STP 226: EXAM 3

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?

0.01

Fill in the blanks to complete the statement. The​ _______ _______ is a statement we are trying to find evidence to support.

The ALTERNATIVE HYPOTHESIS is a statement we are trying to find evidence to support.

Determine whether the following statement is true or false. Sample evidence can prove that a null hypothesis is true.

FALSE

State the conclusion based on the results of the test. The standard deviation in the pressure required to open a certain valve is known to be σ=1.4 psi. Due to changes in the manufacturing​ process, the​ quality-control manager feels that the pressure variability has been reduced. The null hypothesis was not rejected.

There is not sufficient evidence that the standard deviation in the pressure required to open a certain valve has been reduced.

A​ ________ ________ is the value of a statistic that estimates the value of a parameter.

A POINT ESTIMATE is the value of a statistic that estimates the value of a parameter.

The headline reporting the results of a poll​ stated, "Majority of Adults at Personal Best in the​ Morning." The results indicated that a survey of 1300 adults resulted in 55​% stating they were at their personal best in the morning. The​ poll's results were reported with a margin of error of 3​%. Explain why the​ poll's headline is accurate.

All the values within the margin of error are greater than​ 50%.

Explain what ​"90​% ​confidence" means in a 90​% confidence interval.

If 100 different confidence intervals are​ constructed, each based on a different sample of size n from the same​ population, then we expect 90 of the intervals to include the parameter and 10 to not include the parameter.

A trade magazine routinely checks the​ drive-through service times of​ fast-food restaurants. An 80​% confidence interval that results from examining 602 customers in one​ fast-food chain's​ drive-through has a lower bound of 169.5 seconds and an upper bound of 173.1 seconds. What does this​ mean?

One can be 80​% confident that the mean​ drive-through service time of this​ fast-food chain is between 169.5 seconds and 173.1 seconds.

For what type of variable does it make sense to construct a confidence interval about a population​ proportion?

Qualitative with 2 possible outcomes

What happens to the probability of making a Type II​ error, β​, as the level of​ significance, α​, ​decreases? Why?

The probability increases. Type I and Type II errors are inversely related.

If a hypothesis is tested at the α=0.05 level of​ significance, what is the probability of making a type I​ error?

The probability of making a type I error is 0.05.

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.

The procedure for constructing a confidence interval about a mean is ROBUST which means minor departures from normality do not affect the accuracy of the interval.

A​ government's congress has 794 members, of which 197 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 0.218 and an upper bound of 0.278. What is wrong with the​ alien's approach to estimating the proportion of the human race that is​ female?

The sample is not a simple random sample.

Determine if the following statement is true or false. When testing a hypothesis using the​ P-value Approach, if the​ P-value is​ large, reject the null hypothesis.

This statement is false.

State the requirements to perform a​ goodness-of-fit test.

all expected frequencies≥1 at least​ 80% of expected frequencies ≥5

Fill in the blank below. A researcher wants to show the mean from population 1 is less than the mean from population 2 in​ matched-pairs data. If the observations from sample 1 are Xi and the observations from sample 2 are Yi​, and di=Xi−Yi​, then the null hypothesis is H0​: μd=0 and the alternative hypothesis is H1​: μd ​___ 0.

A researcher wants to show the mean from population 1 is less than the mean from population 2 in​ matched-pairs data. If the observations from sample 1 are Xi and the observations from sample 2 are Yi​, and di=Xi−Yi​, then the null hypothesis is H0​: μd=0 and the alternative hypothesis is H1​: < 0.

Explain the difference between an independent and dependent sample.

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.

A sampling method is​ _________ when the individuals selected for one sample are used to determine the individuals in the second sample.

A sampling method is DEPENDENT when the individuals selected for one sample are used to determine the individuals in the second sample.

A sampling method is ___________ when an individual selected for one sample does not dictate which individual is to be in the second sample.

A sampling method is INDEPENDENT when an individual selected for one sample does not dictate which individual is to be in the second sample.

Determine whether the following sampling is dependent or independent. Indicate whether the response variable is qualitative or quantitative. A researcher wishes to compare annual salaries of mathematicians and non-mathematicians. She obtains a random sample of 955 professionals of each category who work and determines each individual's

A. The sampling is independent because an individual selected for one sample does not dictate which individual is to be in the second sample. B. The variable is quantitative because it is a numerical measure.

Some have argued that throwing darts at the stock pages to decide which companies to invest in could be a successful​ stock-picking strategy. Suppose a researcher decides to test this theory and randomly chooses 200 companies to invest in. After 1​ year, 104 of the companies were considered​ winners; that​ is, they outperformed other companies in the same investment class. To assess whether the​ dart-picking strategy resulted in a majority of​ winners, the researcher tested H0​: p=0.5 versus H1​: p>0.5 and obtained a​ P-value of 0.2858. Explain what this​ P-value means and write a conclusion for the researcher.​ (Assume α is 0.1 or​ less.)

