Chpt 9 - Hypothesis Tests Flashcards
What is a hypothesis?
An educated guess about something in the world
What is hypothesis testing?
An important statistical procedure to test a hypothesis. It evaluates the null hypothesis and alternative hypothesis and determines which statement is best supported by the experiment results
What are the two hypothesis set up for hypothesis testing?
Ho: null hypothesis
Ha: alternative hypothesis
Which hypothesis is the outcome you want to prove to be true?
Ha: alternative hypothesis
Why is the null hypothesis assumed to be true?
We cannot prove that anything is actually true, we can only prove a statement is not true. So we test the null hypothesis because we can say it is not true and reject it or that it is “true enough” that it cannot be rejected
What happens that results in a type 1 hypothesis error?
How is a type 1 error notated?
The null hypothesis is true but we rejected it
noted as α (alpha)
What happens that results in a type 2 hypothesis error?
How is a type 2 error notated?
The null hypothesis is false but it is not rejected
Notated as β
What type of error results when a null hypothesis is true but is rejected?
Type 1 α error
What type of error results when a null hypothesis is true but is not rejected?
Nothing, this is what should happen
What type of error results when a null hypothesis is false but it is rejected?
Nothing, this is what should happen
What type of error results when a null hypothesis is false but it is not rejected?
Type 2 β error
If a machine is working properly, the mean volume should be 100 mL. Set up the hypotheses to test if the machine is not working correctly
Ho: μ = 100
Ha: μ ≠ 100
What is the probability of making a type 1 error called?
Significance level α
What is the probability of making a type 2 error called?
Type 2 error probability β
If the sample size is fixed, the smaller the significance level α, the ______ the type II error probability β and vice versa
larger
In a hypothesis test, because type II error probability β is not easy to define, what do we usually control instead?
significance level α
There are 2 methods of hypothesis testing for one population mean and which one we chose depends on if the population standard deviation is known. What are the 2 methods?
Z test when we know the population standard deviation
t test when we do NOT know the population standard deviation
What is the hypothesized mean equal to and why?
μ = μo
because the Ho is assumed to be true
What is the statistic that measures the distance between the sample mean x̄ and the hypothesized mean μo in one population mean hypothesis test called?
Test statistic
What is the relationship between the significance level and the type II error probability?
As the significance level decreases, the type II error probability increases
If the true population of mean μ of X is equal to μo and the population standard deviation is σ, what is the mean and standard deviation of x̄?
Mean = μo
Standard deviation = σ/√n
When sample size n is large enough or the parent distribution is normal, what distribution does the sample mean follow?
approximately normal distribution
What is the test statistic used for hypothesis testing when σ is known? What does this mean for the mean and standard deviation?
σ/√n
Mean of 0, standard deviation of 1
What is the critical value approach to perform hypothesis tests?
It finds a threshold, or critical value, and compares that to the test statistic. It can be two-tailed, left-tailed, and right-tailed depending on the alternative hypothesis
What type of hypothesis test is:
Ho: μo = 0
Ha: μo ≠ 0
How many critical values exist in this case?
When do we reject Ho?
What is the value of α?
What is the critical value?
two-tailed hypothesis
There are two critical values, C1 on the low end and C1 on the upper end.
We reject Ho if they are smaller than C1 or larger than C2
α is the value under the density curve less than C1 and more than C2
C1 & C2 = ±Zα/2
OR
C1 & C2 = ±tα/2
Describe the rejection region in a two tailed hypothesis and what is α?
In a two tailed hypothesis, the area below C1 and above C2 are the rejection regions. their sum is α. The significance level of α is set up prior to doing the test statistic to ensure we set an adequate value rather than make the numbers explain what we want them to.
When we know σ, and the sample size is large enough or parent distribution is normal, the test statistic follows ______distribution and C1 is _____ and C2 is _______.
N(0,1)
C1: -Zα/2
C2: Zα/2
A machine fills lotion bottles whose mean volume is expected to be 100 mL. Exact volumes of 50 bottles were measured. The sample mean of this sample is 98 mL and assume that the population standard deviation is 5 mL. A researcher claims that the machine is NOT working properly. We want to test the statement is correct at the 5% significance level.
What are the hypotheses?
Ho: μ = 100
Ha: μ ≠ 100
A machine fills lotion bottles whose mean volume is expected to be 100 mL. Exact volumes of 50 bottles were measured. The sample mean of this sample is 98 mL and assume that the population standard deviation is 5 mL. A researcher claims that the machine is NOT working properly. We want to test the statement is correct at the 5% significance level.
