Download Statistics Computer Lab 6

Statistics
MINITAB - Lab 11
Small Sample Tests of Hypothesis About a Population Mean
1.
When dealing with large samples we can use the Central Limit Theorem to test
hypotheses about a population mean. What do we do when we are dealing with small
samples (i.e. n < 30) , and therefore the normality of the sampling distribution of the mean
does not follow from the Central Limit Theorem ?
In such cases we may still proceed providing that the underlying distribution from which the
sample is drawn is normal.
However, even when the population from which the sample is drawn may be normal, using
the sample standard deviation, s, as an estimate for , the population standard deviation is
problematic. To correct for the this we conduct our hypothesis test as before except a
distribution called the t distribution (you may see this called the student t distribution in
certain texts) is used in place of the normal distribution for choosing a rejection region.
The test statistic is:
t
x
s
n
and a rejection region is chosen at the desired  level from the t distribution with n-1
degrees of freedom.
How to interpret the Rejection Region:
Reject if t statistic is
Upper tailed test
Lower tailed test
> t-critical
< - t-critical
Two tailed test
> t-critical or
< - t-critical
A major car manufacturer wants to test a new engine to determine whether it meets airpollution standards. The mean emission  of all engines of this type must be less than 20
parts per million of carbon. Ten engines are tested to establish that the population of
engines will be in line with the standards. The results are:
15.6, 16.2, 22.5, 20.5, 16.4, 19.4, 16.6, 17.9, 12.7, 13.9 .
The manufacturer is making the assumption that the relative frequency histogram of the
emission levels for the population of engines of this type is approximately normal.
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Enter this data into MINITAB labelling the variable appropriately and answer the following
question using  = .01.
Do the data supply sufficient evidence to allow the manufacturer to conclude that this type
of engine meets the pollution standards ?
Here are the familiar steps in hypothesis testing - amended to reflect small samples.
Step 1. Choose the population characteristic of interest.  - the population mean
Step 2. Choose the significance level.  = .01 (or 1% level).
Step 3. State null hypothesis. Ho:  = 20 ppm
Step 4. State alternative hypothesis. Ha:  < 20 ppm
Step 5. Choose a test statistic. In this case the test statistic chosen is
t
x
s
n
Step 6. Choose a rejection region
Since the alternative hypothesis includes all means of less than 20 ppm, this is
a one-tailed test. The rejection region will be in the lower tail of the t, df=9
distribution. First get the t quantile (critical value) such that 1% of the t
distribution with n-1 degrees of freedom is to the right and therefore 99% is to
the left with the following command and then take the negative of the answer
since this is a lower tailed test,
MTB > INVCDF .99;
SUBC> t 9.
What is the answer ? _____________
So the null hypothesis will be rejected if:
Step 7. Calculate the test statistic
Fill in the following equation.
t
x x


s
sx
n

 ___________
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Step 8. State Conclusion in the context of the question
Reject / Fail To Reject the Ho: at  = _______, that
__________________________________________________
__________________________________________________
2. MINITAB has a built in function for small sample hypothesis testing about a population
mean Repeat the above test using this function. Go to the STAT>BASIC STATISTICS>
1-Sample t. Specify the Null hypothesis mean in the 'test mean' box and click options.
Choose the correct confidence level and alternative hypothesis and click ok twice.
MINITAB will now print the result of the test in the session window. You will see N - the
number in the sample, Mean - the sample mean, Stdev - the standard deviation of the
sample, SE Mean - the standard error of the mean, 100(1-)% CI or upper or lower
bound (i.e. a two sided or one sided confidence interval depending on the Ha:), t - the
test statistic, P - the actual probability that the sample mean came from the population
specified by the Ho.
What is the P value for this hypothesis test ? ____________.
How to interpret the P-Value:
P<
P>
… Reject the Null Hypothesis and accept the alternative
… Fail to reject the Null Hypothesis
Reject / Fail To Reject the Ho: at  = _______, that
__________________________________________________
__________________________________________________
Change the Ha: to a 2-sided test and redo the test. What is the two sided 99%
confidence interval for the mean ? __________________________________.
Has the P value changed form last time, what is it now ? ________
If so explain why.
______________________________________________________________________
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3. Open the worksheet called Data_Lab11.mtw which is on onlineclasses in the datasets
folder. Read the information below regarding each sample and then using this weeks
and last weeks lab sheets, perform appropriate hypothesis tests for each one.
[ Hint: Generate the summary statistics for each sample first to familiarise yourself with
the data.]
Sample 1:
The average cost of a tin of a loaf of bread is € 0.89. Loaves of bread were bought in ten
shops in Dublin to find evidence against this claim. Sample 1 contains these costs.
[Perform this test at 99% confidence level]
Answer:
H0:
HA:
=
test statistic =
Critical value =
(use tables or Minitab)
p-value =
Conclusion:
Sample 2:
Students at a particular secondary school were complaining about the time it took them
to travel to school on public transport and want the school to organise a private school
bus. The board of management insisted that the journey on public transport only took an
average of 20 minutes. The students conducted a survey of 50 students who travel by
public transport in an effort to prove the journey takes longer. Sample 2 contains the
time in minutes it took each one to travel.
[Perform this test at 95% confidence level]
Answer:
H0:
HA:
=
test statistic =
Critical value =
(use tables or Minitab)
p-value =
Conclusion:
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Sample 3:
An experiment was conducted to examine the weight of a batch of flower bulbs in a
gardening shop. If the average weight is less than 3 grams then the whole batch will be
thrown out. A sample from the batch were taken at random and weighed and recorded in
Sample 3.
[Perform this test at 95% confidence level]
Answer:
H0:
HA:
=
test statistic =
Critical value =
(use tables or Minitab)
p-value =
Conclusion:
REVISION SUMMARY
After this lab you should be able to :
-
Understand the 8 steps of a hypothesis test
-
Look up t-critical values (with appropriate degrees of freedom) in the
Cambridge tables and Minitab
-
Perform a 1-sample t hypothesis test using Minitab’s inbuilt function
-
Interpret a hypothesis test from comparing the test statistic and the critical value
-
Interpret a hypothesis test from comparing the p-value and  level
-
Tell when you should perform a z-test and a t-test
END
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