About 29 in 100 samples will give a sample proportion as high or higher than the one obtained if the population proportion really is 0.5. Because the​ P-value is​ large, do not reject the null hypothesis. There is not sufficient evidence to conclude that the​ dart-picking strategy resulted in a majority of winners.

Explain what a​ P-value is. What is the criterion for rejecting the null hypothesis using the​ P-value approach?

A​ P-value is 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. If P-value<α​, reject the null hypothesis.

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?

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.

Fill in the blank to complete the statement. If we do not reject the null hypothesis when the statement in the alternative hypothesis is​ true, we have made a Type​ _______ error.

Fill in the blank to complete the statement. If we do not reject the null hypothesis when the statement in the alternative hypothesis is​ true, we have made a Type​ 2 error.

Fill in the blank to complete the statement. If we reject the null hypothesis when the statement in the null hypothesis is​ true, we have made a Type​ _______ error.

Fill in the blank to complete the statement. If we reject the null hypothesis when the statement in the null hypothesis is​ true, we have made a Type​ 1 error.

Suppose a researcher is testing the hypothesis H0​: p=0.6 versus H1​: p<0.6 and she finds the​ P-value to be 0.19. Explain what this means. Would she reject the null​ hypothesis? Why?

If the​ P-value for a particular test statistic is 0.19​, she expects results at least as extreme as the test statistic in about 19 of 100 samples if the null hypothesis is true. Since this event is not​ unusual, she will not reject the null hypothesis.

In a survey conducted by the Gallup​ Organization, 1100 adult Americans were asked how many hours they worked in the previous week. Based on the​ results, a​ 95% confidence interval for the mean number of hours worked had a lower bound of 42.7 and an upper bound of 44.5. Provide two recommendations for decreasing the margin of error of the interval.

Increase the sample size. Decrease the confidence level.

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 0.12. Each independently constructed a confidence interval based on the point​ estimate, but​ Jaime's interval has a lower bound of 0.115 and an upper bound of 0.125​, while​ Mariya's interval has a lower bound of 0.103 and an upper bound of 0.218. Which interval is​ wrong? Why?

Mariya​'s interval is wrong because it is not centered on the point estimate.

What are the requirements for the​ chi-square test for​ independence? ​(This is a reading assessment question. Be certain of your answer because you only get one attempt on this​ question.)

No more than​ 20% of the expected counts are less than 5. All expected counts are greater than or equal to 1.

What type of variable is required to construct a confidence interval for a population​ proportion?

Qualitative with 2 possible outcomes

Explain what​ "statistical significance" means.

Statistical significance means that the result observed in a sample is unusual when the null hypothesis is assumed to be true.

Explain the difference between statistical significance and practical significance.

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.

The​ _______ represents the expected proportion of intervals that will contain the parameter if a large number of different samples of size n is obtained. It is denoted​ _______.

The 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. It is denoted (1-a) x 100%.

For the​ following, indicate whether a confidence interval for a proportion or mean should be constructed to estimate the variable of interest. Justify your response. Does chewing your food for a longer period of time reduce​ one's caloric intake of food at​ dinner? A researcher requires a sample of 75 healthy males to chew their food twice as long as they normally do. The researcher then records the calorie consumption at dinner.

The confidence interval for a mean should be constructed because the variable of interest is an individual's reduction in caloric intake, which is a quantitative variable.

For the​ following, indicate whether a confidence interval for a proportion or mean should be constructed to estimate the variable of interest. Justify your response. Researchers within an organization asked a random sample of 1016 adults aged 21 years or​ older, "Right​ now, do you think the state of moral values in the country as a whole is getting​ better, or getting​ worse?"

The confidence interval for a proportion should be constructed because the variable of interest is an individual's opinion, which is a qualitative variable.

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 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, 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 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.

Suppose there are n independent trials of an experiment with k>3 mutually exclusive​ outcomes, where pi represents the probability of observing the ith outcome. What would be the formula of an expected count in this​ situation?

The expected counts for each possible outcome are given by Ei=npi.

Suppose you have two​ populations: Population A—All students at Illinois State University ​(N=​21,000) and Population B—All residents of Homer​ Glen, IL ​(N=​21,000). You want to estimate the mean age of each population using two separate samples each of size n=75. If you construct a​ 95% confidence interval for each population​ mean, will the margin of error for population A be​ larger, the​ same, or smaller than the margin of error for population​ B? Justify your reasoning.

The margin of error for Population A will be smaller because its sample standard deviation will be smaller.

A group conducted a poll of 2084 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 45​% 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.

The margin of error suggests candidate A may receive between 44​% and 54​% of the popular vote and candidate B may receive between 40​% and 50​% of the popular vote. Because the poll estimates overlap when accounting for margin of​ error, the poll cannot predict the winner.