Check assumptions
Normal distribution, in this case we don’t know the parent population distribution, but we do have a sample size of 50 which is large enough to be an approximately normal distribution
Test is a simple random sample
σ is known
A machine fills lotion bottles whose mean volume is expected to be 100 mL. Exact volumes of 50 bottles were measured. The sample mean of this sample is 98 mL and assume that the population standard deviation is 5 mL. A researcher claims that the machine is NOT working properly. We want to test the statement is correct at the 5% significance level.
Find the critical values
Since the significance value is
α = 5% = 0.05
This is a two-tailed hypothesis so:
α/2 = 0.05/2 = 0.025
C1 = -Z0.025 = -1.96
C2 = Z0.025 = 1.96
so the rejection regions are below -1.96 and above 1.96
A machine fills lotion bottles whose mean volume is expected to be 100 mL. Exact volumes of 50 bottles were measured. The sample mean of this sample is 98 mL and assume that the population standard deviation is 5 mL. A researcher claims that the machine is NOT working properly. We want to test the statement is correct at the 5% significance level.
Calculate the test statistic
Zo =
x̄-μo 98-100
——- = ———- = -2.83
σ/√n 5/√50
A machine fills lotion bottles whose mean volume is expected to be 100 mL. Exact volumes of 50 bottles were measured. The sample mean of this sample is 98 mL and assume that the population standard deviation is 5 mL. A researcher claims that the machine is NOT working properly. We want to test the statement is correct at the 5% significance level.
Compare:
C1: -1.96
C2: 1.96
Test: -2.83
Since Zo is -2.83 it falls in the rejection region; therefore, we reject Ho: μ = 100
A machine fills lotion bottles whose mean volume is expected to be 100 mL. Exact volumes of 50 bottles were measured. The sample mean of this sample is 98 mL and assume that the population standard deviation is 5 mL. A researcher claims that the machine is NOT working properly. We want to test the statement is correct at the 5% significance level.
We rejected Ho: μ = 100; so interpret the results in context
At the 5% significance level, the data provides sufficient evidence to conclude that the mean volume is not 100 mL
What type of hypothesis test is:
Ho: μo = > 0
Ha: μo < 0
How many critical values exist in this case?
When do we reject Ho?
What is the value of α?
What is the critical value?
Left tailed hypothesis
1 critical value C1
We reject Ho when the test statistic value is smaller than C1 (so the numbers smaller than C1 are the rejection region)
α is the value under the density curve less than C1
C1 = -Zα or -tα
A researcher claims that the average weight of 5 year old girls is less than 24kg. A simple random sample of 36 of 5 year old girls picked and the weight of each girl is measured. The average weight of the sample is 20kg. Assume that the population standard deviation is 2kg. We are going to use α = 0.01
What are the hypotheses?
Ho: μ => 24
Ha: μ < 24
A researcher claims that the average weight of 5 year old girls is less than 24kg. A simple random sample of 36 of 5 year old girls picked and the weight of each girl is measured. The average weight of the sample is 20kg. Assume that the population standard deviation is 2kg. We are going to use α = 0.01
Check the assumptions
Normal distribution, in this case we don’t know the parent population distribution, but we do have a sample size of 36 which is large enough to be an approximately normal distribution
Test is a simple random sample
σ is known
A researcher claims that the average weight of 5 year old girls is less than 24kg. A simple random sample of 36 of 5 year old girls picked and the weight of each girl is measured. The average weight of the sample is 20kg. Assume that the population standard deviation is 2kg. We are going to use α = 0.01
Find the critical values
α = 0.01
Only 1 critical value in a left sided test so
C1 = -Z0.01 = -2.33
So the rejection region includes values smaller than -2.33
A researcher claims that the average weight of 5 year old girls is less than 24kg. A simple random sample of 36 of 5 year old girls picked and the weight of each girl is measured. The average weight of the sample is 20kg. Assume that the population standard deviation is 2kg. We are going to use α = 0.01
Calculate the test statistic
Zo =
x̄-μo 20-24
——- = ———- = -12
σ/√n 2/√36
A researcher claims that the average weight of 5 year old girls is less than 24kg. A simple random sample of 36 of 5 year old girls picked and the weight of each girl is measured. The average weight of the sample is 20kg. Assume that the population standard deviation is 2kg. We are going to use α = 0.01
Compare:
C1: -2.33
Test statistic: -12
Since Zo -12 falls into the rejection region, we reject Ho
A researcher claims that the average weight of 5 year old girls is less than 24kg. A simple random sample of 36 of 5 year old girls picked and the weight of each girl is measured. The average weight of the sample is 20kg. Assume that the population standard deviation is 2kg. We are going to use α = 0.01
Interpret:
Since Zo -12 falls into the rejection region, we reject Ho
At the 1% significance level, the data provides sufficient evidence to conclude that the average weight of 5 year old girls is less than 24 kg
What type of hypothesis test is:
Ho: μo = < 0
Ha: μo > 0
How many critical values exist in this case?
