Statistics - Chapter 10
Decribe the alternate hypothesis
- It is also called the "research hypothesis". - It is accepted as a consequence of the null hypothesis being rejected.
Correctly describe the null hypothesis.
- We either "reject" or "fail to reject" it, we cannot say that we "accept it" or that it is "true". - It is developed for testing purposes.
How does the use of a two-tailed test effect the size of the rejection area?
Because there are two tails and the total significance remains the same, the area in each tail is half as much.
If the test statistic for a hypothesis test (H0: μ=11) is z = 1.82 and the critical value for 5% significance is zc = 1.96, what should our conclusion be?
Fail to reject the null.
Which of the following are essential elements of hypothesis testing?
-It uses probability theory to determine if a hypothesis is a reasonable statement. -It makes use of sample data.
Why do many statisticians prefer the use of "fail to reject the null hypothesis" instead of "accept the null hypothesis"?
Because only by rejecting the null hypothesis can we calculate the probability of a Type I error. To emphasize that there is always the possibility of a Type II error, which typically cannot be quantified.
What is the correct procedure when you want to do a test of the population mean but the population standard deviation is unknown?
Find the sample standard deviation and use it as an estimate for the population standard deviation. The t-distribution is used in place of the z-distribution, assuming a normal population.
Suppose you are conducting a test with α=0.10, which of the following are true? Select all that apply.
For H0: μ≥25, the lower tail should include a rejection region equal to 0.10. For H0: μ=25, both the upper and lower tails include a rejection region equal to 0.05.
Which one of the following sets of hypotheses requires the use of a two-tailed test?
H0: μ = 5.6, H1: μ ≠ 5.6
Which test hypothesis uses the correct protocol for stating an alternate hypothesis?
H1: μ ≠ 24 H1: μ < 30
Which of the following statements accurately describe the p-value?
If we reject the null, it is the probability of making a Type I error. The null is rejected when the p-value is less than α.
What is the last step in the six step Hypothesis testing procedure?
Interpret the result
Suppose you are performing a hypothesis test with σ unknown, n=29, α=0.01, and the following hypotheses: H0: μ ≤ 24 H1: μ > 24 What is the decision rule?
Reject H0 if the test statistic is greater than 2.467.
Suppose you are performing a hypothesis test with σ unknown, n=23, α=0.10, and the following hypotheses: H0: μ ≥ 24 H1: μ < 24 What is the decision rule?
Reject H0 if the test statistic is less than -1.321.
Suppose you are performing a hypothesis test with σ unknown, n=23, α=0.10, and the following hypotheses: H0: μ ≥ 24 H1: μ < 24 What is the decision rule? Multiple choice question. Reject H0 if the test statistic is less than
Reject H0 if the test statistic is less than -1.321.
What is the p-value?
The probability that a sample value would be as far or further from the expected value, given that the null hypothesis is true.
A soda bottling company fills bottles with 12 ounces of soda. Overfilling causes the company to give away free soda. Underfilling causes the company to cheat the customer. To test to see if the process is working correctly, the company conducts a hypothesis test using the following hypotheses: Ho: μ=12 H1: μ≠12 If we fail to reject the null, what would the interpretation be?
There is insufficient evidence to conclude that the mean is different from 12.
If we fail to reject the null, what would the interpretation be?
There is insufficient evidence to conclude that the mean is different from 12.
Why do many statisticians prefer the use of "fail to reject the null hypothesis" instead of "accept the null hypothesis"? Select all that apply.
To emphasize that there is always the possibility of a Type II error, which typically cannot be quantified. Because only by rejecting the null hypothesis can we calculate the probability of a Type I error.
Which of the following represents the probability of failing to reject the null hypothesis when it is false?
β
What are accurate descriptions of the null hypothesis?
-The null hypothesis is designated H0. -The purpose of the test is to prove the null hypothesis is false. -The term "null" refers to no significant difference.
Find the p-value (to two significant digits) for the following test. H0: μ = 0, H1: μ ≠ 0, σ = 1, z = 2.06 Hint: the population follows the standard normal distribution.
0.04
Find the p-value (to two significant digits) for the following test. H0: μ ≤ 0, H1: μ > 0, σ = 1, z = 1.5 Hint: the population follows the standard normal distribution.
0.07
Suppose a hypothesis test, using α = 0.05, is being conducted with the following null hypothesis: H0: μ = 2. Which one of the following confidence intervals would lead to rejecting the null hypothesis?
A 95% confidence interval of 3 to 4.
Which of the following is correct?
A 99% confidence interval can be used to conduct a two-tailed test with α=0.01.