Suppose the null hypothesis is not rejected. State the conclusion based on the results of the test. Six years​ ago, 11.5​% of registered births were to teenage mothers. A sociologist believes that the percentage has increased since then.

There is not sufficient evidence to conclude that the percentage of teenage mothers has increased.

A national survey of 2000 adult citizens of a nation found that 19​% dreaded​ Valentine's Day. The margin of error for the survey was 2.5 percentage points with 85​% confidence. 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 0.165 and 0.215.

State the conclusion based on the results of the test. The mean of the pressure required to open a certain valve is known to be μ=7.6 psi. Due to changes in the manufacturing​ process, the​ quality-control manager feels that the average pressure has changed. The null hypothesis was not rejected.

There is not sufficient evidence that the mean of the pressure required to open a certain valve has changed.

State the conclusion based on the results of the test. According to the​ report, the standard deviation of monthly cell phone bills was ​$49.43 three years ago. A researcher suspects that the standard deviation of monthly cell phone bills is less today. The null hypothesis is not rejected.

There is not sufficient evidence to conclude that the standard deviation of monthly cell phone bills is less than its level three years ago of $49.43.

Suppose the null hypothesis is not rejected. State the conclusion based on the results of the test. Three years​ ago, the mean price of a​ single-family home was ​$243,747. A real estate broker believes that the mean price has decreased since then.

There is not sufficient evidence to conclude that the mean price of a​ single-family home has decreased.

Suppose the null hypothesis is not rejected. State the conclusion based on the results of the test. Six years​ ago, 12.7​% of registered births were to teenage mothers. A sociologist believes that the percentage has decreased since then.

There is not sufficient evidence to conclude that the percentage of teenage mothers has decreased.

Suppose the null hypothesis is not rejected. State the conclusion based on the results of the test. Six years​ ago, 12.1​% of registered births were to teenage mothers. A sociologist believes that the percentage has increased since then.

There is not sufficient evidence to conclude that the percentage of teenage mothers has increased.

State the conclusion based on the results of the test. According to the Federal Housing Finance​ Board, the mean price of a​ single-family home two years ago was ​$299,200. A real estate broker believes that because of the recent credit​ crunch, the mean price has increased since then. The null hypothesis is rejected.

There is sufficient evidence to conclude that the mean price of a​ single-family home has increased from its level two years ago of $299,200.

Suppose the null hypothesis is rejected. State the conclusion based on the results of the test. Six years​ ago, 12.1​% of registered births were to teenage mothers. A sociologist believes that the percentage has increased since then.

There is sufficient evidence to conclude that the percentage of teenage mothers has increased.

Explain why​ chi-square goodness-of-fit tests are always right tailed.

The​ chi-square goodness-of-fit tests are always right tailed because the numerator in the test statistic is​ squared, making every test​ statistic, other than a perfect​ fit, positive.

Explain why the​ t-distribution has less spread as the number of degrees of freedom increases.

The​ t-distribution has less spread as the degrees of freedom increase​ because, as n​ increases, s becomes closer to σ by the law of large numbers.

An educator wants to determine whether a new curriculum significantly improves standardized test scores for third grade students. She randomly divides 100 third-graders into two groups. Group 1 is taught using the new curriculum, while group 2 is taught using the traditional curriculum. At the end of the school​ year, both groups are given the standardized test and the mean scores are compared. Determine whether the sampling is dependent or independent. Indicate whether the response variable is qualitative or quantitative.

This sampling is independent because the individuals selected for one sample do not dictate which individuals are to be in a second sample. The variable is quantitative because it is a numerical measure.

Determine whether the following statement is true or false. To construct a confidence interval about the​ mean, the population from which the sample is drawn must be approximately normal.

This statement is FALSE.

Determine if the following statement is true or false.​ Why? The expected frequencies in a​ chi-square test for independence are found using the formula below. Expected frequency= (row total)(column total) / table total

True. It is a simplification of multiplying the proportion of a row variable by the proportion of the column variable to find the proportion for a​ cell, then multiplying by the table total.

If the expected count of a category is less than​ 1, what can be done to the categories so that a​ goodness-of-fit test can still be​ performed?

Two of the categories can be​ combined, or the sample size can be increased.

The head of institutional research at a university believed that the mean age of​ full-time students was declining. In​ 1995, the mean age of a​ full-time student was known to be 27.4 years. After looking at the enrollment records of all 4934​ full-time students in the current​ semester, he found that the mean age was 27.1​ years, with a standard deviation of 7.3 years. He conducted a hypothesis of H0​: μ=27.4 years versus H1​: μ<27.4 years and obtained a​ P-value of 0.0020. He concluded that the mean age of​ full-time students did decline. Is there anything wrong with his​ research?

Yes, the head of institutional research has access to the entire​ population, inference is unnecessary. He can say with​ 100% confidence that the mean age has decreased.

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? Justify your answer.

​Matthew's estimate will have the smaller margin of error because the larger sample size more than compensates for the higher level of confidence.