lecture 7 Flashcards
Describe the two types of error possible in a hypothesis test decision.
Type 1 error (occurs if null is rejected but null was true; bad scientist analogy)
Type 2 error (occurs when null is not rejected when it was false; a significant finding was
missed
Which decision results in a Type II Error?
Rejecting the claim p ≤ 0.26 when it is actually the case that p ≤ 0.26.
Not supporting the claim p > 0.26 when it is actually the case that p ≤ 0.26.
Not supporting the claim p > 0.26 when it is actually the case that p > 0.26.
Rejecting the claim p ≤ 0.26 when it is actually the case that p > 0.26.
I don’t know yet
Not supporting the claim p > 0.26 when it is actually the case that p > 0.26.
A type II error is committed when Ho is false and it is not rejected.
The alternative hypothesis (Hₐ) is actually true, but we fail to support it (we stick with H₀).
So, if Hₐ: p > 0.26 is true, but we don’t support it, that’s a Type II Error — we missed detecting a real effect
An environmental action group took six readings to analyze whether the true mean discharge of wastewater per hour exceeded the company claim of 800 gallons.
When a decision is made using a one-sided test using α = 0.01…
Answer
Unselected
If Ho is rejected, the probability that it is actually true is 0.01.
Unselected
The probability that the sample mean equals exactly the observed value is 0.01 if Ho were true.
Unselected
If the plant is exceeding the limit, there is only a 1% chance that we will conclude that they are not exceeding the limit.
Unselected
If the plant is not exceeding the limit, but actually μ = 800, there is only a 1% chance that we will conclude that they are exceeding the limit.
If the plant is not exceeding the limit, but actually μ = 800, there is only a 1% chance that we will conclude that they are exceeding the limit.
Key points about the alternative hypothesis:
It states that a change, difference, or relationship does exist.
It’s what researchers hope to support or prove.
It’s typically written as Hₐ: followed by an inequality (≠, >, <).
Two statisticians, John and Kathryn, used a t-test with the same statistics to perform a hypothesis test.
John tested the claim μ > 50 and Kathryn tested the claim μ ≠ 50.
Knowing that they got the same value for the standardized test statistic, how do their P-values compare?
John’s P-value is twice Kathryn’s.
Kathryn’s P-value is twice John’s
Their P-values are the same.
Kathryn’s two-tailed P-value must be twice John’s one-tailed P-value.
This is because Kathryn’s test looks for a difference in either direction, while John only checks for values greater than 50.
To test the claim μ ≤ 50, it is determined that the requirements to use the standard normal distribution are met.
If α = 0.05, which of the following describes the critical region?
Answer
z₀ > 1.96
z₀ < -1.96
z₀ > 1.645
z₀ < -1.645
z₀ > 1.645
The critical region is determined by the alternative hypothesis, which is μ>50. Thus, this is a right-tailed test.
This means that the critical value is the value of z0 that leaves an area of 0.05 to its right. This means that the critical value is z0 = 1.645, which means the critical region is z0 > 1.645.
An urban planner wants to test the claim that at least 23% of all residents in a large city are in favour of expanding mass transit.
After performing the appropriate hypothesis test, the decision is to reject H₀. What is the appropriate conclusion?
There is not sufficient evidence to reject the claim that at least 23% of all residents in a large city are in favour of expanding mass transit.
There is sufficient evidence to reject the claim that at least 23% of all residents in a large city are in favour of expanding mass transit.
There is not sufficient evidence to support the claim that at least 23% of all residents in a large city are in favour of expanding mass transit.
There is sufficient evidence to support the claim that at least 23% of all residents in a large city are in favour of expanding mass transit.
There is sufficient evidence to reject the claim that at least 23% of all residents in a large city are in favour of expanding mass transit.
The claim is p≥23
, which is a null hypothesis. Therefore, the decision is based on rejecting the claim.
Since the null hypothesis is rejected, there is enough evidence to reject the claim that at least 23
of all residents in a large city are in favor of expanding mass transit.
type 1 error vs type 2
Type I Error (False Positive)
What it is: Rejecting a true null hypothesis (H₀).
Example: You conclude that less than 23% of residents support mass transit when, in fact, 23% do.
Outcome: You think you found something significant, but it’s not actually true.
Probability of making a Type I Error: α (alpha), the significance level (commonly set to 0.05), represents the chance of making a Type I Error.
Type II Error (False Negative)
What it is: Failing to reject a false null hypothesis (H₀).
Example: You conclude that 23% of residents support mass transit when, in fact, less than 23% do.
Outcome: You miss detecting a real effect.
Probability of making a Type II Error: β (beta), and it’s influenced by sample size, variability, and the actual effect size.
The manager of a coffee shop claims that the average amount of time spent waiting for a hot drink is 5.4 minutes.
Which hypothesis test is most appropriate to test this claim? Assume that the sample was collected at random from a normally distributed population.
t-test for a population mean
z-test for a population mean
z-test for a population proportion
χ2-test for a variance or standard deviation
t-test for a population mean
When do you use a t test?
t-test for a Population Mean:
A t-test is used when you want to test a claim about a population mean (μ) and:
You are working with a small sample size (typically n < 30).
