This document contains Michael Parent's answers to 4 questions on a final exam for an ELMP course. The questions assess understanding of key statistical concepts like correlation coefficients, hypotheses testing, and interpreting statistical significance. Michael provides well-explained responses to each question, demonstrating knowledge of how to state hypotheses, perform calculations, make decisions, and draw conclusions based on statistical analyses.
The test used to ascertain whether the difference between estimator & parameter or between two estimator are real or due to chance are called test of hypothesis.
T-test.
Chi-square (휒^2)- test.
F-Test.
ANOVA.
This slideshow is related to testing of hypothesis and goodness of fit of statistics. This may be useful for students, teachers, managers concerned with bio statistics, bioinformatics, data science, etc.
The test used to ascertain whether the difference between estimator & parameter or between two estimator are real or due to chance are called test of hypothesis.
T-test.
Chi-square (휒^2)- test.
F-Test.
ANOVA.
This slideshow is related to testing of hypothesis and goodness of fit of statistics. This may be useful for students, teachers, managers concerned with bio statistics, bioinformatics, data science, etc.
Research methodology - Estimation Theory & Hypothesis Testing, Techniques of ...The Stockker
Fundamentals, Standard Error, Estimation, Interval Estimation, Hypothesis, Characteristics of Hypothesis, Testing The Hypothesis, Type I & Type II error, One tailed & Two tailed test, Tabulated Values, Chi-square (2) Test, Analysis of variance (ANOVA)Introduction, The Sign Test, The rank sum test or The Mann-Whitney U test, Determination of Sample Size
Statistical tests of significance and Student`s T-TestVasundhraKakkar
Statistical tests of significance is explained along with steps involve in Statistical tests of significance and types of significance test are also mentioned. Student`s T-Test is explained
Research methodology - Estimation Theory & Hypothesis Testing, Techniques of ...The Stockker
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Steps of hypothesis testingSelect the appropriate testSo far.docxdessiechisomjj4
Steps of hypothesis testing
Select the appropriate test
So far we’ve learned a couple variation on z- and t-tests
See next slide for how to select
State your research hypothesis and your null hypothesis
State them in English
Then in math
Describe the NULL distribution
Starting here is where you be a skeptic and assume the null is true!
For one-sample tests, you will need to determine μ
(For two-tailed tests, you don’t need to worry about μ)
Compute the relevant standard error
Determine your critical value(s)
Keep in mind whether it is a directional or non-directional test
Compute the test statistic
Compare the test stat to the critical value(s) and make your decision
When to use each test
All of these tests require that the sampling distribution is normal
Either because population is normal or, thanks to central limit theorem, sample size is very large
All of these tests require that the measures be quantitative variables, that is interval/ratio
(Not all quantitative variables are normal, BUT all normal variables are quantitative. So if someone tells you a variable is normal, you know it is also quantitative.)
When to use each test, cont’d
1 Sample z-test
Comparing one sample mean to a population mean
And you do know σ (population SD)
2 sample z-test
Comparing two sample means to each other
And you do know σM1-M2 (standard error of difference of means)
1 sample t-test
Comparing one sample mean to a population mean
You only know s (sample SD)
2 sample t-test
Comparing two sample means to each other
You only know s1 and s2 (sample SDs)
Dependent sample t-test
You have two scores coming from each person, such as if you measured them before and after an experimental manipulation.
Compute the differences between the two scores, then treat like a 1 sample t
What is α?
Put on your skeptic’s hat: you believe the null hypothesis is true
But you’re willing to be convinced you’re wrong
If the test statistic is sufficiently improbable, you will change your mind and decide the null hypothesis is false
What is “sufficiently” improbable?
When your test statistic is more extreme than your critical values
Critical values are selected so that only a small fraction of the entire distribution is more extreme than the critical values
This “small fraction” is called α
Conventionally, α is usually set to .05, that is 5%
Directionality of a test
Is a test simply about whether there a difference, regardless of direction?
If so, it is a non-directed, or undirected, or two-tailed test
Your α must be evenly split between the two tails
For the conventional α = .05, that means each tail should have .025 or 2.5% of the total distribution
Is the test predicting one mean will be bigger than another? Or is it predicting one mean will be less than another?
If so, it a directional, or directed, or one-tailed test
Put all your α in a single tail
Special note on one-tailed tests
Step 3 of our procedure is a little awkward when we have one-tailed tests
How do you descr.