When do we reject Ho?
What is the value of α?
What is the critical value?
Right tailed hypothesis
One, which is C2
We reject Ho when the test statistic is greater than C2 (which is our rejection region)
α is the value under the density curve more than C2
C2 = Zα or tα
The average weight of new born babies in 2014 was 7lbs. Determine whether the average weight in 2015 has increased.
A simple random sample of 100 new born babies are picked and the weight of each baby is measured. The average weight of the sample is 7.4 lbs. Assume that the population standard deviation is 2lb. Suppose we are using a significance level of 1%.
Set up the hypotheses
Ho: μ = < 7
Ha: μ > 7
The average weight of new born babies in 2014 was 7lbs. Determine whether the average weight in 2015 has increased.
A simple random sample of 100 new born babies are picked and the weight of each baby is measured. The average weight of the sample is 7.4 lbs. Assume that the population standard deviation is 2lb. Suppose we are using a significance level of 1%.
Check the assumptions
Normal distribution, in this case we don’t know the parent population distribution, but we do have a sample size of 100 which is large enough to be an approximately normal distribution
Test is a simple random sample
σ is known
The average weight of new born babies in 2014 was 7lbs. Determine whether the average weight in 2015 has increased.
A simple random sample of 100 new born babies are picked and the weight of each baby is measured. The average weight of the sample is 7.4 lbs. Assume that the population standard deviation is 2lb. Suppose we are using a significance level of 1%.
Find the critical values
α = 0.01
Only 1 critical value in a right sided test so
C2 = Z0.01 = 2.33
So the rejection region includes values larger than 2.33
The average weight of new born babies in 2014 was 7lbs. Determine whether the average weight in 2015 has increased.
A simple random sample of 100 new born babies are picked and the weight of each baby is measured. The average weight of the sample is 7.4 lbs. Assume that the population standard deviation is 2lb. Suppose we are using a significance level of 1%.
Calculate the test statistic
Zo =
x̄-μo 7.4-7
——- = ———- = 2
σ/√n 2/√100
The average weight of new born babies in 2014 was 7lbs. Determine whether the average weight in 2015 has increased.
A simple random sample of 100 new born babies are picked and the weight of each baby is measured. The average weight of the sample is 7.4 lbs. Assume that the population standard deviation is 2lb. Suppose we are using a significance level of 1%.
Compare
Test statistic is 2
Critical value is 2.33
Since Zo is 2, it is not in the rejection region, therefore we DO NOT reject Ho
The average weight of new born babies in 2014 was 7lbs. Determine whether the average weight in 2015 has increased.
A simple random sample of 100 new born babies are picked and the weight of each baby is measured. The average weight of the sample is 7.4 lbs. Assume that the population standard deviation is 2lb. Suppose we are using a significance level of 1%.
Interpret
Since Zo is 2, it is not in the rejection region, therefore we DO NOT reject Ho
At the 1% significance level, the data does NOT provide sufficient evidence to conclude that the average weight of new born babies in 2015 is heavier than the new born babies in 2014
What is the P value in a two-tailed alternative hypothesis test?
It is the probability that the test statistic is greater than the absolute value of the observed test statistic value or smaller than the negative absolute value given that Ho: μ = μo is true
Because the rejection regions on both sides are equal, the P value probability is the sum of both OR two times either value so:
P value = P(Z < -Z0) + P(Z > Zo)
OR
P value = 2P(Z < -Zo) (2 x the left value)
OR
P value = 2P(Z > Zo) (2 x the right value)
This holds true when using to instead of Zo
What is the P value in a left-tailed alternative hypothesis test?
The P value is the probability that the test statistic is smaller than the observed test statistic value assuming that Ho: μ = μo is true
P value = P(Z < -Zo)
This holds true when using to instead of Zo
What is the P value in a right-tailed alternative hypothesis test?
The P value is the probability that the test statistic is larger than the observed test statistic value assuming that Ho: μ = μo is true
P value = P(Z > Zo)
This holds true when using to instead of Zo
What assumptions need to be checked for a Z test statistic to be used?
