chapter 9 Flashcards
Explain the Null Hypothesis
making a tentative assumption about a pop parameter
- the result that is hoped to be proven false
Explain the Alternative Hypothesis
- opposite of what is stated in the Null hypothesis
- the result that is hoped to be true
- given by < > or not =
What is hypothesis testing
a method of testing whether or not a claim is valid
What are Two types of Claims
- Proportions - data given by percentages, %
2. Means - data given by data measurements, M
What does developing the Null Hypothesis involve?
collecting a sample and using the sample results to provide evidence for drawing a conclusion
What are we really doing when we are testing the hypothesis
Ha is often what the test is attempting to establish
Ex Hyp Test: WHat is the Ha and Ho
Current system ahs a M of 24 miles we are looking for this new item to be more than 24 miles
Ho M is < or = 24
Ha M > 24
Ex. Hyp test: What is the Ha and Ho
New teaching method developed is believed to be better than current one
Ho - new method is not better
Ha - new method is better
Ex Hyp test: what is the Ha and Ho
New sales force bonus plan developed to increase sales
Ho - new bonus plan does not increase sales
ha - new bonus plan does increase sales
If you disprove Ho you
prove Ha
if you fail to disprove Ho then
Nothing
- it is impossible to provide Ho true nor is it possible to disprove Ha
Ho: M>= M0
Ha: M< Mo
What type of tailed test is this
one tailed test
the line goes to the right of the bell curve
Ho: M<= Mo
Ha: M> Mo
What type of tailed test is this
one tailed
the line goes to the left of the bell curve
Ho: M = Mo
Ha: M does not = Mo
WhaT type of test is this
two tailed test
the lines goes on either end of the bell curve
What is a type I error
probability of rejecting a true Ho
- the prob of committing a type I error is just a significance level a
What are common choices for type I error /a
- 0.05 & 0.01
- this controls the probability of making this type of error
If the cost of making a type I error is not too high,
larger values of a are typically used
What does a significance test control
only controlling type I error
- most applications only control type I error
Due to the uncertainty associated with making a type II error we should say
Do not reject Ho instead of accept Ho
Whenever the prob of making type II error has NOT been determined and controlled use
Do not reject No or Reject Ho
Is controlling type II errors common
no
If proper controls have been established for Type II errors, it can be3 appropriate to use
accept Ho
What are the steps for calcualting the prob of a Type II error
- gather a sample and calculate the sample average and the pop SD
- suppose you know the true average is 58%
- calculate the prob of type II error with a claim at the 1% significance level
- state the Ho and Ha
- calcualte the z score for the claim
What are the steps for calculating the prob of a Type II error
- gather a sample and calculate the sample average and the pop SD
- suppose you know the true average is 58%
- calculate the prob of type II error with a claim at the 1% significance level
- find the area of 0.01 which is a z-score of -2.33 - solve for CV
CV = M+(-2.33)(SE)
- put this number on the curve at the cv line
- the mean for this curve is the M you are trying to find (H0: M) - standardize the number from step 4 the cv
- calculate the z-score for this number M as the actual known mean
- find out what the area is for this z-score
1- this area is the prob of making the type 2 error `
What is the power of the test
prob of correctly rejecting Ho when it is false
what is the formula for power - which type of error is this used for
1-B
- used to find the prob of making a type II error
what is 1 in the power
1 is the prob of correctly rejecting HO
what is B in the power test
prob of making type II error
what do you use in order to calculate the power test
use table 9.7
the prob of a type 2 error depends on what value
depends on the value of the pop mean
for the power test, for values of M near M0, the prob of making teh type II error
can be high
What is the graph for the power called
the power curve
the power curve extends over what values
values of M for which the hypothesis is false
the height of a power curve at any value of M indicates what
the prob of CORRECTLY rejecting HO when Ho is false
Common confidence levels state their significance level and critical value z -score
.90
.95
.99
Confidence Significance CV
.90 .10 1.645
.95 .05 1.96
.99 .01 2.575
applications of hypothesis tests that only control for Type I errors are called
significance tests
What is the p-value
the area for the z-score
definition for p-value
a probability that provides a measure of the evidence against the null hypothesis provided by the sample.
