This document provides an introduction to the t-statistic, which is used to test hypotheses about population means when the population standard deviation is unknown. It describes how the t-statistic is calculated using the sample standard deviation rather than the unknown population standard deviation. It also explains that the t-distribution, which the t-statistic is compared to, depends on the degrees of freedom and becomes closer to a normal distribution as the degrees of freedom increase. The document outlines the four-step process for a hypothesis test using the t-statistic and describes how effect size can be estimated.
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Linearity concept of significance, standard deviation, chi square test, students T- test, ANOVA test , pharmaceutical science, statistical analysis, statistical methods, optimization technique, modern pharmaceutics, pharmaceutics, mpharm 1 unit i sem, 1 year m
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linearity concept of significance, standard deviation, chi square test, stude...KavyasriPuttamreddy
Linearity concept of significance, standard deviation, chi square test, students T- test, ANOVA test , pharmaceutical science, statistical analysis, statistical methods, optimization technique, modern pharmaceutics, pharmaceutics, mpharm 1 unit i sem, 1 year m
pharm, applications of chi square test, application of standard deviation , pharmacy, method to compare dissolution profile, statistical analysis of dissolution profile, important statical analysis, m. pharmacy, graphical representation of standard deviation, graph of chi square test, graph of T test , graph of ANOVA test ,formulation of t test, formulation of chi square test, formula of standard deviation.
A sample design is a definite plan for obtaining a sample from a given population. Researcher must select/prepare a sample design which should be reliable and appropriate for his research study.
What is the need of the t Distribution According to the central.pdfaravlitraders2012
What is the need of the t Distribution?
According to the central limit theorem, the sampling distribution of a statistic (like a sample
mean) will follow a normal distribution, as long as the sample size is sufficiently large.
Therefore, when we know the standard deviation of the population, we can compute a z-score,
and use the normal distribution to evaluate probabilities with the sample mean.
But sample sizes are sometimes small, and often we do not know the standard deviation of the
population. When either of these problems occur, statisticians rely on the distribution of the t
statistic (also known as the t score), whose values are given by:
t = [ x - ? ] / [ s / sqrt( n ) ]
where x is the sample mean, ? is the population mean, s is the standard deviation of the sample,
and n is the sample size. The distribution of the t statistic is called the t distribution or the
Student t distribution.
The t distribution allows us to conduct statistical analyses on certain data sets that are not
appropriate for analysis, using the normal distribution.
Degrees of Freedom
There are actually many different t distributions. The particular form of the t distribution is
determined by its degrees of freedom. The degrees of freedom refers to the number of
independent observations in a set of data.
When estimating a mean score or a proportion from a single sample, the number of independent
observations is equal to the sample size minus one. Hence, the distribution of the t statistic from
samples of size 8 would be described by a t distribution having 8 - 1 or 7 degrees of freedom.
Similarly, a t distribution having 15 degrees of freedom would be used with a sample of size 16.
Solution
What is the need of the t Distribution?
According to the central limit theorem, the sampling distribution of a statistic (like a sample
mean) will follow a normal distribution, as long as the sample size is sufficiently large.
Therefore, when we know the standard deviation of the population, we can compute a z-score,
and use the normal distribution to evaluate probabilities with the sample mean.
But sample sizes are sometimes small, and often we do not know the standard deviation of the
population. When either of these problems occur, statisticians rely on the distribution of the t
statistic (also known as the t score), whose values are given by:
t = [ x - ? ] / [ s / sqrt( n ) ]
where x is the sample mean, ? is the population mean, s is the standard deviation of the sample,
and n is the sample size. The distribution of the t statistic is called the t distribution or the
Student t distribution.
The t distribution allows us to conduct statistical analyses on certain data sets that are not
appropriate for analysis, using the normal distribution.
Degrees of Freedom
There are actually many different t distributions. The particular form of the t distribution is
determined by its degrees of freedom. The degrees of freedom refers to the number of
independent observations in a set of da.
Estimating a Population Mean in Confidence IntervalsCrystal Hollis
This presentation covers Chapter 9.2 in Math 1342 Statistics class (Dallas College) from the textbook, Interactive Statistics: Informed Decisions Using Data 3/e, by Michael J. Sullivan and George Woodbury. It introduces the concept of point estimates, confidence intervals, and the t-distribution in statistical inference. Point estimates provide single values that estimate population parameters, such as the mean. Confidence intervals give a range of values within which the true population parameter is likely to lie, along with a level of confidence. The t-distribution, introduced by William Gosset (aka "Student"), is similar to the standard normal distribution but accounts for the variability introduced by using sample data. It is particularly useful when the population standard deviation is unknown or the sample size is small. The chapter also covers determining t-values and sample sizes necessary for estimating population means with a specified margin of error. Overall, these concepts form the foundation of statistical inference, enabling researchers to make informed decisions and draw conclusions about populations based on sample data.