Simple random sample
σ is known
There is a normally distributed parent population OR a large enough sample (n>30)
What hypothesis test should be done if σ is unknown?
What is the formula for this test statistic?
What distribution does it follow?
T test
x̄-μo to = -------
s/√n
t-distribution with degrees of freedom n-1
The average person watched 5.6 hours of television per day in 2015. the average hours of television per day of a SRS (49 people) in 2016 is 4.8 hours and its standard deviation is 3.6 hours. At the 10% significance level, does the data provide sufficient evidence to conclude that the hours of television watched per day in 2016 by the average person differed from that in 2015
Set up the hypotheses
Ho: μ = 5.6
Ha: μ ≠ 5.6
The average person watched 5.6 hours of television per day in 2015. the average hours of television per day of a SRS (49 people) in 2016 is 4.8 hours and its standard deviation is 3.6 hours. At the 10% significance level, does the data provide sufficient evidence to conclude that the hours of television watched per day in 2016 by the average person differed from that in 2015
Check assumptions
Normal distribution, in this case we don’t know the parent population distribution, but we do have a sample size of 49 which is large enough to be an approximately normal distribution
Test is a simple random sample
σ is unknown so we need to do a t-test
The average person watched 5.6 hours of television per day in 2015. the average hours of television per day of a SRS (49 people) in 2016 is 4.8 hours and its standard deviation is 3.6 hours. At the 10% significance level, does the data provide sufficient evidence to conclude that the hours of television watched per day in 2016 by the average person differed from that in 2015
Find critical value
α = 10% = 0.1
α/2 = 0.05
degrees of freedom = n-1 = 49 -1 = 48
C1 = -tα/2 = -t0.05 = -1.677
C2 = tα/2 = t0.05 = 1.677
The average person watched 5.6 hours of television per day in 2015. the average hours of television per day of a SRS (49 people) in 2016 is 4.8 hours and its standard deviation is 3.6 hours. At the 10% significance level, does the data provide sufficient evidence to conclude that the hours of television watched per day in 2016 by the average person differed from that in 2015
Calculate test statistic
to =
x̄-μo 4.8-5.6
——- = ———–
s/√n 3.6/√49
= -1.56
The average person watched 5.6 hours of television per day in 2015. the average hours of television per day of a SRS (49 people) in 2016 is 4.8 hours and its standard deviation is 3.6 hours. At the 10% significance level, does the data provide sufficient evidence to conclude that the hours of television watched per day in 2016 by the average person differed from that in 2015
Compare:
test statistic is -1.56
C1 = -1.677
C2 = 1.677
Since to is -1.56 it is not in the rejection region; therefore we DO NOT reject Ho
The average person watched 5.6 hours of television per day in 2015. the average hours of television per day of a SRS (49 people) in 2016 is 4.8 hours and its standard deviation is 3.6 hours. At the 10% significance level, does the data provide sufficient evidence to conclude that the hours of television watched per day in 2016 by the average person differed from that in 2015
Interpret;
Since to is -1.56 it is not in the rejection region; therefore we DO NOT reject Ho
At the 10% significance level, the data does not provide sufficient evidence to conclude that the hours of television watched per day in 2016 by the average person differed from that in 2015
The manufacturer of a new car, the Orion, claims that a typical car gets 26 miles per gallon. Some consumers claim that the gas mileage is less. A simple random sample of 100 Orion was selected. Their mean gas mileage is 25.9 mpg and the sample standard deviation is 1.4 mpg. At 5% significance level, does the data provide sufficient evidence to support the consumer’s claim?
Set up the hypotheses
Ho: μ => 26
Ha: μ < 26
The manufacturer of a new car, the Orion, claims that a typical car gets 26 miles per gallon. Some consumers claim that the gas mileage is less. A simple random sample of 100 Orion was selected. Their mean gas mileage is 25.9 mpg and the sample standard deviation is 1.4 mpg. At 5% significance level, does the data provide sufficient evidence to support the consumer’s claim?
Check the assumptions
Normal distribution - there is not information about the parent population, but the sample size is large enough
Simple random sample
σ is unknown so we need to use t-test
The manufacturer of a new car, the Orion, claims that a typical car gets 26 miles per gallon. Some consumers claim that the gas mileage is less. A simple random sample of 100 Orion was selected. Their mean gas mileage is 25.9 mpg and the sample standard deviation is 1.4 mpg. At 5% significance level, does the data provide sufficient evidence to support the consumer’s claim?