what does a smaller p-value indicate
more evidence against the H0
using the p-value, when do you reject the HO
if p-value is <= a
what is the p-value also called
the observed level of significance
How do you calculate the hyp test for p-value approach
- calculate the z score for the sample mean (x bar)
z= x bar - M0 / se - find the area for the z-score (p-value)
- if p-value is larger than the critical value is a= 0.01
do not reject HO
if p-value is smaller than CV - reject Ho
the p-value approach and CV approach will always lead to what conclusion
the same conclusion
what is the advantage of the p-value approach
tells us how significant the results are
if we only use the CV approach, what can we say
we only know that the results are significant at the stated level of significance
what is the rule for computing the p-value in the lower tailed test
compute the prob that z is less or equal to the value of the test statistic (area in the lower tail)
what is the rule for computing the p-value in the upper tailed test
compute the prob that z is greater than or equal to the value of the test statistic (upper tail area)
how do you calculate the hyp test using CV for two tailed test
- a / 2 (example a = 0.05) then it’s 0.025
- find z -score for 0.025
- z = -1.96 and z = 1.96
- calculate the z-score for the test statistic ie z = 1.53
- compare the two z scores by drawing on the bell curve
if the test statistic is is less than -1.96 or greater than 1.96 reject Ho
if it is in between the two values, reject Ho
how do you calculate the hyp test using p-value for two tailed test
- calculate the z-score for the test statistic (using sample mean x bar)
- calculate the area for that z-score
- add
p-value test reject Ho if
p-value is less than or equal to a
critical value approach - reject Ho if
need to know what type of tailed test it is
critical value approach in the lower tailed test, reject HO if
z for test statistic is is less than or equal to -z for the cv
critical value approach in the upper tailed test, reject Ho if
if Z for test statistic is greater than or equal to z for the cv
critical value approach in the two tailed test, reject Ho if
if z for test statistic is less than or equal to the -za/2 for the cv OR if z for test stat is greater than or equal to z a/2
Confidence interval approach to Hyp test for single tailed test
if the confidence interval contains the hypothesis value Mo DO not reject HO
if the confidence interval does not contain the hyp value Mo for single tailed test
reject Ho
for a two tailed hyp test for an interval reject HO if
the confidence interval does not include Mo
by controlling the sample size the user can also control what
the prob of making a type 2 error
level of significance determines what
the prob of making a type I error
controlling the sample size controls what
the prob of making a type II error
sample size for hyp test how do you calculate c
c = Mo - Za x SE
or
C = Ma + Zb x SE
both equations must provide the same value for C
what is the formula to determine the required Sample size to control type 1 and type 2 errors
N = (za +zb)squared x Qsquared / (Mo + Ma) squared
for a given level of significance a, increasing the sample size will
reduce B
for a given sample size decreasing a will
increase B
for a given sample size increasing a will
decrease B
When the prob of type II error is not controlled, this suggests
that one should not choose unnecessarily small values for the level of significance a
When the prob of type II error is not controlled, this suggests
that one should not choose unnecessarily small values for the level of significance a
for a given samples size, choosing a smaller level of signficance means
more exposure to a type 2 error
smaller values of a (significance level) have the disadvantage of
increasing the prob of making a type II error
for a given samples size, choosing a smaller level of significance means
more exposure to a type 2 error
smaller values of a (significance level) have the disadvantage of
increasing the prob of making a type II error
conclusions of the hypothesis testing and interpretation of the results are dependent upon what
a clear and sound definition of the pop parameter being tested (often M or Q)
conclusions of the hypothesis testing and interpretation of the results are dependent upon what
a clear and sound definition of the pop parameter being tested (often M or Q)
with type I and Type 2 errors, a and B are what
probabilities of TYpe 1 and type 2 errors and not the errors
How can we control (or minimize) type 1 and type 2 errors
with a fixed sample size, but not both of them at the same time
- they are conflicting objectives that cannot be achieved simultaneously unless additional data re added by increasing the sample size
Ideally we would like a and B to be what
as small as possible
when the sample size becomes larger and larger and reaches infinity, both a and B become
zero, which is the smallest possible value
- this is what happens when the sample is the whole population
A local pizza store knows the mean amount of time it takes them to deliver an order is 45 minutes after the order is placed. The manager has a new system for processing delivery orders, and they want to test if it changes the mean delivery time. They take a sample of delivery orders and find their mean delivery time is 48 minutes.
