μ2 where μ1 is the mean for population 1 and μ2 is the mean for population 2
The null hypothesis specifies that there is no difference in the population means between the two groups. Researchers collect data on the dependent variable from each of the two samples and use that data to conduct a t-test for independent groups.
Statistical research has shown that if several assumptions are met, then we know that a t– statistic with n1+n2−2 degrees of freedom (n1 is the number of participants in sample 1 and n2 is the number of participants in sample 2) can be calculated using the sample means, sample variances, and sample sizes, and the t-distribution with n–
As an example of a research scenario for which researchers would use a t-test for independent groups, think again about our fertilizer example. The researcher could use a t-test for independent groups to examine if crop yield for plots receiving cow manure (sample 1) is different from crop yield for plots receiving turkey dung (sample 2). Note that crop yield is the dependent variable in this scenario.
Note also that samples 1 and 2 can be considered to be independent of one another; two different sets of plots are being examined in this scenario. This is important to think about because as we’ll see below, researchers use a different type of t-test if the samples are not independent of one another.
Matched Pairs t-Test
The matched pairs t-test is similar to the t-test for independent groups in that both tests compare two means, and the dependent variable for both tests will be continuous. In fact, the null hypothesis for a matched pairs t-test can be stated as HO: μ1 =μ2, the same way that we stated the null hypothesis for the t-test for independent groups.
In a matched pairs t-test, however, the units that produce each set of means are related to one another; they show dependence in some manner.
For example, we may collect data on the same group of individuals at two different time points. Researchers might, for example, collect data from students on both a pre-test and a post-test to see if scores on the test have changed over time.
Note that these two sets of observations will be related to one another, which is why researchers would use a matched pairs t-test and not a t-test for independent groups.
Students’ degree of academic ability is likely to affect their scores on both tests in a similar manner, for example, and there is likely an association between students’ test scores at the two-time points because it is the same group of students taking both tests.
Note
Each student in the study will have two scores for the dependent variable. This differs from studies for which researchers would use a t-test for independent groups because, in those studies, each participant only has one score for the dependent variable.
(Think about the difference between examining test scores from the same group of students at two-time points vs. examining test scores from two separate groups of students. For the first scenario, we would use a matched-pairs t-test, and for the second scenario, we would use a t-test for independent groups.)
Note, also, that for a matched-pairs t-test, a t-statistic is calculated that has n-1 degrees of freedom; if you compare this to what was written above about how t-statistics are calculated for independent groups, you will note that there is a difference between the two types of t-statistics.
| Term | Meaning |
| +∞ | Positive infinity. |
| -.564 | Observed value of the test statistic. |
| -∞ | Negative infinity. |
| .004 | p-value |
| .576 | p-value |
| 2-tailed | The alternative hypothesis states simply that there is a difference between the means but does not specify the direction of the difference. |
| 61 | 61 is the degrees of freedom (df) calculated by n-2 (63-2) |
| alpha | The probability of a type I error. |
| box-plot | A graph that displays key elements of distribution. |
| categorical variables | Variables that have a limited number of possible values; participants in the study get placed into one of a small number of categories for the variable. |
| central limit theorem | regardless of the distribution of the population, if the sample size is relatively large (a rule of thumb is n > 30), the sampling distribution of sample means is close to normal. |
| cohen’s d | A measure of effect size. |
| confidence intervals | A range of values used to specify the likelihood that the population parameter is contained within a specified range. |
| continuous variable | A continuous variable is one based on an interval or ratio level of measurement. Between any two values for the variable, there is another possible value. |
| continuous variables | A continuous variable is one based on an interval or ratio level of measurement. Between any two values for the variable, there is another possible value. |
| control group | The collection of participants in the condition of an experiment who do not receive the treatment. A group receiving an actual treatment can then be compared to the control group. |
| dependent variable | A measure of the outcome that allows us to determine whether the independent variable has an effect. |
| discrete | A variable based on an ordinal, interval, or ratio levels of measurement and has a countable, not infinite, set of possible values. |
| distribution of a population | The distribution of all values for all elements of the population. |
| distribution of a sample | The distribution of actual observations based on the data that you collect. |
| distribution of the sample | Sample distribution (also called distribution of the sample) –for a variable, the distribution of values for the elements of the population that are actually observed. (note that Sample distribution is different from Sampling distribution). |
| element | an entity in the population that may be selected for the sample and then observed. |
| factor | The alternative hypothesis stated simply that there was a difference between the means, and does specify the direction of the difference. |
| frequency distribution | A table or graph that shows the values of a variable and the number (count) of observations associated with each value |
| general rule | Although different sources give slightly different information about assessing the strength of a correlation coefficient, we can use the following as a general rule for interpreting the correlation coefficient:.8 to 1: very strong.6 to .8: strong.4 to .6: moderate.2 to .4: weak0 to .2: very weak to no relationship |
