Stat Chapter 2
The probability of observing the experiment result, a sample mean, for example, or something more unusual just by chance if the null hypothesis is true is the definition of
P value
You are performing a two-tailed t-test with a sample size of 7 If α = 0.2 α = 0.2 , and your test statistic is t = − 2.76 t = - 2.76 , do you:
Reject Null Hypothesis
A student wondered if more than 10% of students enrolled in an introductory Chemistry class dropped before the midterm. Suppose he performed a hypothesis test to test his claim. In the context of the problem, what would happen if the student made a Type II Error?
He claims that 10% (or less) of students in the introductory Chemistry class dropped before the midterm when, in fact, more than 10% really did drop the class.
Benjamin performed a two-tailed one-sample t-test and obtained a p-value equal to 1. What conclusion should he make?
His sample mean must have been exactly equal to his hypothesized value for the population mean.
A medical study was investigating if getting a flu shot actually reduced the risk of developing the flu. A hypothesis test is performed. Which of the following will result in a Type I error?
Researchers said the flu shot reduced the risk of developing the flu when it actually didn't
Reject Null Hypothesis
Test statistics is higher than the alpha
The vice-president of operations wondered if the average strength of wire cables was different between those produced at the company's plant in a rural location and those produced in the company's plant located in a large city. Which of the following is the correct statement of what a Type II Error is in the context of this problem?
The VP did not have evidence to say that there was a difference in the average cable strengths between the two locations when in fact there was a difference in the average strengths.
Jeremy performs a t-test and obtains a t-statistic of 0. Based on this information, which of the following is NOT true.
The sample mean is the same as the hypothesized value of the population mean
Type 2
The statement describes a situation where we fail to reject a false null hypothesis.
Type 1
The statement describes a situation where we reject a valid null hypothesis.
A p-value is the probability that the null hypothesis is true.
This statement is false. The null hypothesis will either be true or it won't be true - there is no probability associated with this fact. A p-value is the probability of observing a sample mean (for example) that we did or something more unusual just by chance if the null hypothesis is true.
Determine if the following statement is true or false. If it is false, explain why. A p-value is the probability of accepting the null hypothesis.
This statement is false. We never accept the null hypothesis no matter what the p-value is. A p-value is the probability of observing a sample mean (for example) or something more unusual just by chance if the null hypothesis is true.
When are conclusions said to be "statistically significant"?
When the p-value is less than a given significance level.
Below is a hypothesis test set up by a student who recently took introductory statistics: H0: x = 5 HA: x ≠ 5 The sample mean of 100 cases used to implement the hypothesis test is x = 4.2. Which of the following statements are accurate? This is a one-sided hypothesis test. There is an error in how these hypotheses were constructed. It would be reasonable to swap "<" for "≠" in the alternative hypothesis.
ii only
A medical study was investigating whether getting a flu shot actually reduced the risk of developing the flu. A hypothesis test is to be performed. Which of the following statements is correct?
A one-tailed test will be performed since the alternative hypothesis states that the parameter is less than the hypothesized value.
The vice-president of operations wondered if the average strength of wire cables was different between those produced at the company's plant in a rural location and those produced in the company's plant located in a large city. Which of the following statements is correct?
A two-tail test will be performed since the alternative hypothesis contains a not equal to.
Researchers conducted a study and obtained a p-value of 0.85. Because the p-value is quite high, there is evidence to accept the null hypothesis.
False. We do not accept a null hypothesis. The large p-value indicates that there is a high probability that we would have gathered our evidence assuming the null is valid. Thus, there is not enough evidence to reject the null hypothesis and so we continue to assume it is valid.
The p-value for a hypothesis test turns out to be 0.0101. At a 2% level of significance, what is the proper decision?
Reject Null hypothesis
A local government office in a rural area conducts a study to determine if elementary schoolers in their district have a longer average one-way commute time. If they determine that the average commute time of students in their district is significantly higher than the commonly cited standard they will invest in increasing the number of school busses to help shorten commute time. What would a Type II error mean in this context?
The local government decides that the data do not provide convincing evidence of an average commute time higher than 30 minutes, when the true average commute time is in fact higher than 30 minutes.
Which of the following facts about the p-value of a test is correct?
The p-value is calculated under the assumption that the null hypothesis is true. The p-value can have values between -1 and 1.
The smaller the p-value, the more likely that our evidence has significance (meaningful results).
True. Smaller p-values indicate that the probability of our evidence being due to chance is unlikely assuming the null is valid. Thus, smaller p-values provide more evidence to reject the null hypothesis.
The vice-president of operations wondered if the average strength of wire cables was different between those produced at the company's plant in a rural location and those produced in the company's plant located in a large city. The VP performed a hypothesis test and obtained a p-value of 0.02. The power of the test was 0.80. He decided to reject the null hypothesis. What type of error could the VP have made?
Type I error.
The claim being assessed in a hypothesis test is called
the null hypothesis.
Which of the following represents the probability of correctly rejecting an invalid null hypothesis?
the power of the test.
A student wondered if more than 10% of students enrolled in an introductory Chemistry class dropped before the midterm. Suppose he performed a hypothesis test to test his claim. In the context of the problem, what would happen if the student made a Type I Error?
He claims that more than 10% of students in the introductory Chemistry class dropped before the midterm when, in fact, 10% (or less) actually dropped.
When is a t-test performed instead of a z-test?
When the population standard deviation is not known.
A p-value is the probability of observing a value of a statistic or a value that is more unusual just by chance.
False, this statement would be true if it included the phrase, assuming the null hypothesis is true.
Researchers can make the results statistically significant by increasing the sample size even if the difference between the sample mean and hypothesized value of the population mean is very small.
True
Although rare, it is possible to get a p-value from a two-sided test greater than 1.
False
The p-value for a hypothesis test turns out to be 0.08714. At a 10% level of significance, what is the proper decision?
Reject
You are performing a left-tailed t-test with a sample size of 17 If α = .05 α = .05 , and your test statistic is t = − 1.76 tterm-15 = - 1.76 , do you:
Reject Null Hypothesis
You are performing a left-tailed z-test If α = .10 α = .10 , and your test statistic is z = − 1.87 z = - 1.87 , do you:
Reject Null Hypothesis
A commonly cited standard for one-way length (duration) of school bus rides for elementary school children is 30 minutes. A local government office in a rural area conducts a study to determine if elementary schoolers in their district have a longer average one-way commute time. If they determine that the average commute time of students in their district is significantly higher than the commonly cited standard they will invest in increasing the number of school busses to help shorten commute time. What would a Type I error mean in this context?
The local government decides that the data provide convincing evidence of an average commute time higher than 30 minutes, when the true average commute time is in fact 30 minutes.
Suppose increasing the sample size will not change the sample mean or the standard deviation. What will happen to the p-value by increasing the sample size?
The p-value will decrease.
A study was conducted based on a sample size of 30 individuals. The p-value was 0.10. Suppose a researcher conducted another study by taking a random sample of 50 individuals from the same population. Suppose they obtained the same sample mean as in the first study with a sample size of 30. (Also assume the population standard deviation is the same for both studies.) Which of the following is true?
The p-value would be smaller for the second study.
p-value is the probability of accepting the null hypothesis.
This statement is false. We never accept the null hypothesis no matter what the p-value is. A p-value is the probability of observing a sample mean (for example) that we did or something more unusual just by chance if the null hypothesis is true.
Failing to reject an invalid null hypothesis (continuing to incorrectly assume a null is valid) is called:
the beta-risk.
