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Assumptions and Conditions One-Sample t-test for the Mean

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Assumptions and Conditions
• Independence Assumption:
– Randomization Condition: The data arise from
a random sample or suitably randomized
experiment. Randomly sampled data
(particularly from an SRS) are ideal.
– 10% Condition: When a sample is drawn
without replacement, the sample should be no
more than 10% of the population.
Assumptions and Conditions (cont.)
– Nearly Normal Condition:
• Normal Population Assumption:
– We can never be certain that the data are
from a population that follows a Normal
model, but we can check the
– Nearly Normal Condition: The data come from
a distribution that is unimodal and symmetric.
• Check this condition by making a histogram or
Normal probability plot.
One-Sample t-test for the Mean
• The conditions for the one-sample t-test for the mean are the
same as for the one-sample t-interval.
• We test the hypothesis H0: μ = μ0 using the statistic
x в€’ Ој0
tn в€’1 =
SE ( x )
• The standard error of the sample mean is
Assumptions and Conditions (cont.)
• The smaller the sample size (n < 15 or so), the
more closely the data should follow a Normal
model.
• For moderate sample sizes (n between 15 and 40
or so), the t works well as long as the data are
unimodal and reasonably symmetric.
• For larger sample sizes, the t methods are safe to
use even if the data are skewed.
Example:
Is the mean weight of female college students still 132
pounds? To test this, you take a random sample of 20
students, finding a mean of 137 pounds with a standard
deviation of 14.2 pounds. Use a significance level of 0.1.
1. Hypothesis
Ој = population mean weight of female college students
SE ( x ) =
s
n
Ho: Ој = 132
Ha: Ој в‰ 132
• When the conditions are met and the null hypothesis is true,
this statistic follows a Student’s t model with n – 1 df. We use
that model to obtain a P-value.
1
Example:
Is the mean weight of female college students still 132
pounds? To test this, you take a random sample of 20
students, finding a mean of 137 pounds with a standard
deviation of 14.2 pounds. Use a significance level of 0.1.
Example:
Is the mean weight of female college students still 132
pounds? To test this, you take a random sample of 20
students, finding a mean of 137 pounds with a standard
deviation of 14.2 pounds. Use a significance level of 0.1.
2. Check Assumptions/Conditions
3. Calculate Test
t=
• SRS is stated
• σ is unknown, use t-distribution
• We will assume an approximately normal distribution
x в€’ Ој0 137 в€’ 132
=
= 1.575
14.2
SE ( x )
20
0.132
P ( x = 137 Ој = 132 ) = 0.264
Example:
Is the mean weight of female college students still 132
pounds? To test this, you take a random sample of 20
students, finding a mean of 137 pounds with a standard
deviation of 14.2 pounds. Use a significance level of 0.1.
4. Conclusion
Since P-value is greater than alpha, we fail to
reject the population mean weight of female
students is 132 pounds.
Example:
A father is concerned that his teenage son is watching too much
television each day, since his son watches an average of 2 hours per
day. His son says that his TV habits are no different than those of his
friends. Since this father has taken a stats class, he knows that he can
actually test to see whether or not his son is watching more TV than his
peers. The father collects a random sample of television watching times
from boys at his son's high school and gets the following data
1.9 2.3 2.2 1.9 1.6 2.6 1.4 2.0 2.0 2.2
Is the father right? That is, is there evidence that other boys average
less than 2 hours of television per day?
1. Hypothesis
Ој = population mean number of hours watching TV
Ho : Ој = 2
Ha : Ој < 2
Example:
Example:
A father is concerned that his teenage son is watching too much
television each day, since his son watches an average of 2 hours per
day. His son says that his TV habits are no different than those of his
friends. Since this father has taken a stats class, he knows that he can
actually test to see whether or not his son is watching more TV than his
peers. The father collects a random sample of television watching times
from boys at his son's high school and gets the following data
1.9 2.3 2.2 1.9 1.6 2.6 1.4 2.0 2.0 2.2
Is the father right? That is, is there evidence that other boys average
less than 2 hours of television per day?
A father is concerned that his teenage son is watching too much
television each day, since his son watches an average of 2 hours per
day. His son says that his TV habits are no different than those of his
friends. Since this father has taken a stats class, he knows that he can
actually test to see whether or not his son is watching more TV than his
peers. The father collects a random sample of television watching times
from boys at his son's high school and gets the following data
1.9 2.3 2.2 1.9 1.6 2.6 1.4 2.0 2.0 2.2
Is the father right? That is, is there evidence that other boys average
less than 2 hours of television per day?
2. Check Assumptions/Conditions
3. Calculate Test
SRS is stated
Пѓ is unknown, use t-distribution
Based on the linearity of the
normal probability plot, we have
an approximately normal
distribution.
t=
x в€’ Ој0 2.01 в€’ 2
=
= 0.0918
0.345
SE ( x )
10
0.5356
P ( x = 2.01 Ој = 2 ) = 0.5356
2
Example:
A father is concerned that his teenage son is watching too much
television each day, since his son watches an average of 2 hours per
day. His son says that his TV habits are no different than those of his
friends. Since this father has taken a stats class, he knows that he can
actually test to see whether or not his son is watching more TV than his
peers. The father collects a random sample of television watching times
from boys at his son's high school and gets the following data
1.9 2.3 2.2 1.9 1.6 2.6 1.4 2.0 2.0 2.2
Is the father right? That is, is there evidence that other boys average
less than 2 hours of television per day?
4. Conclusion
О± = 0.05
Since the P-value is greater than alpha, we fail to reject the
population mean number of hours watching TV by the other boys
is 2.
3
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