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t Test for One Sample
Why use a t test?
 The sampling distribution of t represents the
distribution that would be obtained if a value of t were
calculated for each sample mean for all possible
random samples of a given size from some population.
 Just like the z sampling distribution!!
Why use a t test?
The figure indicates where the control and treatment group means are
located. The question the t-test addresses is whether the means are
statistically different.
Why use a t test?
When we are looking at the differences between scores for two groups, we
have to judge the difference between their means relative to the spread or
variability of their scores.
Gas Mileage Investigation (269)
Hypothesis test summary for t test for a population mean
Research Problem
Does the mean gas mileage for some population of cars drop below the legally required minimum of 45 mpg?
Statistical Hypothesis
H0: µ ≥ 45
H1: µ < 45
Decision Rule
Reject H0 at the .01 level of significance if t≤ -3.365 (from Table B, Appendix C, given df = n-1 = 6 – 1 = 5).
Calculations
Given X = 43, sx = 0.89 (See table 13.1 on page 275 for computations)
t = 43 – 45 = -2.25
0.89
Decision
Retain H0 at .01 level of significance because t = -.2.25 is less negative than -3.365
Interpretation
The population mean gas mileage could equal the required 45 mpg or more. The manufacturer shouldn’t be penalized.
Table of critical values for t
 Table B of Appendix C page 520
Missing df
 If the desired number of degrees of freedom doesn’t
appear in the df column of Table B, use the row in the
table with the next smallest number of degrees of
freedom.
t ratio for a single population mean
t = sample mean – hypothesized population mean
estimated standard error
or X - µhyp
sx
Estimating the standard error
 If the population standard deviation is unknown, it
must be estimated from the sample.
sx = _s_
√n
Where s = SS
√df
Sum of squares for X
SSx = Σ(X)2 – (ΣX)2
n
Calculations for a t test
1.
2.
3.
4.
5.
6.
7.
8.
9.
10.
Define n
Sum all X scores
Find mean of X
Square each X score
Sum the squared Xs
Calculate SS
Calculate s
Calculate sx
Assign value to µhyp
Calculate t
In class
 Progress Check 13.3
Confidence interval (p 276)
X ± (tconf )(sx)
 Find value of tconf in Table B
Homework
 Review question 13.7, a and b, due Tuesday 4/7
 Show your work, answers in the back of the book.
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