What are appropriate hypotheses for their significance test?
H 0:μ=45 minutes
H a:μ not =45 minutes
A ketchup company regularly receives large shipments of tomatoes. For each shipment that is received, a supervisor takes a random sample of 500 tomatoes to see what percent of the sample is bruised and performs a significance test. If the sample shows convincing evidence that more than 10%, percent of the entire shipment of tomatoes is bruised, then they will request a new shipment of tomatoes.
What are appropriate hypotheses for the company’s significance test?
Ha:p: > .10
the average running time for a computer was 36 hours. It is believed that a new program reduced the average hours per week. What is the hypothesis
HO:M>= 36
Ha: M< 36
The average cost of heating per month for a business is $700. Because of the weather change, it is believed that there is a decrease in the average cost per month.
what is the null and alternative hypothesis
Ho:M > 700
Ha: M <700
A manufacturing process fills cans with 8.2 oz of liquid. the manufacturer does not want over or under filling to happen. What is the null and alternative hypothesis
HO:M = 8.2
Ha:M does not = 8.2
The average rate of pay for a labourer with 2 years of experience is $25/hr. Because of the high demand for labourers, it is believed there has been a significant increase in the average wage of labourers. What is the null and alternative hypothesis
HO: M < or = 25
Ha: M >25
A friend thinks the average grade on a final exam is at least 75. You take a sample to test this belief. What is the null and alternative hypothesis
HO: m >= 75
Ha: m does not <75%
In the past 25% of people who visit the job site went to see the demo phase. Recently the co undertook some extensive advertising showing the demo of their projects. They are interested in determining whether the advertising actually increased the proportion of visitors who visited the demo phase at job sites. what is the null and alternative hyp
HO:p<= 75%
Ha: P>75%
The average tire produced by CDNT has been 40,000 miles. Mgmt thinks that due to new production process, the life expectancy of their tires has increased.
what is the null and alternative hyp,
Ho: m<=40000
Ha:m>40,000
A company fills liquid detergent bottles with 12 oz. any over or under filling causes a machine to be readjusted and the department shuts down. To determine if the machine is running properly, the null and alternative hyp is
Ho: M = 12
Ha: M does not = 12
A school is thinks that at least 35% of all students will need to attend tutoring classes after school.
what is the correct null and alternative hyp
Ho:p >= 35
Ha: p< 35
If you were looking at increasing a mean from its current avg. lets say of 5 what is the null and alternative hyp
HO: M <=5
Ha: M> 5
if you are expecting an avg to be AT LEAST 10%. what is the null and alternative hyp
HO:M>= .10
Ha: M<10
if you are expecting an avg to be 80 OR LESS. What is the mull and alternative hyp
Ho: M <= 80
Ha: M > 80
If you are expecting a proportion to be AT LEAST 30%. waht is the null and alternative hyp
Ho: p >=.30
Ha: p
the p-value must be a number between
zero and 1
what type of error occurs if you fail to reject HO when it is not true
Type 2
THe prob of committing a Type I error when the null hypothesis is true is
the level of significance
In hypothesis testing, the smaller the type I error, the (what type 2 error)
the larger the type 2 error will be
In hype testing the tentative assumption about the pop parameter is
the null hypothesis HO
for a lower tailed test, the p-value is the prob of obtaining a value for the test statistic at least as
small as the that provided by the sample
the p-value is a prob that measures the support (or lack of support) for the
null hypothesis HO
the p-value is a
probability
for a two -tailed test, the p-value is the prob of obtaining a value for the test statistic as
unlikely as that provided by the sample
the level of significance is a maximum allowable
probability of a type 1 error
the power curve provides the prob of
correctly rejecting the null hypothesis
a type II error is committed when
a true alternative hypothesis is mistakenly rejected
the error of rejecting a true null hypothesis is
a type I error
In hype testing, if the null hypothesis is rejected when the altnernative is true
the correct decision has been made
as the test statistic becomes larger, the p-value
gets smaller
the p-value ranges between
zero to one
if a hyp is rejected at the 5% level of significance, it will be (rejected/accepted) at the 1% level
may be rejected or not rejected at the 1% level
if a hype is NOT rejected at the 5% level of significance, it will be (rejected/accepted) at the 1% level
will also not be rejected at the 1% level
