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.
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Objectives
#1: Obtain a Point Estimate for the population Mean
#2: Understand the properties of Student's t-Distribution
#3: determine t-Values
#4: create and Interpret Confidence Intervals for a Population Mean
#5: create sample size to estimate a population mean (within a margin
of error)
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01. Formulasand Explanations
• Point estimates are educated guesses about the population
parameters based on sample data
• Point estimate of the population mean µ is the sample mean x
#1: Obtain a Point Estimate for the population Mean
µ = x
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• Origin of Student's T-Distribution
◦ 1900s - William Gosset experiments at Guinness brewery, limited to
small data sets
◦ Developed a model to account for additional variability in confidence
intervals to include population mean.
◦ Published results under pseudonym "Student" in Student's
t-distribution.
• Why it was a game-changer: The t-distribution made statistical
inference more accurate for small samples and is used extensively in
hypothesis testing, confidence interval construction, and many more
statistical analyses.
01. Formulasand Explanations
#2: Understand the properties of Student's t-Distribution
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• Student's t-distribution formula
◦ n= simple random sample size from a normal distribution population
◦ x= Sample mean
◦ µ = Population mean
◦ s= sample standard deviation
• Similar to z-score, the t-statistic represents the
number of sample standard errors x is from µ.
01. Formulasand Explanations
#2: Understand the properties of Student's t-Distribution
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Six Properties of Student's t-Distribution
1. The t-distribution varies based on the degrees of freedom.
2. It's symmetric and centered around 0.
3. The total area under the curve is always 1. The area to the right of 0 equals
the area to the left of 0, which is 0.5.
4. As degrees of freedom increase or decrease without bound, the t-distribution
approaches but never reaches zero.
5. The tails of the t-distribution have slightly more area than the tails of the
standard normal distribution due to using the sample standard deviation as
an estimate of the population standard deviation.
6. As sample size increases, the t-distribution curve approaches the standard
normal distribution curve.
01. Formulasand Explanations
#2: Understand the properties of Student's t-Distribution
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• Let tɑ = the t-value with an area under the t-distribution to the
right of it as ɑ.
• The shape of the t-distribution is influenced by the sample
size n, and the degrees of freedom, denoted by df.
• The value of tɑ depends on both ɑ and the degrees of freedom
df.
• In the table, the degrees of freedom are listed in the left
column, and the area under the t-distribution to the right of
certain t-values is represented in the top row.
• If df is not on the chart, choose the closest df number. If df is
greater than 1000, use z-values because it starts to behave like
a standard normal distribution as n increases.
01. Formulasand Explanations
#3: determine t-Values
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Constructing a Confidence Interval for a population mean
• Given: Simple random sample data, size n is small relative to the population size (n ≤ 0.05N), data is from a
normal distribution population / large sample size
• "t-interval": ( 1 - ɑ) * 100% Confidence Interval for µ
• If the requirements to compute a t-interval are not met
◦ Increase the sample size, use nonparametric procedures, or use resampling methods
01. Formulasand Explanations
#4: create and Interpret Confidence Intervals for a Population Mean
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• Formula: Margin of error (E) in constructing confidence
interval about population mean is E = t*(s/√n)
• If sample size (n) is not given to determine degrees of
freedom:
◦ t-distribution approaches standard normal z-distribution
as sample size increases.
◦ Use z-scores for large samples or approximate
t-distribution with z-distribution.
◦ Substitute z-score for t and use sample standard
deviation (s) to determine sample size (n). E = z*(s/√n)
• Allows determining necessary sample size without prior
knowledge of population standard deviation (σ).
01. Formulasand Explanations
#5: create the required sample size to estimate a population mean within a given
margin of error
