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  • In a hypothesis test problem, you may see words such as "the level of significance is 1%." The "1%" is the preconceived or preset α .
  • The statistician setting up the hypothesis test selects the value of α to use before collecting the sample data.
  • If no level of significance is given, a common standard to use is α = 0.05.
  • When you calculate the p -value and draw the picture, the p -value is the area in the left tail, the right tail, or split evenly between the two tails. For this reason, we call the hypothesis test left, right, or two tailed.
  • The alternative hypothesis , H a , tells you if the test is left, right, or two-tailed. It is the key to conducting the appropriate test.
  • H a never has a symbol that contains an equal sign.
  • Thinking about the meaning of the p -value    : A data analyst (and anyone else) should have more confidence that he made the correct decision to reject the null hypothesis with a smaller p -value (for example, 0.001 as opposed to 0.04) even if using the 0.05 level for alpha. Similarly, for a large p -value such as 0.4, as opposed to a p -value of 0.056 (alpha = 0.05 is less than either number), a data analyst should have more confidence that she made the correct decision in not rejecting the null hypothesis. This makes the data analyst use judgment rather than mindlessly applying rules.

The following examples illustrate a left-, right-, and two-tailed test.

H o : μ = 5, H a : μ <5

Test of a single population mean. H a tells you the test is left-tailed. The picture of the p -value is as follows:

Normal distribution curve of a single population mean with a value of 5 on the x-axis and the p-value points to the area on the left tail of the curve.
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H 0 : μ = 10, H a : μ <10

Assume the p -value is 0.0935. What type of test is this? Draw the picture of the p -value.

left-tailed test

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H 0 : p ≤ 0.2   H a : p >0.2

This is a test of a single population proportion. H a tells you the test is right-tailed . The picture of the p -value is as follows:

Normal distribution curve of a single population proportion with the value of 0.2 on the x-axis. The p-value points to the area on the right tail of the curve.
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H 0 : μ ≤ 1, H a : μ >1

Assume the p -value is 0.1243. What type of test is this? Draw the picture of the p -value.

right-tailed test

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H 0 : p = 50   H a : p ≠ 50

This is a test of a single population mean. H a tells you the test is two-tailed . The picture of the p -value is as follows.

Normal distribution curve of a single population mean with a value of 50 on the x-axis. The p-value formulas, 1/2(p-value), for a two-tailed test is shown for the areas on the left and right tails of the curve.
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H 0 : p = 0.5, H a : p ≠ 0.5

Assume the p -value is 0.2564. What type of test is this? Draw the picture of the p -value.

two-tailed test

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Full hypothesis test examples

Jeffrey, as an eight-year old, established a mean time of 16.43 seconds for swimming the 25-yard freestyle, with a standard deviation of 0.8 seconds . His dad, Frank, thought that Jeffrey could swim the 25-yard freestyle faster using goggles. Frank bought Jeffrey a new pair of expensive goggles and timed Jeffrey for 15 25-yard freestyle swims . For the 15 swims, Jeffrey's mean time was 16 seconds. Frank thought that the goggles helped Jeffrey to swim faster than the 16.43 seconds. Conduct a hypothesis test using a preset α = 0.05. Assume that the swim times for the 25-yard freestyle are normal.

Set up the Hypothesis Test:

Since the problem is about a mean, this is a test of a single population mean .

H 0 : μ = 16.43   H a : μ <16.43

For Jeffrey to swim faster, his time will be less than 16.43 seconds. The "<" tells you this is left-tailed.

Determine the distribution needed:

Random variable: X = the mean time to swim the 25-yard freestyle.

Distribution for the test: X is normal (population standard deviation is known: σ = 0.8)

X ¯ ~ N ( μ , σ X n ) Therefore, X ¯ ~ N ( 16.43 , 0.8 15 )

μ = 16.43 comes from H 0 and not the data. σ = 0.8, and n = 15.

Calculate the p -value using the normal distribution for a mean:

p -value = P ( x ¯ <16) = 0.0187 where the sample mean in the problem is given as 16.

p -value = 0.0187 (This is called the actual level of significance .) The p -value is the area to the left of the sample mean is given as 16.

