Final Exam
Step 1: Make a statement regarding the nature of the population. Step 2: Collect evidence (sample data) to test the statement. Step 3: Analyze the data to assess the plausibility of the statement.
Steps in hypothesis testing
The margin of error suggests candidate A may receive between 44% and 54% of the popular vote and candidate B may receive between 41% and 51% of the popular vote. Because the poll estimates overlap when accounting for margin of error, the poll cannot predict the winner.
A group conducted a poll of 2081 likely voters just prior to an election. The results of the survey indicated that candidate A would receive 49% of the popular vote and candidate B would receive 46% of the popular vote. The margin of error was reported to be 5%. The group reported that the race was too close to call. Use the concept of a confidence interval to explain what this means.
There is 85% confidence that the proportion of the adult citizens of the nation that dreaded Valentine's Day is between 0.132 and 0.288.
A national survey of 1500 adult citizens of a nation found that 21% dreaded Valentine's Day. The margin of error for the survey was 7.8 percentage points with 85% confidence. Explain what this means.
The sampling is dependent because an individual selected for one sample does dictate which individual is to be in the second sample. The variable is qualitative
A researcher wishes to compare the personality types of brothers and sisters. She obtains a random sample of 807 pairs of siblings who take a personality inventory and determines each sibling's personality type.
There is sufficient evidence to conclude that the mean monthly cell phone bill is less than its level three years ago of $48.59.
According to the report, the mean monthly cell phone bill was $48.59 three years ago. A researcher suspects that the mean monthly cell phone bill is less today. The null hypothesis is rejected.
.95
As the number of samples increases, the proportion of 95% confidence intervals that include the population proportion approaches
The correct procedure is a hypothesis test for a single mean. The comparison is between the mean IQ of the class and the national average IQ. The class is a sample of the population, so it is not a comparison between two population means. The objective is to find whether the sample mean is higher than the population mean, so it is a hypothesis test and not a confidence interval
Assuming all model requirements for conducting the appropriate procedure have been satisfied, is the mean IQ of the students in the professor's statistics class higher than that of the general population, 100? Explain what statistical procedure should be used for this research objective.
The correct procedure is a confidence interval for a single proportion. The goal is to determine the proportion of the population that favors a tax increase. There is no comparison being made and there is only one population, so rather than hypothesis testing, it is appropriate to use a confidence interval
Assuming all model requirements for conducting the appropriate procedure have been satisfied, what proportion of registered voters is in favor of a tax increase to reduce the federal debt? Explain which statistical procedure would most likely be used for the research objective given.
The sample is not a simple random sample
A government's congress has 311 members, of which 58 are women. An alien lands near the congress building and treats the members of congress as as a random sample of the human race. He reports to his superiors that a 95% confidence interval for the proportion of the human race that is female has a lower bound of 0.144 and an upper bound of 0.230. What is wrong with the alien's approach to estimating the proportion of the human race that is female?
left tailed
Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed. What parameter is being tested? H0: p=0.76 H1: p< 0.76
right tailed, M (pop mean)
Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed. What parameter is being tested? H0: p=0.76 H1: p> 0.76
two tailed, population proportion
Determine whether the hypothesis test is left-tailed, right-tailed, or two-tailed. What parameter is being tested? H0:p= 0.4 H1:p≠ 0.4
A sample is independent when an individual selected for one sample does not dictate which individual is to be in the second sample. A sample is dependent when an individual selected for one sample dictates which individual is to be in the second sample
Explain the difference between an independent and dependent sample
Statistical significance means that the sample statistic is not likely to come from the population whose parameter is stated in the null hypothesis. Practical significance refers to whether the difference between the sample statistic and the parameter stated in the null hypothesis is large enough to be considered important in an application
Explain the difference between statistical significance and practical significance
If 100 different confidence intervals are constructed, each based on a different sample of size n from the same population, then we expect 98 of the intervals to include the parameter and 2 to not include the parameter
Explain what "98% confidence" means in a 98% confidence interval
A matched-pairs t-test on the difference of means is most likely appropriate because the mean price is a good measure of how much a hotel chain charges, the research objective involves a comparison of two things, and one would likely select hotels paired by location
For the study given below, explain which statistical procedure would most likely be used for the research objective. Assume all model requirements for conducing the appropriate procedure have been satisfied. Does hotel chain A charge more than hotel chain B for a one-night stay?