1
Running head: RESEARCH PROJECT PROPOSAL
1
3
RESEARCH PROJECT PROPOSAL
Research Project Proposal
Name Here
South University Online
Research Project Proposal
Provide a brief, general introduction to the topic and importance to your role (do not use first person anywhere in the paper). Conclude with a thesis statement.
Background and Significance of the Problem
Discuss here.
Statement of the Problem and Purpose of the Study
Discuss here.
Literature Review
Discuss here.
Research Questions, Hypothesis, and Variables
Discuss here.
Theoretical Framework
Overview and Guiding Propositions
Discuss here.
Application of Theory to Study Focus
Discuss here.
Methodology
Sample/Setting
Discuss here.
Sampling Strategy
Discuss here.
Research Design
Discuss here.
Extraneous Variables
Discuss here.
Instruments
Discuss here.
Description of the Intervention
Discuss here.
Data Collection Procedures
Discuss here.
Data Analysis Plans
Discuss here.
Ethical Issues
Discuss here.
Limitations of Proposed Study
Discuss here.
Implications for Practice
Discuss here.
References
Appendices
Lecture 6
(Additional information on t-tests and hypothesis testing)
Lecture 5 focused on perhaps the most common of the t-tests, the two sample assuming
equal variance. There are other versions as well; Excel lists two others, the two sample assuming
unequal variance and the paired t-test. We will end with some comments about rejecting the null
hypothesis.
Choosing between the t-test options
As the names imply each of the three forms of the t-test deal with different types of data
sets. The simplest distinction is between the equal and unequal variance tests. Both require that
the data be at least interval in nature, come from a normally distributed population, and be
independent of each other – that is, collected from different subjects.
The F-test for variance.
To determine if the population variances of two groups are statistically equal – in order to
correctly choose the equal variance version of the t-test – we use the F statistic, which is
calculated by dividing one variance by the other variance. If the outcome is less than 1.0, the
rejection region is in the left tail; if the value is greater than 1.0, the rejection region is in the
right tail. In either case, Excel provides the information we need.
To perform a hypothesis test for variance equality we use Excel’s F-Test Two-Sample for
Variances found in the Data Analysis section under the Data tab. The test set-up is very similar
to that of the t-test, entering data ranges, checking Labels box if they are included in the data
ranges, and identifying the start of the output range. The only unique element in this test is the
identification of our alpha level.
Since we are testing for equality of variances, we have a two sample test and the rejection
region is again in both tails. This means that our rejection region in each tail is 0.25. The F-test
id ...
Section 1 Data File DescriptionThe fictional data represents a te.docxbagotjesusa
Section 1: Data File Description
The fictional data represents a teacher's recording of student demographics and performance on quizzes and a final exam across three sections of the course. Each section consists of 35 students which totals to 105 students (sample size, N = 105). The dataset has 21 variables but in this case only two variables will be analyzed. These are: gender and gpa variables. The gender variable is categorized as nominal since the numbers are arbitrarily assigned to represent group membership. The ‘gpa’ variable belongs to the interval data group since it has a true zero which is meaningful. Alternatively, gender could be categorized as a categorical variable while ‘gpa’ ‘as a continuous variable.
Section 2: Testing Assumptions
1. Articulate the assumptions of the statistical test.
Paste SPSS output that tests those assumptions and interpret them. Properly integrate SPSS output wher1e appropriate. Do not string all output together at the beginning of the section.
All statistical tests operate under a set of assumptions. For the t test, there are three assumptions:
· The first assumption is independence of observations.
· The outcome variable Y is normally distributed.
· The variance of Y scores is approximately equal across groups (homogeneity of variance assumption)
Figure 1: histogram of GPA
The histogram above shows that the variable is probably not normally distributed. The bell shape is absent and two peaks are evident.
Table 1: descriptives
Descriptives
Statistic
Std. Error
GPA
Mean
2,78
,075
95% Confidence Interval for Mean
Lower Bound
2,63
Upper Bound
2,93
5% Trimmed Mean
2,80
Median
2,72
Variance
,583
Std. Deviation
,764
Minimum
1
Maximum
4
Range
3
Interquartile Range
1
Skewness
-,052
,236
Kurtosis
-,811
,467
With reference to the table 1 above, the ‘GPA’ variable is in the ideal range for skewness due to the fact that its absolute value for skewness are is less than .50 (approximately symmetric). The GPA variable is not ideal but acceptable since its kurtosis value is greater than .50 but less than 1. This new information gives mixed signals about the data being normal and only a normality test could iron out the differences.
