Stat Chapter 18
unlikely
5% to 10% is an ____________ event.
simple While there exist inference methods for other types of random samples, they are beyond the scope of this course.
A __________ random sample is required for inference with z-procedures.
convicting an innocent person. A Type I error is when the null hypothesis is wrongly rejected. In this case, it would be arriving at a guilty verdict when the person is really innocent of the crime.
A statistical hypothesis test is analogous to a criminal trial in the American justice system. There, the null hypothesis is that the defendant is innocent. In this case, a Type I error is
letting a guilty person go free. A Type II error is when the null hypothesis is wrongly NOT rejected. Here, the null hypothesis is that the person is innocent.
A statistical hypothesis test is analogous to a criminal trial in the American justice system. There, the null hypothesis is that the defendant is innocent. In this case, a Type II error is
I A Type I error is when the null hypothesis is wrongly rejected. Here, the null hypothesis is that the person is innocent.
A statistical hypothesis test is analogous to a criminal trial in the American justice system. There, the null hypothesis is that the defendant is innocent. In this case, convicting an innocent person is a Type ___ error.
II A Type II error is when the null hypothesis is wrongly NOT rejected. Here, the null hypothesis is that the person is innocent.
A statistical hypothesis test is analogous to a criminal trial in the American justice system. There, the null hypothesis is that the defendant is innocent. In this case, letting a guilty person go free is a Type ___ error.
p-values
Cautions about significance tests: When H0 represents an assumption that is widely believed, small __-________ are needed.
rejecting
Cautions about significance tests: When there are strong consequences of ____________ H0 in favor of HA, we need strong evidence. Either way, strong evidence of rejecting H0 requires small p-values.
decreases Decreasing the confidence level results in a decrease in z* which decreases the margin of error and the width of the confidence interval.
Decreasing the level of confidence ___________ the margin of error.
increases Decreasing the sample size will increase the margin of error.
Decreasing the sample size ____________ the margin of error.
significance level
Denoted by α, the probability of a Type I error of any fixed level test.
Power will increase. Increasing the sample size will make the standard deviation of the sample statistic smaller, and will increase the chance of detecting a difference when it is actually there.
For a given effect size and level of significance, increasing the sample size will have what effect on the power of the test?
decrease power. Decreasing α results in rejecting a false null hypothesis less often, thus automatically decreasing power.
For a specified sample size, decreasing α will _____________ __________.
Larger margin of error. A larger standard deviation would results in a larger margin of error and a wider width of the confidence interval.
If the population standard deviation were 10 instead of 5, what effect would this new standard deviation have on margin of error?
decreased. Smaller standard deviations will make the margin of error smaller, because we are changing σ in the formula for the margin of error.
If the population standard deviation were 2 instead of 5, the margin of error would be ____________.
power
Measures the ability of a significance test to detect an alternative hypothesis. Against a specific alternative, this is the probability that the test will reject H0 at a chosen significance level α when the specified alternative value of the parameter is true.
likely
More than 10% is a _________ event.
z confidence interval
The z confidence interval for the mean of a Normal population will have a specified margin of error m when the sample size is n = (z* standard deviation / m)^2
H0
Very large samples can yield small p-values that lead to rejection of _____.
effect size
the departure from a null hypothesis that results in practical significance.
statistically significant 0.0007 is a small P-value. We would reject a null hypothesis with a P-value like this for any typical α level.
A group of researchers wanted to know if there was a difference in average yearly income taxes paid between residents of two very large cities in the Midwestern United States. The average for the first city was $6,505 and for the second city, it was $6,511. The difference provided a P-value of 0.0007. The difference is _______________ ____________ because the P-value is small.
incorrectly rejecting the null hypothesis.
A type I error is...
decrease Increasing the sample size while keeping the confidence level fixed will decrease the margin of error and the confidence interval width because the sample size is in the denominator of the margin of error formula.
An airline wants to know the average time it takes their passengers to claim their luggage. The time to claim luggage for this airline is known to be Normally distributed with mean μ and standard deviation σ = 5 minutes. The margin of error for a 95% confidence interval would ___________ if the airline took a simple random sample of 100 passengers instead of 10 passengers.
null
Analogous to how we use high levels of confidence for confidence intervals, we need strong evidence and very small p-values to reject ______ hypotheses.
details
Cautions about confidence intervals: Consider the _________ of a study before completely trusting a confidence interval
source
Cautions about confidence intervals: The __________ of the data is of utmost importance.
sampling errors
Cautions about confidence intervals: The margin of error covers only _____________ _________.
biases
Cautions about confidence intervals: Undercoverage, nonresponse, or other __________ are not reflected in margins of error.
fixed
Cautions about significance tests: Be careful about conducting multiple analyses for a _________ a... It is preferred to just run a single test and reach a decision.
10
Cautions about significance tests: Depending on the situation, p-values that are below ____% can lead to rejecting H0.
half
Cautions about significance tests: The P-value for a one-sided test is ______ of the P-value for the two-sided test of the same null hypothesis and of the same data.
two
Cautions about significance tests: The _____-sided case combines two equal areas.
one
Cautions about significance tests: The _____-sided case has one of those areas PLUS an inherent supposition by the researcher of the direction of the possible deviation from H0.
5
Cautions about significance tests: Unless stated otherwise, researchers assume the de facto significance level of ___%.
decrease Increasing the desired margin of error will require a smaller sample size. The margin of error is in the denominator of the formula for sample size. Making that larger will lower the result of the calculation.
Increasing the desired margin of error will _____________ the required sample size.
decrease Increasing the desired margin of error will require a smaller sample size. The margin of error is in the denominator of the formula for sample size. Making that larger will lower the result of the calculation.
