One-sample
and
Two-sample
t-test
When population standard deviation (∂) is NOT
KNOWN
When ∂ ia NOT
given, we use the tstatistic
x- µ
____
t=
s/√n
Example: performing a one-sample tDiet colas use artificial sweeteners to avoid sugar.
test
These sweeteners gradually lose their sweetness over
time. Manufacturers therefore test new colas for loss
of sweetness before marketing them. Trained tasters
sip the cola along with drinks of standard sweetness
and score the cola on a “sweetness scale” of 1 to 10.
The cola is then stored for a month at high
temperature to imitate the effect of four months'
storage at room temperature. Each taster scores the
cola again after storage. Our data are the differences
(score before storage minus score after storage) in the
tasters' scores. The bigger these differences, the
bigger the loss of sweetness. Here are the sweetness
losses for a new cola, as measured by 10 trained
tasters:
I. Hypotheses:
Ho: the mean sweetness loss is
µ=0
Ha: the mean sweetness loss is positive µ>0
II. Conditions:
SRS: the tasters are considered to be TRAINED so
we can rely that the 10 tasters are taken from
random. This is a matter of judgement
Normality: the boxplot shows skewness so we will
proceed with caution.
Independence: since N≥100 taken from the entire
population, we can view these samples as
III Calculations
One-sample t-test
t=
1.02 - 0
1.196/√10
= 2.70
df = 9
p-value = .0123
Syntax: tcdf (t-value, n, df)
IV. Conclusion
Since the p-value of .0123 is small, it
gives a strong evidence against the
null-hypothesis making our test
significant. Therefore we have
reason to believe that the mean
sweetness
loss
is
positive.
However, since our sampling
distribution is skewed, we will
proceed with caution.
Your turn!
Healthy bones Here are estimates of the daily
intakes of calcium (in milligrams) for an SRS of 38
women between the ages of 18 and 24 years who
participated in a study of women's bone health:
Suppose that the recommended daily allowance (RDA) of
calcium for women in this age range is 1200 milligrams.
Doctors involved in the study suspected that participating
subjects had significantly lower calcium intakes than the
RDA.Test the doctors' claim at the α = 0.05 significance
level
Ho: the mean daily calcium intake is µ=1200 mg
Ha: the mean daily calcium intake is µ<1200 mg
SRS: taken from SRS of 29 women ages 1824
Normality: boxplot shows distribution is
skewed to the right.
Independence: N≥10(39)
One-sample t-statistic:
t-value = -3.95
p-value =
.00017
Since our p-value is less than the 5% significance level, we will reject
the null hypothesis making our test significant. Therefore the mean
daily intake of calcium of women is significantly less than the RDA
recommendation. However, we will proceed with caution since our
distribution is skewed.
Two-sample
t-test
We hear that listening to Mozart improves
students' performance on tests. Perhaps
pleasant odors have a similar effect. To test
this idea, 21 subjects worked a paper-andpencil maze while wearing a mask. The
mask was either unscented or carried a floral
scent. The response variable is their average
time on three trials. Each subject worked the
maze with both masks, in a random order. Is
there an improvement in time of solving the
maze with the scented mask? Carry out an
appropriate test to prove this claim.
Text
Ho: the mean difference between wearing scented
mask and unscented mask is µ=0
Ha: the mean difference between wearing scented
mask and unscented mask is positive µ>0
SRS: sample came from an SRS of 21 subjects
Normality: The boxplot shows an approx. symmetrical
distribution
Independence: We are willing to assume that the 21
subjects' are independent observations.
one-sample
t-statistic:
t-value =
.349
p-value =
.25
Since our p-value is greater than the 5% or even 10%
significance level, we will NOT reject the null hypothesis
making our test NOT significant. Therefore the floral
scented mask did not help improve the average time in
solving the maze.
homework
Right versus left: The design of controls and instruments affects how easily
people can use them. A student project investigated this effect by asking 25
right-handed students to turn a knob (with their right hands) that moved an
indicator by screw action in seconds.The project designers hoped to show that
right-handed people find right-hand threads easier to use. Carry out a
significance test at the 5% significance level to investigate this claim.
Answer:
Ho: the mean difference between the left hand and
the right hand speed is µ=0
Ha: the mean difference between the left hand and
the right hand speed is µ<0
T-value = -2.9037
P-value=.0039
Since our p-value is less than the 5% significance level, we
will reject the null hypothesis making our test significant.
Therefore the mean difference between the left hand the the
right hand is negative. However, we will proceed with caution
since our distribution is skewed.