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