Zoonar/Thinkstock
chapter 5
The t-Test
Learning Objectives
After reading this chapter, you will be able to. . .
1. explain the advantage of the one-sample t-test over the z-test.
2. compare the one-sample t-test to the independent t-test.
3. distinguish between one-sample and one-tailed t-tests.
4. explain hypothesis testing in statistical analysis.
5. determine practical significance.
6. construct a confidence interval for the difference between the means.
7. discuss research applications for the t-tests.
8. present results of a t-test and draw conclusions based on hypotheses.
9. interpret t-test results and report in APA format.
10. discuss nonparametric Mann-Whitney U-test compared to the t-test.
CN
CO_LO
CO_TX
CO_NL
CT
CO_CRD
suk85842_05_c05.indd 137 10/23/13 1:17 PM
CHAPTER 5Section 5.1 Estimating the Standard Error of the Mean
The z-test (Chapter 4) is more than just an expansion of the z score to groups because it introduces statistical significance. Those who work with quantitative data need to
be able to distinguish between outcomes that probably occurred by chance and those that
are likely to emerge each time the data is gathered and analyzed. When data indicates
that a group of clients, each grieving over the loss of a loved one, becomes more positive
and peaceful with time, the therapist needs to know whether this would have happened
anyway with the passage of time, or whether it has something to do with the treatment
the therapist provided. The z-test answers such questions.
The z-test has important limitations. The greatest difficulty is the need to have a value
for the population standard error of the mean, sM. It is not the type of information that
tends to come up as a matter of course, and it can be fairly difficult to calculate when the
researcher does not have access to population data.
The second limitation is that the z-test allows only one type of comparison: a sample to a
population. What if two therapists want to compare their respective groups of grieving cli-
ents to see if one type of therapy for grief counseling is better than the other? The z-test pro-
vides nowhere to go with such questions, which is where William Sealy Gosset comes in.
Gosset worked for Guinness Brewing during the early part of the 20th century. Part of
his responsibility was quality control, and he studied ways to make sure that day-to-
day brewing remained consistent with Guinness’s standards. A man of remarkable ability,
Gosset devised procedures for quantifying product quality and then testing the consis-
tency of the quality over time. To this end, he developed t-tests.
Gosset had some sense of how important the t-tests were and wanted to publish so that
others could benefit. The roadblock was a no-publishing policy at Guinness, a policy that
was begun after one of Gosset’s predecessors published what the Guinness people consid-
ered trade secrets. Although he felt that Guinness wo ...
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
A sample design is a definite plan for obtaining a sample from a given population. Researcher must select/prepare a sample design which should be reliable and appropriate for his research study.
What is the need of the t Distribution According to the central.pdfaravlitraders2012
What is the need of the t Distribution?
According to the central limit theorem, the sampling distribution of a statistic (like a sample
mean) will follow a normal distribution, as long as the sample size is sufficiently large.
Therefore, when we know the standard deviation of the population, we can compute a z-score,
and use the normal distribution to evaluate probabilities with the sample mean.
But sample sizes are sometimes small, and often we do not know the standard deviation of the
population. When either of these problems occur, statisticians rely on the distribution of the t
statistic (also known as the t score), whose values are given by:
t = [ x - ? ] / [ s / sqrt( n ) ]
where x is the sample mean, ? is the population mean, s is the standard deviation of the sample,
and n is the sample size. The distribution of the t statistic is called the t distribution or the
Student t distribution.
The t distribution allows us to conduct statistical analyses on certain data sets that are not
appropriate for analysis, using the normal distribution.
Degrees of Freedom
There are actually many different t distributions. The particular form of the t distribution is
determined by its degrees of freedom. The degrees of freedom refers to the number of
independent observations in a set of data.
When estimating a mean score or a proportion from a single sample, the number of independent
observations is equal to the sample size minus one. Hence, the distribution of the t statistic from
samples of size 8 would be described by a t distribution having 8 - 1 or 7 degrees of freedom.
Similarly, a t distribution having 15 degrees of freedom would be used with a sample of size 16.
Solution
What is the need of the t Distribution?