Table 2: Normality test
Tests of Normality
Kolmogorov-Smirnova
Shapiro-Wilk
Statistic
df
Sig.
Statistic
df
Sig.
GPA
,091
105
,033
,956
105
,001
a. Lilliefors Significance Correction
Looking at the table above, the p-value is less than 0.05. Therefore, the null hypothesis is rejected and thus it can be concluded that the variable is not normally distributed. However, since the sample size is sufficiently large, one does not need to worry about this violation. On the other hand, Levene’s test provides a p =0.566 (table 3) meaning that the null hypothesis should not be rejected. Thus the homogeneity of variances assumption is not violated. It’s also assumed that proper research procedures that maintain independence of observations were followed. Two of the th.
PAGE
O&M Statistics – Inferential Statistics: Hypothesis Testing
Inferential Statistics
Hypothesis testing
Introduction
In this week, we transition from confidence intervals and interval estimates to hypothesis testing, the basis for inferential statistics. Inferential statistics means using a sample to draw a conclusion about an entire population. A test of hypothesis is a procedure to determine whether sample data provide sufficient evidence to support a position about a population. This position or claim is called the alternative or research hypothesis.
“It is a procedure based on sample evidence and probability theory to determine whether the hypothesis is a reasonable statement” (Mason & Lind, pg. 336).
This Week in Relation to the Course
Hypothesis testing is at the heart of research. In this week, we examine and practice a procedure to perform tests of hypotheses comparing a sample mean to a population mean and a test of hypotheses comparing two sample means.
The Five-Step Procedure for Hypothesis Testing (you need to show all 5 steps – these contain the same information you would find in a research paper – allows others to see how you arrived at your conclusion and provides a basis for subsequent research).
Step 1
State the null hypothesis – equating the population parameter to a specification. The null hypothesis is always one of status quo or no difference. We call the null hypothesis H0 (H sub zero). It is the hypothesis that contains an equality.
State the alternate hypothesis – The alternate is represented as H1 or HA (H sub one or H sub A). The alternate hypothesis is the exact opposite of the null hypothesis and represents the conclusion supported if the null is rejected. The alternate will not contain an equal sign of the population parameter.
Most of the time, researchers construct tests of hypothesis with the anticipation that the null hypothesis will be rejected.
Step 2
Select a level of significance (α) which will be used when finding critical value(s).
The level you choose (alpha) indicates how confident we wish to be when making the decision.
For example, a .05 alpha level means that we are 95% sure of the reliability of our findings, but there is still a 5% chance of being wrong (what is called the likelihood of committing a Type 1 error).
The level of significance is set by the individual performing the test. Common significance levels are .01, .05, and .10. It is important to always state what the chosen level of significance is.
Step 3
Identify the test statistic – this is the formula you use given the data in the scenario. Simply put, the test statistic may be a Z statistic, a t statistic, or some other distribution. Selection of the correct test statistic will depend on the nature of the data being tested (sample size, whether the population standard deviation is known, whether the data is known to be normally distributed).
The sampling distribution of the test statistic is divided into t.
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Hypothesis Testing Definitions A statistical hypothesi.docxwilcockiris
Hypothesis Testing
Definitions:
A statistical hypothesis is a guess about a population parameter. The guess may or not be
true.
The null hypothesis, written H0, is a statistical hypothesis that states that there is no
difference between a parameter and a specific value, or that there is no difference between
two parameters.
The alternative hypothesis, written H1 or HA, is a statistical hypothesis that specifies a
specific difference between a parameter and a specific value, or that there is a difference
between two parameters.
Example 1:
A medical researcher is interested in finding out whether a new medication will have
undesirable side effects. She is particularly concerned with the pulse rate of patients who
take the medication. The research question is, will the pulse rate increase, decrease, or
remain the same after a patient takes the medication?
Since the researcher knows that the mean pulse rate for the population under study is 82
beats per minute, the hypotheses for this study are:
H0: µ = 82
HA: µ ≠ 82
The null hypothesis specifies that the mean will remain unchanged and the alternative
hypothesis states that it will be different. This test is called a two-tailed test since the
possible side effects could be to raise or lower the pulse rate. Notice that this is a non
directional hypothesis. The rejection region lies in both tails. We divide the alpha in two
and place half in each tail.