Determine the critical values
α = 5% = 0.05
degrees of freedom = n-1 = 100-1 = 99
C1 = -t0.05 = -1.661
The manufacturer of a new car, the Orion, claims that a typical car gets 26 miles per gallon. Some consumers claim that the gas mileage is less. A simple random sample of 100 Orion was selected. Their mean gas mileage is 25.9 mpg and the sample standard deviation is 1.4 mpg. At 5% significance level, does the data provide sufficient evidence to support the consumer’s claim?
Calculate the test statistic
to =
x̄-μo 25.9-26
——- = ————- = -0.714
s/√n 1.4/√100
The manufacturer of a new car, the Orion, claims that a typical car gets 26 miles per gallon. Some consumers claim that the gas mileage is less. A simple random sample of 100 Orion was selected. Their mean gas mileage is 25.9 mpg and the sample standard deviation is 1.4 mpg. At 5% significance level, does the data provide sufficient evidence to support the consumer’s claim?
Compare
Test statistic is -0.714
C1 is -1.661
Since to is 0.714 is is not within the rejection region, therefore we do not reject Ho
The manufacturer of a new car, the Orion, claims that a typical car gets 26 miles per gallon. Some consumers claim that the gas mileage is less. A simple random sample of 100 Orion was selected. Their mean gas mileage is 25.9 mpg and the sample standard deviation is 1.4 mpg. At 5% significance level, does the data provide sufficient evidence to support the consumer’s claim?
Interpret
Since to is 0.714 is is not within the rejection region, therefore we do not reject Ho
At the 5% significance level, the data does not provide sufficient evidence to support the consumers’ claim
The supplier claims that the main life time of their light bulbs is more than 15 000 hours. Steve randomly samples 36 bulbs and recorded their life times. The sample mean of the sample is 16 000 hours and the sample standard deviation is 2.8 thousand hours. At the 10% significance level, is there strong enough evidence to support the supplier’s claim?
Set up the hypotheses
Ho: μ =< 15
Ha: μ > 15
The supplier claims that the main life time of their light bulbs is more than 15 000 hours. Steve randomly samples 36 bulbs and recorded their life times. The sample mean of the sample is 16 000 hours and the sample standard deviation is 2.8 thousand hours. At the 10% significance level, is there strong enough evidence to support the supplier’s claim?
Check the assumptions
Normal distribution: there is not parent population information, but the sample is greater than n=30 so it will be approximately normal
simple random sample
σ is unknown, so need a t-test
The supplier claims that the main life time of their light bulbs is more than 15 000 hours. Steve randomly samples 36 bulbs and recorded their life times. The sample mean of the sample is 16 000 hours and the sample standard deviation is 2.8 thousand hours. At the 10% significance level, is there strong enough evidence to support the supplier’s claim?
Calculate the critical value
α = 10% = 0.1
degrees of freedom = n-1 = 36-1 = 35
C2 = t0.1 = 1.306
The supplier claims that the main life time of their light bulbs is more than 15 000 hours. Steve randomly samples 36 bulbs and recorded their life times. The sample mean of the sample is 16 000 hours and the sample standard deviation is 2.8 thousand hours. At the 10% significance level, is there strong enough evidence to support the supplier’s claim?
Calculate the test statistic
to =
x̄-μo 16-15
——- = ————- = 2.14
s/√n 2.8/√36
The supplier claims that the main life time of their light bulbs is more than 15 000 hours. Steve randomly samples 36 bulbs and recorded their life times. The sample mean of the sample is 16 000 hours and the sample standard deviation is 2.8 thousand hours. At the 10% significance level, is there strong enough evidence to support the supplier’s claim?
Compare
test statistic is 2.14
C2 is 1.306
Since to is 2.14, it is in the rejection region, therefore we reject Ho
The supplier claims that the main life time of their light bulbs is more than 15 000 hours. Steve randomly samples 36 bulbs and recorded their life times. The sample mean of the sample is 16 000 hours and the sample standard deviation is 2.8 thousand hours. At the 10% significance level, is there strong enough evidence to support the supplier’s claim?
Interpret
Since to is 2.14, it is in the rejection region, therefore we reject Ho
At the 10% significance level, the data provides sufficient evidence to support the supplier’s claim
What are the 6 steps to a hypothesis test?
- Set up hypotheses
- Check assumptions
- Determine significance level and find critical value(s)
- Calculate the test statistic
- Compare the test statistic and critical value(s)
- Interpret the results in context of the question
