Download t statistic

PRED 354 TEACH. PROBILITY &
STATIS. FOR PRIMARY MATH
Lesson 8
Introduction to the t statistics
t statistic
Use sample mean to test hypotheses about a
population.
X
X
z
1. It is expected more or less to approximate its
population mean (µ).
2. The standard error provides a measure of how
well a sample mean approximates the population
mean.
3. Compare sample mean with the hypothesized
population mean by computing a z-score test
statistic.
t statistic: a substitute for z
When the variability for the population is not
known, sample variability is used in its
place.
SS SS
s 

n  1 df
s
sX 
n
2
Estimated
standard error
It is used as an
estimate of  X
when the value of σ
is unknown.
t statistic: a substitute for z
X 
t
sX
Both z and t
formulas are used
for hypothesis
testing.
When you know the value σ2, use zscore. If σ2 is unknown, use t statistic.
The larger the value of df is , the more closely the t
distribution approximates a normal distribution.
Determining probabilities for t
distributions
t distribution table is used to find probabilities for t
statistics.
TABLE….
Hypothesis tests with the t statistic
Step 1. The hypotheses are stated, and the alpha level
is set.
Step 2. Critical region is located.
Step 3. The sample data are collected, and the test
statistic is computed.
Step 4. The null hypothesis is evaluated.
Example
A professor of math hypothesizes that an introductory course in
logic will help college student with their other studies. To test
the hypothesis, a random sample of n = 25 freshman is
selected. These students are required to complete a logic
course during their freshman year. At the time of graduation,
the final grade point average is computed for each of these
students. The mean GPA for this sample X  2.83 with SS=6.
Can the professor conclude that the grades for the sample
were significantly different from the rest of the graduating
class, which had an average GPA of µ=2.58? Test at α=.05.
a.
State the hypotheses.
b.
Determine the value of df, and locate the critical region.
c.
Compute the test statistic.
d.
Make a decision regarding H0.
Example
Fifteen years ago, the average weight of Turkish
women the ages of 30 and 40 was µ=67 kg.
A researcher would like to determine whether
there has been any change in this figure
during the past 15 years. A sample of n=100
women is obtained. The average weight for
this sample is 61 with standard deviation of
s=12. On the basis of these data, can the
researcher conclude that there has been a
significant change in weight? Use a two-tailed
test with α=.01.
Hypothesis tests with two independent
samples: Assumptions
1.
2.
The values in the sample must consist
independent observations.
The population sampled must be normal.
of
PRED 354 TEACH. PROBILITY &
STATIS. FOR PRIMARY MATH
Lesson 9
Hypothesis tests with two
independent samples
Hypothesis tests with two independent
samples
Population A
Population B
Sample B
Sample A
Hypothesis tests with two independent
samples
Sample statistic – population parameter
t
( X 1  X 2 )  ( 1  2 )
t
sX  X
obtained difference
difference exp ected by chance
1
2
Estimated error
sX  X 
1
2
s
2
p

s
2
p
n1 n2
Weighted variance
(Pooled variance)
SS1  SS2
s 
df1  df 2
2
p
Degrees of freedom for the t statistic is df=df1+df2
Hypothesis tests with two independent
samples
Step 1. The hypotheses are stated, and the alpha level
is set.
Step 2. Critical region is located.
Step 3. The sample data are collected, and the test
statistic is computed.
Step 4. The null hypothesis is evaluated.
Examples
In recent years, psychologist have demonstrated repeatedly
that using mental images can greatly improve memory. A
hypothetical experiment, designed to examine this
phenomenon, is presented here. The psychologist first
prepares a list of 40 pairs of nouns (for example,
dog/bicycle, grass/door, lamp/piano). Next, two groups
of subjects are obtained (two separate samples).
Subjects in the first group are given the list for 5
minutes and instructed memorize the 40 noun pairs.
Subjects in the second group receive the same list of
words, but in addition to the regular instructions, these
people are told to form a mental image for each pair of
nouns (imagine a dog riding a bicycle, for example).
Examples
Later each group given a memory test, and the psychologist
records the number of words correctly recalled for each
individual. The data from this experiment are as follows.
On the basis of these data, can the psychologist
conclude that mental images affected memory?
Number of words recalled
Group 1 (No images)
Group 2 (Images)
24, 23, 16, 17, 19, 13,
17, 20, 15, 26
18, 19, 23, 29, 30, 31,
29, 26, 21, 24
Hypothesis tests with two independent
samples: Assumptions
1. The observations within each sample must be
independent.
2. The two population from which the samples are
selected must be normal.
3. The two populations from which the samples are
selected must have equal variances.