Example 2:
An entrepreneur invents an additive to increase the life of an automobile battery. If the
mean lifetime of the automobile battery is 36 months, then his hypotheses are:
H0: µ ≤ 36
HA: µ > 36
Here, the entrepreneur is only interested in increasing the lifetime of the batteries, so his
alternative hypothesis is that the mean is greater than 36 months. The null hypothesis is
that the mean is less than or equal to 36 months. This test is one-tailed since the interest
is only in an increased lifetime. Notice that the direction of the inequality in the alternate
hypothesis points to the right, same as the area of the curve that forms the rejection
region.
Example 3:
A landlord who wants to lower heating bills in a large apartment complex is considering
using a new type of insulation. If the current average of the monthly heating bills is $78,
his hypotheses about heating costs with the new insulation are:
H0: µ ≥ 78
HA: µ < 78
This test is also a one-tailed test since the landlord is interested only in lowering heating
costs. Notice that the direction of the inequality in the alternate hypothesis points to the
left, same as the area of the curve that forms the rejection region.
Study Design:
After stating the hypotheses, the researcher’s next step is to design the study. In designing
the study, the researcher selects an appropriate statistical test, chooses a level of
significance, and formulates a plan for conducting the study..
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1. Michael Parent
ELMP 6005
Final Exam
September 30, 2007
Red=Suggested improvements from QE Review session with Dr. Walsh
QUESTION 1
Explain what a correlation coefficient is
a) Strength of Relationship:
There is a strong relationship between the two variables. Why? Use the Rule of
Thumb chart. What does the Pearson r mean?
b) Direction of the Relationship:
There is a negative relationship between the two variables. An increase in one
variable causes a decrease in another variable. What variables? Use the
language.
c) Interpretation of the level of significance:
The reported level of significance is .002. This means that if I were to conduct a
similar analysis, 2 out of 1000 times the direction of the line would flip-flop.
Therefore, this correlation can be confidently applied. Anything greater than .05,
then it cannot be confidently applied.
d) Interpretation of r2:
R2=.737881, which translates as 74% of the variation in averages can be
explained by variations in hours of television viewed nightly.
What can you say about the policy and practice after this analysis?
QUESTION 2
1. Research Problem:
The purpose of this study is to determine if new Math GEPA preparation activities are
making a difference in an elementary school. Is there a difference, on average, between
the mean scores of Group 1 and the mean score of Group 2?
2. State the Null Hypothesis:
Ho: μ1 - μ2 = 0 (No Difference between two samples means) Do not try to use
language if you can’t make perfect sense of it. Use only the equation!
3. State the Alternate Hypothesis:
H1 : μ1 - μ2 ≠ 0 (there is a difference between the two sample means)
4. Decision Rule:
I’ll reject Ho at .05 level of significance (95% level of confidence) if t is equal or more
positive than the critical value of t (.05) or if t is equal or more negative than 1.97,given
df = 200+. I will also reject Ho if the calculated significance of a 2-tailed test is equal or
less than .05. Include the type of t-test being used. (single, paired, etc.) The p value
2. rule is NECESSARY, not the t value.
5. Calculations:
Degrees of freedom df are calculated as the sum of samples minus 2 (df = n1 + n2 – 2).
According to the Table, at .05 level of significance and 200+ degrees of freedom, the
critical value of t, or tcrit = ±1.97.
The t value calculated from the data is 2.51. The p value is .025. This value doesn’t fall
between my 95% probability or .05 significance level. If it doesn’t, it falls in the critical
area.
6. Decision:
There is enough evidence for suspecting the null hypothesis. I reject the Null
Hypothesis Ho at .05 level of significance because the calculated t value of 2.251 is
more negative or more positive than the t critical value of 1.97. The calculated t value of
2.251 falls in the critical or rejection area. The probability of the t value 2.251 beyond
tcrit = 1.97 equals .05. It happens about 5 times in 100; this is a rare outcome signifying
that something special is probably happening in the underlying populations.
Also, I reject Ho at .05 level of significance because the 2-tailed level of significance
of .025 is less than the .05 level of significance, it’s below the threshold .05 that I require
as stated in the decision rule.
7. Interpretative Statement:
I found strong evidence to suggest that there is significant difference between the mean
scores of students in Group 1 and students in Group 2.
8. PPR:
Based on the data given and the evidence of statistical significance between the mean
scores of the two paired groups, it is apparent that the GEPA preparation activities are
working well. We might consider this pilot project a success and expand it to the control
group. I recommend that we continue to monitor the mean scores of students and
possible expand the preparation program to the lower grades.