According to the central limit theorem, the sampling distribution of a statistic (like a sample
mean) will follow a normal distribution, as long as the sample size is sufficiently large.
Therefore, when we know the standard deviation of the population, we can compute a z-score,
and use the normal distribution to evaluate probabilities with the sample mean.
But sample sizes are sometimes small, and often we do not know the standard deviation of the
population. When either of these problems occur, statisticians rely on the distribution of the t
statistic (also known as the t score), whose values are given by:
t = [ x - ? ] / [ s / sqrt( n ) ]
where x is the sample mean, ? is the population mean, s is the standard deviation of the sample,
and n is the sample size. The distribution of the t statistic is called the t distribution or the
Student t distribution.
The t distribution allows us to conduct statistical analyses on certain data sets that are not
appropriate for analysis, using the normal distribution.
Degrees of Freedom
There are actually many different t distributions. The particular form of the t distribution is
determined by its degrees of freedom. The degrees of freedom refers to the number of
independent observations in a set of da.
Estimating a Population Mean in Confidence IntervalsCrystal Hollis
This presentation covers Chapter 9.2 in Math 1342 Statistics class (Dallas College) from the textbook, Interactive Statistics: Informed Decisions Using Data 3/e, by Michael J. Sullivan and George Woodbury. It introduces the concept of point estimates, confidence intervals, and the t-distribution in statistical inference. Point estimates provide single values that estimate population parameters, such as the mean. Confidence intervals give a range of values within which the true population parameter is likely to lie, along with a level of confidence. The t-distribution, introduced by William Gosset (aka "Student"), is similar to the standard normal distribution but accounts for the variability introduced by using sample data. It is particularly useful when the population standard deviation is unknown or the sample size is small. The chapter also covers determining t-values and sample sizes necessary for estimating population means with a specified margin of error. Overall, these concepts form the foundation of statistical inference, enabling researchers to make informed decisions and draw conclusions about populations based on sample data.
Zoonar/Thinkstock
chapter 5
The t-Test
Learning Objectives
After reading this chapter, you will be able to. . .
1. explain the advantage of the one-sample t-test over the z-test.
2. compare the one-sample t-test to the independent t-test.
3. distinguish between one-sample and one-tailed t-tests.
4. explain hypothesis testing in statistical analysis.
5. determine practical significance.
6. construct a confidence interval for the difference between the means.
7. discuss research applications for the t-tests.
8. present results of a t-test and draw conclusions based on hypotheses.
9. interpret t-test results and report in APA format.
10. discuss nonparametric Mann-Whitney U-test compared to the t-test.
CN
CO_LO
CO_TX
CO_NL
CT
CO_CRD
suk85842_05_c05.indd 137 10/23/13 1:17 PM
CHAPTER 5Section 5.1 Estimating the Standard Error of the Mean
The z-test (Chapter 4) is more than just an expansion of the z score to groups because it introduces statistical significance. Those who work with quantitative data need to
be able to distinguish between outcomes that probably occurred by chance and those that
are likely to emerge each time the data is gathered and analyzed. When data indicates
that a group of clients, each grieving over the loss of a loved one, becomes more positive
and peaceful with time, the therapist needs to know whether this would have happened
anyway with the passage of time, or whether it has something to do with the treatment
the therapist provided. The z-test answers such questions.
The z-test has important limitations. The greatest difficulty is the need to have a value
for the population standard error of the mean, sM. It is not the type of information that
tends to come up as a matter of course, and it can be fairly difficult to calculate when the
researcher does not have access to population data.
The second limitation is that the z-test allows only one type of comparison: a sample to a
population. What if two therapists want to compare their respective groups of grieving cli-
ents to see if one type of therapy for grief counseling is better than the other? The z-test pro-
vides nowhere to go with such questions, which is where William Sealy Gosset comes in.
Gosset worked for Guinness Brewing during the early part of the 20th century. Part of
his responsibility was quality control, and he studied ways to make sure that day-to-
day brewing remained consistent with Guinness’s standards. A man of remarkable ability,
Gosset devised procedures for quantifying product quality and then testing the consis-
tency of the quality over time. To this end, he developed t-tests.
Gosset had some sense of how important the t-tests were and wanted to publish so that
others could benefit. The roadblock was a no-publishing policy at Guinness, a policy that
was begun after one of Gosset’s predecessors published what the Guinness people consid-
ered trade secrets. Although he felt that Guinness wo ...
