Chapter 11 wrap-up

1
Lecture 4

• Chapter 11 wrap-up
• Chapter 12.2 - Inference about the mean when the
  s.d. is unknown
• Chapter 12.3 Inference about a population
  proportion

2
Hypothesis Testing Basic Steps

1. Set up alternative and null hypotheses
2. Calculate test statistic, e.g. z-score
3. Find critical values and compare the test
   statistic to critical value (rejection region
   method) or find p-value (p-value method)
4. Make substantive conclusions.

3
Right-, Left, Two-Sided Tests

• Right-sided
• Left-sided
• 2-sided

4
Summary Steps in Testing

• Determine and (right//left//2sided),
  and decide on a significance level .
• Rejection region method calculate and
  reject if //
  //
• P-value method calculate gt P(Zgtz)
  // P(Zltz) // P(Zgtz) from z-tables,
  or Probgtz // Probltz //
  Probgtz from JMP reject if p-value .
• Interpret the result and tell a story.

5
Relationship Between CIs and Hypothesis Tests

• There is a duality between confidence intervals
  and hypothesis tests
• We can construct a level hypothesis test
  based on a level confidence
  interval by rejecting if and
  only if is not in the confidence interval
• We can construct a level
  confidence interval based on a level
  hypothesis test by including in the
  confidence interval if and only if the test does
  not reject

6
Calculation of Type II error

1. State alternative for which you want to find
   P(Type II error).
2. Find rejection region in terms of unstandardized
   statistic (sample mean)
3. Find the probability of the sample mean falling
   outside the rejection region if the alternative
   under consideration is true (use standardization
   relative to the alternative hypothesis mean to
   calculate this probability).

7
Summary Power Calculations

• Works only for the rejection region method, and
  we dont do it for 2-sided tests.
• Calculate for level-
  test.
• Right-sided P(Zltz) from z-table P(Zltz)
• Left-sided P(Zgtz) from z-table P(Zgtz)

8
Frequent -values
0.10 0.05 0.025 0.01 0.005
1.28 1.64 1.96 2.33 2.58
9
Practice Problems

• 11.68,11.84,12.40,12.46

10
Chapter 12

• In this chapter we utilize the approach developed
  before to describe a population.
• Identify the parameter to be estimated or tested.
• Specify the parameters estimator and its
  sampling distribution.
• Construct a confidence interval estimator or
  perform a hypothesis test.

11
12.2 Inference About a Population Mean When the
Population Standard Deviation Is Unknown

• Recall that when s is known we use the
  following
• statistic to estimate and test a population
  mean
• When s is unknown, we use its point estimator s,
  and the z-statistic is replaced then by the
  t-statistic

12
t-Statistic

• When the sampled population is normally
  distributed, the t statistic is Student t
  distributed with n-1 degrees of freedom.
• Confidence Interval
  where is the quantile of
  the Student t-distribution with n-1 degrees of
  freedom.

13
t-Statistic

• When the sampled population is normally
  distributed, the t statistic is Student t
  distributed with n-1 degrees of freedom.
• Confidence Interval
  where is the quantile of
  the Student t-distribution with n-1 degrees of
  freedom.

14
The t - Statistic
t
s
The degrees of freedom, (a function of the
sample size) determine how spread
the distribution is (compared to the normal
distribution)
The t distribution is mound-shaped, and
symmetrical around zero.
d.f. v2
d.f. v1
v1 lt v2
0
15
tA
t.100
t.05
t.025
t.01
t.005
16
Testing m when s is unknown

• Example 12.1
• In order to determine the number of workers
  required to meet demand, the productivity of
  newly hired trainees is studied.
• It is believed that trainees can process and
  distribute more than 450 packages per hour within
  one week of hiring.
• Fifty trainees were observed for one hour. In
  this sample of 50 trainees, the mean number of
  packages processed is 460.38 and s38.82.
• Can we conclude that the belief is correct, based
  on the productivity observation of 50 trainees?