Increasing the desired margin of error will ______________ the required sample size.
decreases
Increasing the sample size ___________ the margin of error.
5
Less than ____% is an extremely unlikely event.
Phenomena that are "statistically significant" are not always "practically significant."
Phenomena that are "statistically significant" are not always "practically significant." Copy.
90, 95, 99
Standard levels of confidence are ____%, ____%, and ____%.
10, 5, 1
Standard levels of significance are ____%, ____%, and ____%.
effect size.
The difference between the hypothesized parameter value and the actual parameter value is called the _________ _______.
voluntary response The number of respondents was indeed large, but the respondents made their own decision about whether to participate.
The local television station invited viewers to call in to respond to a poll on the local election. They later stated "Over 2500 viewers responded to our poll about the sales tax question on the election ballot. 75% of our viewers are opposed to the proposal for an additional penny sales tax. The poll has a margin of error of plus or minus 3%." Their results as reported are not reliable because the sample was obtained through ____________ ___________.
voluntary response. The number of respondents was indeed large, but the respondents made their own decision about whether to participate.
The local television station invited viewers to call in to respond to a poll on the local election. They later stated "Over 2500 viewers responded to our poll about the sales tax question on the election ballot. 75% of our viewers are opposed to the proposal for an additional penny sales tax. The poll has a margin of error of plus or minus 3%." Their results as reported are not reliable because the sample was obtained through ______________ _____________.
True
True or False: A Type II error is failing to reject the null hypothesis when it is wrong.
False. It is often difficult to assess Normality of a sample, especially when the sample size is small. We really are relying on the sample mean having a Normal distribution. This can always be guaranteed when?
True or False: A requirement of inference with z-procedures is that the sample has a Normal distribution.
True. Since increasing the confidence level from 95% to 99% will increase the margin of error, they need to do something to decrease the margin of error. Increasing the sample size will do this.
True or False: An airline wants to know the average time it takes their passengers to claim their luggage. The time to claim luggage for this airline is known to be Normally distributed with mean μ and standard deviation σ = 5 minutes. The airline took a simple random sample of 10 passengers and calculated a 95% confidence interval to be 21.9 to 28.1 minutes. Suppose the airline decides to increase the confidence level to 99% and they don't want to increase their margin of error. They can accomplish this by increasing their sample size.
True. An α level of 5% means results this extreme will happen 5% of the time by chance. Sometimes we get the "unlucky" sample (one in the tails of the distribution). Finding only three "significant" results is not strong evidence of an impact on the students.
True or False: At a certain university, the National Science Foundation awarded a large grant to create environmental science laboratory courses. The purpose of these was to educate students about the impacts of certain activities on the environment. In assessing the impact of the courses on students' attitudes, a special survey was administered the first few semesters the courses were taught. The data from these were analyzed using multiple hypothesis tests. In all, 51 tests were performed to attempt to connect student demographics with increased environmental awareness. Three of the test results were significant at the 5% level. We should exercise caution in drawing a conclusion of the effectiveness of the courses from these results because at the 5% level, we'll expect 0.05*51 = 2.55 "significant" test results even if all the null hypotheses of "no difference" are true.
False This might have been the case, but it is not the issue here and there is no evidence of this stated in the problem.
True or False: At a certain university, the National Science Foundation awarded a large grant to create environmental science laboratory courses. The purpose of these was to educate students about the impacts of certain activities on the environment. In assessing the impact of the courses on students' attitudes, a special survey was administered the first few semesters the courses were taught. The data from these were analyzed using multiple hypothesis tests. The data from these were analyzed to attempt to connect student demographics with increased environmental awareness. In all, 51 hypothesis tests were performed. Three of the test results were significant at the 5% level. Based on three tests being significant, should we conclude that an important effect has been discovered?
True. The sample size appears in the formula in the denominator as a square root. Increasing the sample size will decrease the margin of error, because dividing by a larger quantity makes the result smaller.
True or False: Increasing the sample size decreases the margin of error.
True. The farther the actual parameter is from the hypothesized parameter value, the more likely we are to reject a false null hypothesis (find the difference is there).
True or False: The larger the effect size the larger the power.
False. The significance level is the probability of rejecting the null hypothesis when it is, in fact, correct (making a Type I error).
True or False: The significance level of a test, α, is the probability of getting significant results when there is really large effect in the population.
False. This is σ, which is part of the formula for margin of error.
True or False: The standard deviation does NOT affect margin of error.
False. You would get this value if you use 10% in one tail of the Normal distribution. Confidence goes in the center; the remaining area is split equally between the two tails. Refer to Table C.
True or False: When computing a sample size for a 90% confidence interval, we'll use z* = 1.28.
type ii error
Type of error made when we fail to reject the null hypothesis when the alternative hypothesis is true... the probability of not rejecting H0, when we should have rejected it.
type i error
Type of error made when we reject the null hypothesis when in fact it is true... the maximum allowable "error" of a falsely rejected H0 (also the significance level, a).
A simple random sample
What type of sample is required for inference with z-procedures?
will not reliably reflect the opinions of all students because only about 25% responded. Almost 75% of those selected failed to reply. This is a very large nonresponse. Because we cannot infer the attitudes of those who don't respond, these data are not reliable to reflect the opinions of all students.
Your college is interested in obtaining student opinions about activities such as drinking alcohol at parties. They select a random sample of 2000 students; 516 of those actually complete the survey. The information obtained...
stratified When separate simple random samples are taken from subgroups of the population, the resulting sample is called a stratified random sample.
Your college wants to know about student opinions on a proposed change in graduation requirements. They poll a random sample of 10% of the members of each class. We cannot use methods learned to perform inference on the overall sample because it is a _____________ random sample.
margin of error
expressed as x- +/- m where m = z* standard deviation/square root n