Graph:

Normal distribution curve for the average time to swim the 25-yard freestyle with values 16, as the sample mean, and 16.43 on the x-axis. A vertical upward line extends from 16 on the x-axis to the curve. An arrow points to the left tail of the curve.

μ = 16.43 comes from H 0 . Our assumption is μ = 16.43.

Interpretation of the p -value: If H 0 is true , there is a 0.0187 probability (1.87%)that Jeffrey's mean time to swim the 25-yard freestyle is 16 seconds or less. Because a 1.87% chance is small, the mean time of 16 seconds or less is unlikely to have happened randomly. It is a rare event.

Compare α and the p -value:

α = 0.05 p -value = 0.0187 α > p -value

Make a decision: Since α > p -value, reject H 0 .

This means that you reject μ = 16.43. In other words, you do not think Jeffrey swims the 25-yard freestyle in 16.43 seconds but faster with the new goggles.

Conclusion: At the 5% significance level, we conclude that Jeffrey swims faster using the new goggles. The sample data show there is sufficient evidence that Jeffrey's mean time to swim the 25-yard freestyle is less than 16.43 seconds.

The p -value can easily be calculated.

Press STAT and arrow over to TESTS . Press 1:Z-Test . Arrow over to Stats and press ENTER . Arrow down and enter 16.43 for μ 0 (null hypothesis), .8 for σ , 16 for the sample mean, and 15 for n . Arrow down to μ : (alternate hypothesis) and arrow over to< μ 0 . Press ENTER . Arrow down to Calculate and press ENTER . The calculator not only calculates the p -value ( p = 0.0187) but it also calculates the test statistic ( z -score) for the sample mean. μ <16.43 is the alternative hypothesis. Do this set of instructions again except arrow to Draw (instead of Calculate ). Press ENTER . A shaded graph appears with z = -2.08 (test statistic) and p = 0.0187 ( p -value). Make sure when you use Draw that no other equations are highlighted in Y = and the plots are turned off.

When the calculator does a Z -Test, the Z-Test function finds the p -value by doing a normal probability calculation using the central limit theorem :

P ( x ¯ < 16 ) = 2nd DISTR normcdf ( 10 ^ 99 , 16 , 16.43 , 0.8 / 15 ) .

The Type I and Type II errors for this problem are as follows:

The Type I error is to conclude that Jeffrey swims the 25-yard freestyle, on average, in less than 16.43 seconds when, in fact, he actually swims the 25-yard freestyle, onaverage, in 16.43 seconds. (Reject the null hypothesis when the null hypothesis is true.)

The Type II error is that there is not evidence to conclude that Jeffrey swims the 25-yard free-style, on average, in less than 16.43 seconds when, in fact, he actually does swim the 25-yard free-style, on average, in less than 16.43 seconds. (Do not reject the null hypothesis when the null hypothesis is false.)

