<< Chapter < Page Chapter >> Page >
This course is a short series of lectures on Introductory Statistics. Topics covered are listed in the Table of Contents. The notes were prepared by EwaPaszek and Marek Kimmel. The development of this course has been supported by NSF 0203396 grant.

Tests about one mean and one variance

In the previous paragraphs it was assumed that we were sampling from a normal distribution and the variance was known. The null hypothesis was generally of the form H 0 μ μ 0 .

There are essentially tree possibilities for the alternative hypothesis, namely that μ has increased,

  • H 1 μ >   μ 0 ; μ has decreased,
  • H 1 μ <   μ 0 ; μ has changed, but it is not known if it has increased or decreased, which leads to a two-sided alternative hypothesis
  • H 1 ; μ μ 0 .

To test H 0 ; μ = μ 0 against one of these tree alternative hypotheses, a random sample is taken from the distribution, and an observed sample mean, x ¯ , that is close to μ 0 supports H 0 . The closeness of x ¯ to μ 0 is measured in term of standard deviations of X ¯ , σ / n which is sometimes called the standard error of the mean . Thus the statistic could be defined by

Z = X ¯ μ 0 σ 2 / n = X ¯ μ 0 σ / n , and the critical regions, at a significance level α , for the tree respective alternative hypotheses would be:

  • z z α
  • z z α
  • | z | = z α / 2

In terms of x ¯ these tree critical regions become

  • x ¯ μ 0 + z α σ / n ,
  • x ¯ μ 0 z α σ / n ,
  • | x ¯ μ 0 | z α σ / n

These tests and critical regions are summarized in TABLE 1 . The underlying assumption is that the distribution is N ( μ , σ 2 ) and σ 2 is known. Thus far we have assumed that the variance σ 2 was known. We now take a more realistic position and assume that the variance is unknown. Suppose our null hypothesis is H 0 ; μ = μ 0 and the two-sided alternative hypothesis is H 1 ; μ μ 0 . If a random sample X 1 , X 2 , ... , X n is taken from a normal distribution N ( μ , σ 2 ) ,let recall that a confidence interval for μ was based on T = X ¯ μ S 2 / n = X ¯ μ S / n .

Table 1
H 0 H 1 Critical Region
μ = μ 0 μ > μ 0 z z α or x ¯ μ 0 + z α σ / n
μ = μ 0 μ < μ 0 z z α or x ¯ μ 0 z α σ / n
μ = μ 0 μ μ 0 | z | z α / 2 or | x ¯ μ 0 | z α / 2 σ / n

This suggests that T might be a good statistic to use for the test H 0 ; μ = μ 0 with μ replaced by μ 0 . In addition, it is the natural statistic to use if we replace σ 2 / n by its unbiased estimator S 2 / n in ( X ¯ μ 0 ) / σ 2 / n in a proper equation. If μ = μ 0 we know that T has a t distribution with n -1 degrees of freedom. Thus, with μ = μ 0 ,

P [ | T | t α / 2 ( n 1 ) ] = P [ | X ¯ μ 0 | S / n t α / 2 ( n 1 ) ] = α .

Accordingly, if x ¯ and s are the sample mean and the sample standard deviation, the rule that rejects H 0 ; μ = μ 0 if and only if | t | = | x ¯ μ 0 | s / n t α / 2 ( n 1 ) .

Provides the test of the hypothesis with significance level α . It should be noted that this rule is equivalent to rejecting H 0 ; μ = μ 0 if μ 0 is not in the open 100 ( 1 α ) % confidence interval ( x ¯ t α / 2 ( n 1 ) s / n , x ¯ + t α / 2 ( n 1 ) s / n ) .

Table 2 summarizes tests of hypotheses for a single mean, along with the three possible alternative hypotheses, when the underlying distribution is N ( μ , σ 2 ) , σ 2 is unknown, t = ( x ¯ μ 0 ) / ( s / n ) and n 31 . If n >31, use table 1 for approximate tests with σ replaced by s .

Table 2
H 0 H 1 Critical Region
μ = μ 0 μ > μ 0 t t α ( n 1 ) or x ¯ μ 0 + t α ( n 1 ) s / n
μ = μ 0 μ < μ 0 t t α ( n 1 ) or x ¯ μ 0 t α ( n 1 ) s / n
μ = μ 0 μ μ 0 | t | t α / 2 ( n 1 ) or | x ¯ μ 0 | t α / 2 ( n 1 ) s / n

Let X (in millimeters) equal the growth in 15 days of a tumor induced in a mouse. Assume that the distribution of X is N ( μ , σ 2 ) . We shall test the null hypothesis H 0 : μ = μ 0 = 4.0 millimeters against the two-sided alternative hypothesis is H 1 : μ 4.0 . If we use n =9 observations and a significance level of α =0.10, the critical region is | t | = | x ¯ 4.0 | s / 9 t α / 2 ( 8 ) = t 0.05 ( 8 ) = 1.860.