The population standard deviation (σ) is unknown.
The data is assumed to come from a normally distributed population.
When to use a z test for a pop mean?
The population standard deviation (σ) is known:
The key difference between a z-test and a t-test is that for a z-test, you must know the population standard deviation (σ). In contrast, for a t-test, the population standard deviation is unknown, and we use the sample standard deviation (s) as an estimate.
The sample size is large (typically n ≥ 30):
If the sample size is large, the Central Limit Theorem suggests that the sampling distribution of the sample mean will be approximately normal, even if the population distribution is not. Therefore, a z-test can be used.
If the sample size is smaller than 30, and the population standard deviation is known, you may still use a z-test, but the population should be approximately normally distributed.
The data is approximately normally distributed (especially important if the sample size is small, n < 30):
Which of these claims would result in a left-tailed test?
The average number of sick days taken by each employee last year was at least 6.
On average, a family of 4 spends $300 per week on groceries.
Most employees are in favor of having more days per week to work at home.
A left-tailed test occurs when the alternative hypothesis (Ha) contains a “<” symbol.
The statement “The average number of sick days taken by each employee last year was at least 6. “ is a null hypothesis, written μ≥6
. Thus, the alternative hypothesis is μ<6
, which results in a left-tailed test.
The statement “Most employees are in favor of having more days per week to work at home. “ is an alternative hypothesis, written p > 0.5 (most means more than 50%, which is a statement about a proportion). This results in a right-tailed test.
The statement “On average, a family of 4 spends $300 per week on groceries. “ is a null hypothesis, written μ=300
. Thus, the alternative hypothesis is μ≠300
, which results in a two-tailed test.
why do we work with alternative hypothesis when finding out what tailed test
When determining whether to use a one-tailed or two-tailed test, we focus on the alternative hypothesis (Hₐ) because it indicates the direction of the effect or relationship we are trying to test for. The alternative hypothesis specifies what we believe is true or what we want to prove, and this guides whether the test should focus on one side (one-tailed) or both sides (two-tailed) of the distribution.
A hypothesis test for a population mean yields a standardized test statistic of t = -2.13.
Which of the following are possible claims for the hypothesis test (Select all that apply)?
Select All That Apply
μ>50
μ=50
μ<50
μ=50
μ<50
If the test statistic is negative, the test is either left-tailed or two-tailed.
When the test is left-tailed, the sample mean is less than the population mean, which produced a negative t-value.
When the test is two-tailed, the sample mean could be either more or less than the population mean, which means a negative test statistic is possible.
When to use a z-test for a population proportion
%
The claim μ = 10 was tested using the normal distribution. Rounded to two decimal places, the standardized test statistic is z = 2.13.
What is the P-value for this test?
0.0166
0.9834
0.9668
0.0332
0.0332
The claim μ=10
is a null hypothesis with alternative hypothesis μ≠10 , which results in a two-tailed test.
Since z = 2.13 is positive, the P-value is the area to the right of z = 2.13, then doubled.
If you look up z = 2.13 in the z-chart, you obtain 0.9834 as the area to the left.
Then, the area to the right is 1 - 0.9834 = 0.0166.
Finally, the P-value is 2(0.0166) = 0.0332.
Which statement is not true about all one-sample hypothesis tests from this chapter?
If the standardized test statistic is not in the critical region, fail to reject the null hypothesis.
If n ≥ 30, then a z-test is used to test the claim.
Unselected
The null hypothesis is the statement that allows the parameter to be equal to a specified value
If P ≤ α, reject the null hypothesis.
b
A manufacturer wants to test a claim that the variance of the production times is no more than 40.
A sample of 12 production times gave a mean of 5 minutes and a variance of 30.
To test this claim using α = 0.05, what is the value of the test statistic?
8.25
For variance tests, we use the Chi-squared test statistic formula:
Each statement below its true about hypothesis testing except….
Since decisions are made based on samples, there is some risk (error) involved.
Unselected
The smaller the P-value, the more evidence there is to reject the null hypothesis.
Unselected
If the goal is to support a claim, write it as an alternative hypothesis.
Hypothesis tests are used to prove whether a claim is true or false.
What is a hypothesis test?
A process that uses sample statistics to test a claim about the value of a population parameter. This is a formalized way of
using statistics to infer our predictions and research findings.
To test a population parameter, you will state a pair of hypotheses what do they each represent?
- One represents the claim, and the other is the complement (recall from probability)
- Either hypothesis—the null hypothesis or the alternative hypothesis—may represent
the original claim.
Null hypothes ( Ho) is
<, =, or >.
alternative hypothesis Ha is….
the complement of the null hypothesis. It is
a statement that must be true if H0 is false and it contains a statement of
strict inequality, such as >, not=, or <.
Write each claim as a mathematical sentence. State the null and alternative hypotheses, and identify which represents the claim.
a) A car dealership announces that the mean time for their oil change service is less than 15 minutes
Null: u>_15
Alternate: u <15 (claime)