A Strategic Approach: GenAI in EducationPeter Windle
Artificial Intelligence (AI) technologies such as Generative AI, Image Generators and Large Language Models have had a dramatic impact on teaching, learning and assessment over the past 18 months. The most immediate threat AI posed was to Academic Integrity with Higher Education Institutes (HEIs) focusing their efforts on combating the use of GenAI in assessment. Guidelines were developed for staff and students, policies put in place too. Innovative educators have forged paths in the use of Generative AI for teaching, learning and assessments leading to pockets of transformation springing up across HEIs, often with little or no top-down guidance, support or direction.
This Gasta posits a strategic approach to integrating AI into HEIs to prepare staff, students and the curriculum for an evolving world and workplace. We will highlight the advantages of working with these technologies beyond the realm of teaching, learning and assessment by considering prompt engineering skills, industry impact, curriculum changes, and the need for staff upskilling. In contrast, not engaging strategically with Generative AI poses risks, including falling behind peers, missed opportunities and failing to ensure our graduates remain employable. The rapid evolution of AI technologies necessitates a proactive and strategic approach if we are to remain relevant.
Biological screening of herbal drugs: Introduction and Need for
Phyto-Pharmacological Screening, New Strategies for evaluating
Natural Products, In vitro evaluation techniques for Antioxidants, Antimicrobial and Anticancer drugs. In vivo evaluation techniques
for Anti-inflammatory, Antiulcer, Anticancer, Wound healing, Antidiabetic, Hepatoprotective, Cardio protective, Diuretics and
Antifertility, Toxicity studies as per OECD guidelines
Model Attribute Check Company Auto PropertyCeline George
In Odoo, the multi-company feature allows you to manage multiple companies within a single Odoo database instance. Each company can have its own configurations while still sharing common resources such as products, customers, and suppliers.
Welcome to TechSoup New Member Orientation and Q&A (May 2024).pdfTechSoup
In this webinar you will learn how your organization can access TechSoup's wide variety of product discount and donation programs. From hardware to software, we'll give you a tour of the tools available to help your nonprofit with productivity, collaboration, financial management, donor tracking, security, and more.
The French Revolution, which began in 1789, was a period of radical social and political upheaval in France. It marked the decline of absolute monarchies, the rise of secular and democratic republics, and the eventual rise of Napoleon Bonaparte. This revolutionary period is crucial in understanding the transition from feudalism to modernity in Europe.
For more information, visit-www.vavaclasses.com
Read| The latest issue of The Challenger is here! We are thrilled to announce that our school paper has qualified for the NATIONAL SCHOOLS PRESS CONFERENCE (NSPC) 2024. Thank you for your unwavering support and trust. Dive into the stories that made us stand out!
2. 2
The t Statistic
• The t statistic allows researchers to use sample
data to test hypotheses about an unknown
population mean.
• The particular advantage of the t statistic, is that
the t statistic does not require any knowledge of
the population standard deviation.
• Thus, the t statistic can be used to test
hypotheses about a completely unknown
population; that is, both μ and σ are unknown,
and the only available information about the
population comes from the sample.
3. 3
The t Statistic (cont.)
• All that is required for a hypothesis test
with t is a sample and a reasonable
hypothesis about the population mean.
• There are two general situations where
this type of hypothesis test is used:
4. 4
The t Statistic (cont.)
1.The t statistic is used when a researcher wants
to determine whether or not a treatment causes
a change in a population mean. In this case you
must know the value of μ for the original,
untreated population. A sample is obtained from
the population and the treatment is administered
to the sample. If the resulting sample mean is
significantly different from the original population
mean, you can conclude that the treatment has
a significant effect.
5. 5
The t Statistic (cont.)
2. Occasionally a theory or other prediction
will provide a hypothesized value for an
unknown population mean. A sample is
then obtained from the population and the
t statistic is used to compare the actual
sample mean with the hypothesized
population mean. A significant difference
indicates that the hypothesized value for μ
should be rejected.
6. 6
The Estimated Standard Error and
the t Statistic
• Whenever a sample is obtained from a
population you expect to find some discrepancy
or "error" between the sample mean and the
population mean.
• This general phenomenon is known as
sampling error.
• The goal for a hypothesis test is to evaluate the
significance of the observed discrepancy
between a sample mean and the population
mean.
7. 7
The Estimated Standard Error and
the t Statistic (cont.)
The hypothesis test attempts to decide between
the following two alternatives:
1.Is it reasonable that the discrepancy between M
and μ is simply due to sampling error and not
the result of a treatment effect?
2.Is the discrepancy between M and μ more than
would be expected by sampling error alone?
