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