| independent variable | The variable that is studied to see if it causes a change in a dependent variable. |
| interval | The level of measurement that addresses differences, or intervals, between entities. |
| interval estimates | A range of values that is likely to contain the population parameter. |
| levels of confidence | The probability that the population parameter is contained within a specified range of values. Usually, the level of confidence is 0.95 or 95%. |
| levels of measurement | Also called scale of measurement, describes the amount and type of information (nominal, ordinal, interval, and ratio) that is conveyed by the numbers or words assigned to real-world objects during the measurement process. |
| levene’s test | Tests the null hypothesis that the two populations show equal variance. |
| margin of error | The amount of estimated error in the point estimate of a population parameter determined by the level of confidence and the sampling distribution for the sample statistic. In estimating the population means, the margin of error equals a critical value for statistic times the standard error of the mean, e.g., Zα2*σn. |
| mean | The average of the scores for a variable. |
| median | An appropriate measure of central tendency when a measurement is at the ordinal, interval, or ratio level. |
| mode | The most frequently occurring value in the data set. |
| n | n = sample size |
| n1 | n1 = the number of participants in sample 1 |
| n2 | n2 = the number of participants in sample 2 |
| negative skew | This refers to the tail of the distribution appearing longer on the left-hand side of the distribution. |
| nominal | The lowest level of measurement, which addresses naming—identifying or categorizing objects using a name. |
| one-tailed | The alternative hypothesis is directional and states that one mean is greater than the other. |
| ordinal | The level of measurement above nominal that addresses ordering real-world entities. |
| outliers | Observation points that are distant from other observations. |
| p <.01 | This indicates that the p-value (.000) is less than .01 and that the correlation test is statistically significant. |
| p-value | The probability of obtaining a result equal to or “more extreme” than what was actually observed, when the null hypothesis is true. |
| pictogram | A graphic character used in picture writing. |
| point estimate | An estimate of the unknown parameter of interest using a single value. |
| population | The set of all possible elements (entities and observations) to which the researcher wishes to generalize. |
| population distribution | for a variable, the distribution of all values for all elements of the population. |
| positive skew | This refers to the tail of the distribution appearing longer on the right side of the distribution. |
| qualitative | A variable based on nominal measurement. |
| quantitative | A variable with an ordinal, interval or ratio level of measurement. |
| r | r is the symbol indicating a Pearson’s correlation coefficient |
| r-squared | The proportion of variability in the dependent variable that is accounted for by your model. |
| random assignment | Random assignment is placing experimental units in treatment conditions or control conditions by use of a random process. |
| random sampling | The selection of experimental units so that each element in the population has the same chance of being selected for the sample. |
| random variable | A variable whose value is determined by a random process such as being selected in a survey or being observed in an experiment. |
| ratio | The level of measurement that addresses proportion, or ratios between entities. |
| ratio level | The level of measurement that addresses proportion, or ratios, between entities. |
| relative frequency distribution | A table or graph that shows the values of a variable and the proportion of observations associated with each value using decimal fractions or percentages. |
| research design | The overall plan for how a researcher will collect data. |
| sample | A subset of all possible observations. |
| sampling distribution | The distribution of a sample statistic. |
| sampling distribution of the sample mean | The distribution of values for the sample mean for all possible random samples of size n. |
| sampling error | The absolute value of a statistic minus the parameter being estimated. |
| simple random sampling | Each unit in the population has an equal chance of being selected into the sample. |
| statistical analyses | The use of probabilistic models to analyze data. |
| statistical inferences | the process of using sample information to make statements about population parameters. |
| statistical power | The probability of rejecting a null hypothesis if the null is false (i.e., the alternative is true). |
| statistically significant | Statistical significance means a null hypothesis has been rejected. |
| t-test for two independent groups | A statistical test used to examine whether two independent groups have different means on a dependent variable. This test is also sometimes referred to as an independent samples t-test. |
| two-tailed | The alternative hypothesis states simply that there is a difference between the means but does not specify the direction of the difference. |
| type i error | Rejecting the null hypothesis if the null is actually true. |
| type ii error | Incorrectly retaining a false null hypothesis (a “false negative”). |
| unit of analysis | The real-world entity that is observed and for which data are recorded and used in statistical analysis. |
| value | A single observation defined for a variable. |
| variable | The mathematical representation of the real-world entity being measured. |
| variance | Variance is a measure of variability in a set of observations based on the approximate average of squared deviations from the mean. |
| visual displays of data | Help researchers communicate the distribution and other key information (the story they are telling with their data) both effectively and efficiently. |
| µ1 | mean for population 1 |
| µ2 | mean for population 2 |
| β | β is the symbol researchers use when they report a standardized regression coefficient. |
| μ not primed | This indicates the population means for the “not primed” condition. |
| μ not primed – μ primed >0 | The alternative hypothesis specifies that the “not primed” condition will score higher than the “primed” condition. |
| μ primed | This indicates the population means for the “primed” condition. |