Qualitative, 2 outcomes
For what type of variable does it make sense to construct a confidence interval about a population proportion?
The probability of making a type I error is 0.05
If a hypothesis is tested at the α=0.05 level of significance, what is the probability of making a type I error?
1.96
If a 95% confidence interval results in a sample proportion that does not include the population proportion, then the sample proportion is more than ___ standard errors from the population proportion.
we would expect 98 of the intervals to capture the unknown parameter
If one hundred 98% confidence intervals are constructed for a population parameter
.01
If the consequences of making a Type I error are severe, would you choose the level of significance, α, to equal 0.01, 0.05, or 0.10?
less than
If the normality requirement is not satisfied (that is, np(1−p) is not at least 10), then a 95% confidence interval about the population proportion will include the population proportion in_____ 95% of the intervals.
Type II Error
If we do not reject the null hypothesis when the alternative hypothesis is true
Matthew's estimate will have the smaller margin of error because the larger sample size more than compensates for the higher level of confidence.
Katrina wants to estimate the proportion of adults who read at least 10 books last year. To do so, she obtains a simple random sample of 100 adults and constructs a 95% confidence interval. Matthew also wants to estimate the proportion of adults who read at least 10 books last year. He obtains a simple random sample of 400 adults and constructs a 99% confidence interval. Assuming both Katrina and Matthew obtained the same point estimate, whose estimate will have the smaller margin of error?
robust
The procedure for constructing a confidence interval about a mean is _________ which means minor departures from normality do not affect the accuracy of the interval.
symmetric about 0
The Student's t-distribution is
There is not sufficient evidence to conclude that the mean price of a single-family home has increased.
Three years ago, the mean price of a single-family home was $243,767. A real estate broker believes that the mean price has increased since then.
False
True or False: A 95% confidence interval may be interpreted by saying there is a 95% probability that the interval includes the unknown parameter.
False
True or False: Sample evidence can prove that a null hypothesis is true
False
True or False: To construct a confidence interval about the mean, the population from which the sample is drawn must be approximately normal.
False
True or False: When testing a hypothesis using the P-value Approach, if the P-value is large, reject the null hypothesis.
Jaime's interval is wrong because it is not centered on the point estimate.
Two researchers, Jaime and Mariya, are each constructing confidence intervals for the proportion of a population who is left-handed. They find the point estimate is 0.25. Each independently constructed a confidence interval based on the point estimate, but Jaime's interval has a lower bound of 0.238 and an upper bound of 0.322, while Mariya's interval has a lower bound of 0.191 and an upper bound of 0.309. Which interval is wrong?
The probability increases. Type I and Type II errors are inversely related
What happens to the probability of making a Type II error, β, as the level of significance, α, decreases? Why?
Make an assumption about reality, and collect sample evidence to determine whether it contradicts the assumption
What is at the "heart" of hypothesis testing in statistics?
Qualitative with 2 possible outcomes
What type of variable is required to construct a confidence interval for a population proportion?
statistically significant
When observed results are unlikely under the assumption that the null hypothesis is true, the result is statistically significant and we reject the statement in the null hypothesis
null hypothesis
denoted H0, is a statement to be tested, and is a statement of no change, no effect, or no difference
alternative hypothesis
denoted H1, is a statement that the researcher is trying to find evidence to support.
point estimate
is the value of a statistic that estimates the value of a parameter
(1-a) 100%
level of confidence is denoted as
level of confidence
represents the expected proportion of intervals that will contain the parameter if a large number of different samples of size n is obtained
P-value
the probability of observing a sample statistic as extreme or more extreme than the one observed under the assumption that the statement in the null hypothesis is true
independent sampling method
when an individual selected for one sample does not dictate which individual is to be in the second sample
dependent sampling method
when the individuals selected for one sample are used to determine the individuals in the second sample