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Questions & Answers

According to my understanding, statistics are numerical data or information presented in charts or graphs
Kristen Reply
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Said
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Elisha Reply
how to find critical angle
Muhammad Reply
what is statistics ?
Samad
Statistics is deal with the collection, analysis, interpretation, presentation of data to drawn out conclusions.
Ajao
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Samad
In my own perspective statistics is deal with the collection, analysis, interpretation, presentation of data before will drawn our conclusions
Ajao
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Samad
In simple way it is the technique of problem solving by providing probabilistic answer to questions otherwise whose exact answer is not possible..besides it is the support to every subject you study.
being
data
Deng
Data is information that has been translated into a form that is efficient for movement or processing
Mansour
group the scores below into a class interval of seven and obtain the following: 1.. i..class interval ii..tally iii..frequency iv..mid point v...cummulative frequency less than vi..cummulative frequency more than vii..relative frequency viii...relative percentage
Icecool
2..draw a histogram and frequency polygon to sick it the statistical table
Icecool
20,30,39,25,50,26,39,32,43,20,30,40,50,10,16,12,19,31,32,26,29,27,28,42,63,48,11,26,35,36,37,38,39,39,35,29,53,37,32,24,36,43,48,42,51,37,39,39,24,7,8,13,62,52,69,68,37,38,39,40,41,39,54
Icecool
2.draw a histogram and frequency polygon to sick it the statistical
Deng
2.draw a histogram and frequency polygon to dipict the statistical table
Icecool
Describe the organizational setup of bureau of statics government of sindh give its past achievement in brief
Awais Reply
Life Table and its construction
Awais
only asking what is hospital statistics
Suly Reply
What's the meaning Of statistic
Farouk Reply
what's the meaning of statistic
Musa
The mean is the average of the numbers. It is easy to calculate: add up all the numbers, then divide by how many numbers there are. In other words it is the sum divided by the count.
khama Reply
pls I want to know the characterize of mean
Trixie Reply
I want solution to; x,x+1,3.5,3.4 and 2.5 is 3.1. find the value of x
Trixie
mean is nothing but average.
NAMDEV
(x+x+1+3.5+3.4+2.5)/5=3.1 2x+10.4=3.1*5 x=(15.5-10.4)/2 x=2.55
NAMDEV
How. Pls explain or break it down thanks
Icecool
the sum of all five divided by the 5 which is the average is equal to 3.1 so you sum them, divide and equate to 3.1. then solve for X by making X the subject
Pascale
Think of it as your grade, or gpa. how do you find the average of it?
La
great
Rushikesh
nicely explained
Rushikesh
What is the meaning of statistics and it's relevance in the social sciences
Icecool Reply
statistics is the branch of mathematics that using math& statistics theories to interpret data to information which is supporting the decision makers in social sciences and applied too. statistics is decision making science now aday
Ishag
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Enny Reply
I'm not seeing the Math problem?
Debra
the data below where from 1 miles. miles 1:8380, 8210,3836,7840,7910,7510,7720,7750,8100,7690 test wherever their difference is significance at 5% level with population of #8000.
Enny
1*1
Enny
me too
New
enny i don't see any math problems
New
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ott
help me the interquartile range and semi interquartile range of grouped data
Ishraaq
what is the difference between macro and micro data?
Sani Reply
Macrodata consist of 'headcounts', Microdata contain data on individuals,
Cherry
macro is a big business and micro is small business
Mukhtar
macro means whole or big and micro means part or small therefore in relation to data, macro data will be big or whole data and micro data will be small or individual part of a data. for example taking the whole data of an organisation or economy that will be macro data and taking the data on employe
Pascale
employees will be micro data
Pascale
How to determine appropriate statistics tool for my data
Jayr Reply
hi
Juliana
hi
Emy
You cannot know everything! Statistics is a vast subject so don’t be afraid to say that you don’t know. Ask others for help or look up information on the internet to help. Don’t assume that the student knows anything about the technique they are suggesting!
Kalu
well
Tahir
my class starts at February 4th
Elaine
P is the probability that a given die shows even number. To test H0 : P =1/2 against H1 : P=1/3, following procedure is adopted. Toss the die twice and accept H0 if both times it shows even number. Find the probabilities of type I and type II errors
Loyolite Reply
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bilal Reply
what is data
natnael
It Is the collection, collation,Analysing Interpretation,and presentation of data....
Tosayo
statistics is a science that studies numerical values assigned to objects, attributes or variables in order to characterize and make the variables more meaningful.
Okosa
statistics is the science of collection, analysis, Interpretation & presentation of numerical data
Kishan
Numeric data
Tosayo
if you think about normal number and operations of math such as fractions and percent values, statisitics is when it comes down to it a lot of just that - counting and then fractions, then that data and those counts (and ratios)/fractions that were used or counted in one group will represent a pop.
Jesus
An example is, lets say you are in charge of making large orders of supplies and that based on your need of X bricks = X feet of building - now thats the math, well stats is using that math in a way the next person making orders can base those orders of what they will need based on saying "in this
Jesus
..case, when Bilal needed 75 feet of building walls that 150 bricks were used" - now, that helps to get one job done but if we then watch those jobs one at a time, and count that 5/5 times the count was 150 bricks total - the new person could order 150 bricks with a high probability of being right
Jesus
If X is a Uniform random variable in [ -2, 2 ], find the pdf of Y X  and E Y[ ].
Kezang Reply
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Anji
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Shahjee
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Anaya
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abhishek
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Source:  OpenStax, Introductory statistics. OpenStax CNX. May 06, 2016 Download for free at http://legacy.cnx.org/content/col11562/1.18
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