Download Stat 280 Lab 9: Hypothesis Testing

Stat 280 Lab 9: Significance Tests and Confidence
Intervals
Objectives: This lab is designed to introduce hypothesis testing and such
diagnostics as significance tests and confidence intervals.
Directions: Follow the instructions below, answering all questions. Your answers
should be in the form of a brief report (MS Word), to be handed in to the instructor
before you leave. Please include plots and descriptive statistics in your report.
Confidence Interval: A level C confidence interval for a parameter is an interval
computed from sample data by a method that has probability C of producing an
interval containing the true value of the parameter. There are two important
features of confidence intervals:
i)
ii)
It is an interval of the form (a,b) where a and b are numbers computed from
the data.
It has a property called a confidence level that gives the probability that the
interval covers the parameter.
We can choose the confidence level, most often 90% or higher because we most
often want to be quite sure of our conclusions.
Confidence Interval for a Population Mean: Choose a simple random sample (SRS)
of size n from a population having unknown mean  and known standard deviation
. A level C confidence interval for  is
xbar  z*/sqrt(n),
where xbar is the sample mean. This confidence interval is of the form
xbar  margin of error.
The confidence interval for a population mean will have a specified margin of error,
m, when the sample size is
n= (z*/ m)^2.
Example: You want to estimate the mean SAT-Math score for the more than
250,000 high school seniors in California. You know better than to trust data from
the students who choose to take the SAT. Only about 45% of California students
take the SAT. These self-selected students are planning to attend college and are
not representative of all California seniors. Suppose you give the test to a simple
random sample of 500 California high school seniors. The mean score for your
sample is xbar=461. What can you say about the mean score  in the population of
all 250,000 seniors?
The sample mean, xbar, is the natural estimator of the unknown population mean .
Suppose we know the that if the entire population of SAT scores has mean  and
standard deviation , then in repeated samples of size 500 the sample mean xbar
follows the Normal(,/sqrt(500)) distribution. The standard deviation  of SATMath scores in our California population is  = 100. So xbar = 100/sqrt(500) = 4.5.
The 95% confidence interval is then
xbar  z* /sqrt(n) = 461  1.96*4.5 = 461  8.82  461  9.
Consider:



The 68-95-99.7 rule says that the probability is about 0.95 that xbar will be
within 9 points (two standard deviations of xbar) of the population mean .
To say that xbar lies within 9 points of  is the same as saying  is within 9
points of xbar.
So 95% of all samples will capture the true  in the interval from xbar – 9 to
xbar + 9.
A 95% confidence interval for , means that if we take a number of simple random
samples from the same population and construct confidence intervals for each of
these samples, in the long run, 95% of all the samples will give an interval that
covers the true mean.
In our example, we say that we are 95% confident that the unknown mean score for
all California seniors lies between (452, 470). To understand the grounds for
confidence realize that there are only two possibilities:
1. The interval between 452 and 470 contains the true .
2. Our SRS was one of the few sample for which xbar is not within 9 points or
the true . Only 5% of all samples give such inaccurate results.
Tests of Significance: A significance test is a formal procedure for comparing
observed data with a hypothesis whose truth we want to assess. The hypothesis is a
statement about the parameters in a population or model. The statement being
tested in a test of significance is called the null hypothesis. The test of significance is
designed to assess the strength of the evidence against the null hypothesis. Usually
the null hypothesis is a statement of “no effect” or “no difference”.
The probability, computed assuming that the null hypothesis, Ho, is true, that the
test statistic would take a value as extreme or more extreme than that actually
observed is called the p-value of the test. The smaller the p-value, the stronger the
evidence against Ho provided by the data. The decisive value is called the
significance level, denoted by . If the p-value is as small or smaller than , we say
that the data are statistically significant at level .
The four steps for conducting a test of significance are as follows:
1. State the null hypothesis, Ho and the alternative hypothesis Ha. The test is
designed to assess the strength of the evidence against Ho; Ha is the statement
that we will accept if the evidence enables us to reject Ho.
2. Calculate the value of the test statistic on which the test will be based. This
statistic usually measure how far the data are from Ho.
3. Find the p-value for the observed data. This is the probability, calculated
assuming that Ho is true, that the test will weight against Ho at least as
strongly as it does for these data.
4. State a conclusion. One way to do this is to choose a significance level . If
the p-value is less than or equal to , you conclude that the alternative
hypothesis is true; if it is greater than , you conclude that the data do not
provided sufficient evidence to reject the null hypothesis. Another way to
arrive at a conclusion is to use your 100(1-)% confidence interval. If your
hypothesized value of the parameter in the null hypothesis falls outside the
interval, then you reject Ho.
1. Answer the following questions for the data set pulse.mtw (pulse.xls). These are
pulse rates for a bunch of students in a statistics class.
(a) What is the estimate of the population mean?
(b) Construct a 95% confidence interval for the true population mean.
(c) What sample size do we need in order to reduce the above interval width
(2*margin of error) by half?
(d) Repeat part (a) &( b) above for a 90% confidence interval.
(e) What can you say about the effect of changing the level of confidence & the
sample size on the width of the confidence interval?
2. Imagine choosing n=16 women at random from a large population and measuring
their heights. Assume the heights of the women in this population are normally
distributed with mean 64 inches and standard deviation 3 inches. Suppose you then
test the null hypothesis that the population mean is 64 against the alternative that it
is different from 64 using level of significance 0.1. Simulate the results of doing this
test 20 times by choosing Calc->Random Data->Normal and generating 16 rows of
data in C1-C20 with 64 as the mean and 3 as the standard deviation.
(i) Do hypothesis tests using Stat->Basic->Statistics->1-Sample Z with the Test mean
option and specify a mean of 64 and a sigma of 3.
(ii) Construct 90% C.I.s and 95% C.I.s(hint: the procedure is the same as in (1) only
change the Test mean option into confidence interval option and specify the
confidence level 90.0 & 95.0).
(a) What are the null hypothesis and alternative (research) hypothesis?
(b) In how many tests did you fail to reject the null hypothesis ? That is, how
many times did you make the "correct decision"?
(c) How many times did you make an "incorrect decision"(that is, reject the null
hypothesis)? On the average how many times out of 20 would you expect to
make the wrong decision?
(d) How many 90% confidence intervals cover the true mean value 64?
(e) Is the frequency that you make the correct decision the same as the
frequency that the 90% C.I.s cover 64?
(f) Plot a histogram of the p-values and comment on its shape.
(g) Suppose you used alpha=0.05 instead of alpha=0.10. Does this change any of
your decisions to reject or not? Should it in some cases?