If we are given that n =9, x ¯ =4.3, and s =1.2, we see that t = 4.3 4.0 1.2 / 9 = 0.3 0.4 = 0.75.

Thus | t | = | 0.75 | < 1.860 and we accept (do not reject) H 0 : μ = 4.0 at the α =10% significance level. See Figure 1 .

Rejection region at the α = 10 % significance level.
Got questions? Get instant answers now!
In discussing the test of a statistical hypothesis, the word accept might better be replaced by do not reject . That is, in Example 1 , x ¯ is close enough to 4.0 so that we accept μ =4.0, we do not want that acceptance to imply that μ is actually equal to 4.0. We want to say that the data do not deviate enough from μ =4.0 for us to reject that hypothesis; that is, we do not reject μ =4.0 with these observed data, With this understanding, one sometimes uses accept and sometimes fail to reject or do not reject , the null hypothesis.

In this example the use of the t -statistic with a one-sided alternative hypothesis will be illustrated.

In attempting to control the strength of the wastes discharged into a nearby river, a paper firm has taken a number of measures. Members of the firm believe that they have reduced the oxygen-consuming power of their wastes from a previous mean μ of 500. They plan to test H 0 : μ = 500 against H 1 : μ < 500 , using readings taken on n =25 consecutive days. If these 25 values can be treated as a random sample, then the critical region, for a significance level of α =0.01, is t = x ¯ 500 s / 25 t 0.01 ( 24 ) = 2.492.

The observed values of the sample mean and sample standard deviation were x ¯ =308.8 and s =115.15. Since t = 308.8 500 115.15 / 25 = 8.30 < 2.492 , we clearly reject the null hypothesis and accept H 1 : μ < 500 . It should be noted, however, that although an improvement has been made, there still might exist the question of whether the improvement is adequate. The 95% confidence interval 308.8 ± 2.064 ( 115.15 / 5 ) or [ 261 .27, 356 .33 ] for μ might the company answer that question.

Got questions? Get instant answers now!

Questions & Answers

what is a market demand schedule
Blessed Reply
What is demand curve
mohamed Reply
Types of demand curve
mohamed
demand curve is a showing the aggregate of demand whether falling from right to the left in the table above
BULAMA
demand curve is the graphical representation of various quantities of a commodity bought at various prices
Blessed
What is the formula for calculating elasticity
Hafsat Reply
what is demand
Anefor Reply
demand is the quantity of commodity that consumers wil be able and wiling to buy at a given price at a given time
Delly
what is international trade?
Friday Reply
it is the exchange of goods and services across international borders.
Tweneboah
what is taxation
Eunice Reply
the definition of economic by Adam Smith
Geon Reply
Adam Smith define economic as the science of wealth
Hafsat
What is enconomics
mohamed
what is meant by pisp in economics?
Amara Reply
payment initiation service provider
Khuselo
what is demand
Joy Reply
demand
Demand refers to the consumers' desire to purchase goods at given prices. Demand can mean either demand for a specific good or aggregate demand for the total of all goods in an economy.
What is economc
Joseph Reply
What is economc
nahurira
What is Economic
Vicky Reply
What's the price index of consumption?
Pun Reply
what population
Hackman Reply
I think it's the sum total of people in a goegraphical area at a given time
Ruth
it refers to the number of people living in a particular area over a given period of time
Vanessa
decribe law of demand
Lovely Reply
the higher the price , the higher the quantity demanded
Ruth
what is economic growth
Metuge Reply
what is an economic system
Metuge
Economic system is the mechanism which deals with the production, distribution and consumption of good and services in a particular society
Happy
of course
Maesela
what is a weight
Maesela
What is a weight
mohamed
in a comparison of the stages of meiosis to the stage of mitosis, which stages are unique to meiosis and which stages have the same event in botg meiosis and mitosis
Leah Reply
Got questions? Join the online conversation and get instant answers!
Jobilize.com Reply

Get Jobilize Job Search Mobile App in your pocket Now!

Get it on Google Play Download on the App Store Now




Source:  OpenStax, Introduction to statistics. OpenStax CNX. Oct 09, 2007 Download for free at http://cnx.org/content/col10343/1.3
Google Play and the Google Play logo are trademarks of Google Inc.

Notification Switch

Would you like to follow the 'Introduction to statistics' conversation and receive update notifications?

Ask