QUESTION 3
1. Research Problem:
The purpose of this study is to determine how the students’ mean scores on the
NDGKS compare with the mean scores of all students in the state who took the
NDGKS.
2. State the Null Hypothesis:
Ho: μ = 280
3. State the Alternate Hypothesis:
3. H1 : μ ≠ 280
4. Decision Rule:
To be 95% confident that the change in the one sample mean score didn’t happen by
random chance, I’ll reject Ho at .05 level of significance (95% level of confidence) if t
equals or is greater than the critical value of t 1.98, given df = n – 1: 109 – 1 = 100. I
will also reject Ho if the calculated significance of a 2-tailed test is equal or less than .
05. Include the type of t-test being used. (single, paired, etc.) The p value rule is
NECESSARY, not the t value.
5. Calculations:
The SPSS has provided the same t value = -15.827, and a two-tailed level of
significance (p) of .000. According to the Table at .05 level of significance and 100
degrees of freedom, the critical value of t is 1.98 .
This value doesn’t fall between my 95% probability. It falls in the critical area.
6. Decision:
Based on the data given, I reject the Null Hypothesis Ho at .05 level of significance
because the calculated t value is greater than the t critical value. I also reject Ho at .05
level of significance because the 2-tailed level of significance is less than the .05 level of
significance that it’s required as stated in the decision rule.
7. Interpretative Statement:
There is strong evidence to reject the null hypothesis. There is a significant difference
between the students’ mean score at this high school with the mean score statewide.
The 2-tailed level of significance suggests that a sample mean score of 280 is unlikely
to occur. The 2-tailed level of significance of .000 is below the threshold .05.
The difference between the hypothesized mean of 280 and the one-sample mean of
235.40 is significant; it didn’t happen by random chance, something happened to the
underlying population.
8. PPR:
The data provided is evidence to suggest that the students at this high school are
performing significantly poorer that other students in North Dakota. The principal might
consider implementing an improvement plan to improve students’ mean scores. I
suggest that the school continue to monitor students’ mean scores during any reform
measure.
QUESTION 4
4. 1. Research Problem:
Is there a difference between the doctor’s SCORES? Is there a significant difference
between those who took the pil and those who did not? The purpose of this study is to
determine if there is a difference between doctors’ abilities to focus during surgery after
having ingested the “Alert” pill.
2. State the Null Hypothesis:
Ho: μD = 0 (no difference between each pair of mean scores in two repeated samples)
The null hypothesis states that the doctors did not show any statistical difference in
ability to focus after having ingested the “Alert” pill. Do not try to use language if you
can’t make perfect sense of it. Use only the equation!
3. State the Alternate Hypothesis:
H1: μD ≠ 0
The alternate hypothesis states that the doctors did show a statistical difference in
ability to focus after having ingested the “Alert” pill. Do not try to use language if you
can’t make perfect sense of it. Use only the equation!
4. Decision Rule:
The reported mean is 5.02. The calculated t value provided in the SPSS output is 1.577
I’ll reject Ho at .05 level of significance (95% level of confidence) if t equals or is greater
than 2.01, the critical value of t, given df = n – 1 =50 – 1 = 49.
I will also reject Ho if the calculated significance of a 2-tailed test is equal or less than .
05. Include the type of t-test being used. (single, paired, etc.) The p value rule is
NECESSARY, not the t value.
5. Calculations:
The SPSS has provided me with the t value of 1.577 and the two-tailed level of
significance (p) of .121. This t value 1.577 does fall between 95% probability. It falls
within the range of acceptance.
6. Decision:
I fail to reject the Null Hypothesis Ho at .05 level of significance because the calculated t
value of 1.577 is not greater than the t critical value of 2.01. The calculated t value falls
in the normal area.
Also, I fail to reject Ho at .05 level of significance because the 2-tailed level of
significance of .121 is greater than the .05 level of significance that I require as stated in
the decision rule, and commonly accepted as the threshold of significance for social
science research.
7. Interpretative Statement:
I found evidence that there is no difference between the paired means of physician’s
who took the “Alert” pill pre-test and the pill’s post-test.
5. 8. PPR:
The company should consider further testing to determine if the “Alert” pill can have a
significant impact on surgeons. The company might reconsider its claim that their pill
improves focus. Because this is a medical test, the company should consider
establishing a level of significance at the .01 level on future studies.