Transcript D_C_Statistics_Week_22_files

Assumptions and Conditions
– Randomization Condition: The data arise from
a random sample or suitably randomized
experiment. Randomly sampled data
(particularly from an SRS) are ideal.
– 10% (Independence) Condition: When a
sample is drawn without replacement, the
sample should be no more than 10% of the
population.
Assumptions and Conditions (cont.)
• 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 OR assume that it is true.
Assumptions and Conditions (cont.)
– Nearly Normal Condition:
• 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.
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:  = 0 using the statistic
x  0
tn 1 
SE  x 
• The standard error of the sample mean is
s
SE  x  
n
• 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.
Example:
Is the mean weight of 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
 = population mean weight of college students
Ho:  = 132
Ha:   132
Example:
Is the mean weight of 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
• Random sample is stated
• Assume population of college students > 200
• Assume population is approx. normally distributed
•  is unknown, use t-distribution
Example:
Is the mean weight of 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.
3. Calculate Test
x  0 137  132
t

 1.575
14.2
SE  x 
20
0.066
t-critical = 1.729
p-value = 0.066*2 = 0.132
Example:
Is the mean weight of 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 t-statistic is less than t-critical, we fail to
reject the population mean weight of students is
132 pounds. --OR-Since P-value is greater than alpha, we fail to
reject the population mean weight of students is
132 pounds.
Thus, there is NO evidence to support a claim
that the true mean weight of college students
has changed.
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
 = population mean number of hours boys at the high school
spend watching TV
Ho:  = 2
Ha:  < 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?
2. Check Assumptions/Conditions
Random sample is stated
Assume boy population at the high
school > 100
Based on the linearity of the normal
probability (quantile) plot, we have
approx normal data.
 is unknown, use t-distribution
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?
3. Calculate Test
x  0 2.01  2
t

 0.0918
0.345
SE  x 
10
t-critical = -1.833
P  x  2.01   2   0.5356
0.5356
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
 = 0.05
Since the t-statistic is greater than the t-critical, we fail to reject that the
population mean number of hours watching TV by the HS boys is 2.
--OR—
Since the p-value is greater than alpha, we fail to reject that the
population mean number of hours watching TV by the HS boys is 2.
Thus, we do not have evidence that supports the father’s claim.