Which one of the following best describes Hypothesis Testing?
A procedure based on sample evidence and probability to see if a hypothesis is a reasonable statement.
A statement that is accepted if there is significant evidence that the null hypothesis is false.
A statement about a population parameter subject to verification.
Choose the best definition of the Alternate Hypothesis, H1?
A statement that is accepted if there is significant evidence that the null hypothesis is false.
In the context of hypothesis testing, what is a test statistic?
A value, determined from sample information, used to test the null hypothesis.
We wish to test H0: μ ≤ 12 and H1: μ > 12 at the 0.05 level of significance. Which of these statements are correct?
Fail to reject H0 if z < 1.65 Reject H0 if z > 1.65
We wish to test H0: μ ≥ 30 and H1: μ < 30 at the 0.05 level of significance. Which of these statements are correct? Select all that apply.
Fail to reject H0 if z > -1.65 Reject H0 if z < -1.65
A bag of potatoes has a stated weight of 10 pounds with σ = 0.75 pound. A sample of 30 bags has an average weight of 10.3 pounds. Conduct a hypothesis test using a 1% significance level for: H0: μ = 10 H1: μ ≠ 10
Fail to reject the null hypothesis
A bag of potatoes has a stated weight of 10 pounds with σ = 0.75 pound. A sample of 30 bags has an average weight of 10.3 pounds. Conduct a hypothesis test using a 1% significance level for: H0: μ = 10 H1: μ ≠ 10 Multiple choice question. Reject the null hypothesis
Fail to reject the null hypothesis
A brand of chocolate bar has a stated weight of 6 oz. with σ = 0.25 oz. A sample of 9 bars has an average weight of 6.05 oz. Test H0: μ = 6 oz. H1: μ ≠6 oz. at the 5% significance level.
Fail to reject the null hypothesis
Which of the following sets of hypotheses require the use of a one-tailed test? Select all that apply.
H0: μ ≤ -25, H1: μ > -25 H0: μ ≥ 7.5, H1: μ < 7.5
There is insufficient evidence to conclude that the mean is different from 12.
H0: μ ≥ 7.5, H1: μ < 7.5 H0: μ ≤ -25, H1: μ > -25
The result of a hypothesis test was traditionally reported as either "reject H0" or "fail to reject H0". What is the purpose of reporting the p-value in addition?
It provides additional insight into the strength of the decision.
A model of car claims mileage of 24 mpg. with σ = 4 mpg. A sample of 4 cars got an average of 20.5 mpg. Test H0: μ = 24 H1: μ ≠ 24 at the 10% significance level.
Reject the null hypothesis
A paint manufacturer claims that a gallon of their paint will cover at least 1200 square feet of smooth wall with σ = 80 square feet. Thirty-six gallons of paint were tested and the average square feet covered was 1175. Conduct a hypothesis test using a 5% significance level for: H0: μ ≥ 1200 H1: μ < 1200
Reject the null hypothesis
What is the "critical value" for a hypothesis test?
The dividing point between rejecting and failing to reject the null hypothesis.
A tire manufacturer claims its new tire has an average tread life of 80,000. To test to see if the process is true, the company conducts a hypothesis test using the following hypotheses: Ho: μ=80,000 H1: μ≠80,000 If the null is rejected, what would the interpretation be?
The manufacturer's claim is not true - the population average is not 80,000
What is the meaning of "level of significance" in the context of hypothesis testing?
The probability of rejecting the null hypothesis when it is true.
Choose the statement that best explains the use of the sample standard deviation in tests of the mean where the population standard deviation is not known.
The sample standard deviation is used as an estimate for the population standard deviation.
Why do many statisticians prefer the use of "fail to reject the null hypothesis" instead of "accept the null hypothesis"? Select all that apply.
When the null hypothesis is not rejected when it should have been rejected, there is always the chance if a Type II error. When the null hypothesis is rejected when it should not be rejected, there is always the chance if a Type I error.
A one-tailed and a two-tailed test have different critical values at the same significance level. Match the critical values to the to the correct test.
z=1.96 or -1.96 H0:μ = 0, H1:μ ≠ 0, α=0.05 H0:μ = 0, H1:μ ≠ 0, α=0.05 z=-2.33 H0:μ≥0, H1:μ<0, α=0.01 H0:μ≥0, H1:μ<0, α=0.01 z=2.58 or -2.58 H0:μ=0, H1:μ≠0, α=0.01 H0:μ=0, H1:μ≠0, α=0.01 z=1.65 H0:μ≤0, H1:μ>0, α=0.05 H0:μ≤0, H1:μ>0, α=0.05