That is, is the sample mean significantly different
from the population mean?
8. 8
The Estimated Standard Error and
the t Statistic (cont.)
• The critical first step for the t statistic hypothesis
test is to calculate exactly how much difference
between M and μ is reasonable to expect.
• However, because the population standard
deviation is unknown, it is impossible to compute
the standard error of M as we did with z-scores
in Chapter 8.
• Therefore, the t statistic requires that you use
the sample data to compute an estimated
standard error of M.
9. 9
The Estimated Standard Error and
the t Statistic (cont.)
• This calculation defines standard error exactly
as it was defined in Chapters 7 and 8, but now
we must use the sample variance, s2, in place of
the unknown population variance, σ2 (or use
sample standard deviation, s, in place of the
unknown population standard deviation, σ).
• The resulting formula for estimated standard
error is
s2 s
sM = ── or sM = ──
n n
10. 10
The Estimated Standard Error and
the t Statistic (cont.)
• The t statistic (like the z-score) forms a ratio.
• The top of the ratio contains the obtained
difference between the sample mean and the
hypothesized population mean.
• The bottom of the ratio is the standard error
which measures how much difference is
expected by chance.
obtained difference M μ
t = ───────────── = ─────
standard error sM
11. 11
The Estimated Standard Error and
the t Statistic (cont.)
• A large value for t (a large ratio) indicates
that the obtained difference between the
data and the hypothesis is greater than
would be expected if the treatment has no
effect.
12. 12
The t Distributions and
Degrees of Freedom
• You can think of the t statistic as an "estimated
z-score."
• The estimation comes from the fact that we are
using the sample variance to estimate the
unknown population variance.
• With a large sample, the estimation is very good
and the t statistic will be very similar to a z-
score.
• With small samples, however, the t statistic will
provide a relatively poor estimate of z.
13.
14. 14
The t Distributions and
Degrees of Freedom (cont.)
• The value of degrees of freedom, df = n - 1, is
used to describe how well the t statistic
represents a z-score.
• Also, the value of df will determine how well the
distribution of t approximates a normal
distribution.
• For large values of df, the t distribution will be
nearly normal, but with small values for df, the t
distribution will be flatter and more spread out
than a normal distribution.
15. 15
The t Distributions and
Degrees of Freedom (cont.)
• To evaluate the t statistic from a hypothesis test,
you must select an α level, find the value of df
for the t statistic, and consult the t distribution
table.
• If the obtained t statistic is larger than the critical
value from the table, you can reject the null
hypothesis.
• In this case, you have demonstrated that the
obtained difference between the data and the
hypothesis (numerator of the ratio) is
significantly larger than the difference that would
be expected if there was no treatment effect (the
standard error in the denominator).
16. 16
Hypothesis Tests with the t Statistic
The hypothesis test with a t statistic follows the same
four-step procedure that was used with z-score tests:
1. State the hypotheses and select a value for α.
(Note: The null hypothesis always states a specific
value for μ.)
2. Locate the critical region. (Note: You must find the
value for df and use the t distribution table.)
3. Calculate the test statistic.
4. Make a decision (Either "reject" or "fail to reject"
the null hypothesis).
17.
18. 18
Measuring Effect Size with the t
Statistic
• Because the significance of a treatment
effect is determined partially by the size of
the effect and partially by the size of the
sample, you cannot assume that a
significant effect is also a large effect.
• Therefore, it is recommended that a
measure of effect size be computed along
with the hypothesis test.
19. 19
Measuring Effect Size with the t
Statistic (cont.)
• For the t test it is possible to compute an
estimate of Cohen=s d just as we did for
the z-score test in Chapter 8. The only
change is that we now use the sample
standard deviation instead of the
population value (which is unknown).
mean difference M μ
estimated Cohen=s d = ─────────── = ──────
standard deviation s
20.
21. 21
Measuring Effect Size with the t
Statistic (cont.)
• As before, Cohen=s d measures the size of the
treatment effect in terms of the standard
deviation.
• With a t test it is also possible to measure effect
size by computing the percentage of variance
accounted for by the treatment.
• This measure is based on the idea that the
treatment causes the scores to change, which
contributes to the observed variability in the
data.
22. 22
Measuring Effect Size with the t
Statistic (cont.)
• By measuring the amount of variability that
can be attributed to the treatment, we
obtain a measure of the size of the
treatment effect. For the t statistic
hypothesis test,
t2
percentage of variance accounted for = r2 = ─────
t2 + df