17
(No Transcript)
18
Checking the required conditions

• In deriving the test and confidence interval, we
  have made two assumptions (i) the sample is a
  random sample from the population (ii) the
  distribution of the population is normal.
• The t test is robust the results are still
  approximately valid as long as the population is
  not extremely nonnormal. Also if the sample size
  is large, the results are approximately valid.
• A rough graphical approach to examining normality
  is to look at the sample histogram.

19
JMP Example

• Problem 12.45 Companies that sell groceries over
  the Internet are called e-grocers. Customers
  enter their orders, pay by credit card, and
  receive delivery by truck. A potential e-grocer
  analyzed the market and determined that to be
  profitable the average order would have to exceed
  85. To determine whether an e-grocer would be
  profitable in one large city, she offered the
  service and recorded the size of the order for a
  random sample of customers. Can we infer from
  the data that e-grocery will be profitable in
  this city at significance level 0.05?

20
12.3 Inference About a Population Variance

• Sometimes we are interested in making inference
  about the variability of processes.
• Examples
• The consistency of a production process for
  quality control purposes.
• Investors use variance as a measure of risk.
• To draw inference about variability, the
  parameter of interest is s2.

21
12.3 Inference About a Population Variance

• The sample variance s2 is an unbiased, consistent
  and efficient point estimator for s2.
• The statistic has a
  distribution called Chi-squared, if the
  population is normally distributed.

d.f. 5
d.f. 10
22
Confidence Interval for Population Variance

• From the following probability statement P(c21-
  a/2 lt c2 lt c2a/2) 1-awe have (by substituting
  c2 (n - 1)s2/s2.)

23
Testing the Population Variance

• Example 12.3 (operation management application)
• A container-filling machine is believed to fill 1
  liter containers so consistently, that the
  variance of the filling will be less than 1 cc
  (.001 liter).
• To test this belief a random sample of 25 1-liter
  fills was taken, and the results recorded
  (Xm12-03). s20.8659.
• Do these data support the belief that the
  variance is less than 1cc at 5 significance
  level?
• Find a 99 confidence interval for the variance
  of fills.

24
(No Transcript)
25
JMP implementation of two-sided test
26
12.4 Inference About a Population Proportion

• When the population consists of nominal data
  (e.g., does the customer prefer Pepsi or Coke),
  the only inference we can make is about the
  proportion of occurrence of a certain value.
• When there are two categories (success and
  failure), the parameter p describes the
  proportion of successes in the population. The
  probability of obtaining X successes in a random
  sample of size n from a large population can be
  calculated using the binomial distribution.

27
12.4 Inference About a Population Proportion

• Statistic and sampling distribution
• the statistic used when making inference about p
  is

28
Testing and Estimating the Proportion

• Test statistic for p

• Interval estimator for p (1-a confidence level)

29
Testing the Proportion

• Example 12.5 (Predicting the winner in election
  day)
• Voters are asked by a certain network to
  participate in an exit poll in order to predict
  the winner on election day.
• The exit poll consists of 765 voters. 407 say
  that they voted for the Republican network.
• The polls close at 800. Should the network
  announce at 801 that the Republican candidate
  will win?

30
(No Transcript)
31
Selecting the Sample Size to Estimate the
Proportion

• Recall The confidence interval for the
  proportion is
• Thus, to estimate the proportion to within W, we
  can write
• The required sample size is

32
Sample Size to Estimate the Proportion

• Example
• Suppose we want to estimate the proportion of
  customers who prefer our companys brand to
  within .03 with 95 confidence.
• Find the sample size needed.
• Solution
• W .03 1 - a .95,
• therefore a/2 .025,
• so z.025 1.96

Since the sample has not yet been taken, the
sample proportion is still unknown.
We proceed using either one of the following two
methods
33
Sample Size to Estimate the Proportion

• Method 1
• There is no knowledge about the value of
• Let . This results in the largest
  possible n needed for a 1-a
  confidence interval of the form .
• If the sample proportion does not equal .5, the
  actual W will be narrower than .03 with the n
  obtained by the formula below.
• Method 2
• There is some idea about what will turn out
  to be.
• Use a probable value of to calculate the
  sample size

34
Practice Problems

• 12.40, 12.46, 12.58, 12.77, 12.98