Statistics Exam #3

If the consequences of making a Type I error are​ severe, would you choose the level of​ significance, alpha​, to equal​ 0.01, 0.05, or​ 0.10?

.01

left handed people and point estimate

2nd because not centered on point estimate

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 200 200 companies to invest in. After 1​ year, 104 104 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 Upper H 0 H0​: p equals =0.5 versus Upper H 1 H1​: p greater than >0.5 and obtained a​ P-value of 0.2858 0.2858. Explain what this​ P-value means and write a conclusion for the researcher.​ (Assume alpha α is 0.1 or​ less.)

About 29 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 this probability is not​ small, do not reject the null hypothesis. There is not sufficient evidence to conclude that the​ dart-picking strategy resulted in a majority of winners.

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 Upper H 0​: pequals0.5 versus Upper H 1​: pgreater than0.5 and obtained a​ P-value of 0.2858. Explain what this​ P-value means and write a conclusion for the researcher.​ (Assume alpha 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 this probability is not​ small, do not reject the null hypothesis. There is not sufficient evidence to conclude that the​ dart-picking strategy resulted in a majority of winners.

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 54​% 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 a​ P-value is. Choose the correct answer below.

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 less than a​, reject the null hypothesis.

What is a P-Value? 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-valueless than a, reject the null hypothesis.

Cross section

Data at specific time and was observational Variable of interest? Whether person has gum disease or not Qualitative

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 difference and practical difference?

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

Explain why tdistribution is less spread as n increases?

Sample evidence can prove that a null hypothesis is true.

FALSE

To construct a confidence interval about the​ mean, the population from which the sample is drawn must be approximately normal

FALSE

When testing a hypothesis using the​ P-value Approach, if the​ P-value is​ large, reject the null hypothesis.

FALSE

What does ​" 98 98​% ​confidence" mean in a 98 98​% 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 98 98 of the intervals to include the parameter and 2 2 to not include the parameter.

Suppose a researcher is testing the hypothesis Upper H 0​: pequals0.3 versus Upper H 1​: pless than0.3 and she finds the​ P-value to be 0.31. Explain what this means. Would she reject the null​ hypothesis? Why?

If the​ P-value for a particular test statistic is 0.31​, she expects results at least as extreme as the test statistic in about 31 of 100 samples if the null hypothesis is true Since this event is not​ unusual, she will not reject the null hypothesis.

Suppose a researcher is testing the hypothesis Upper H 0​: pequals0.3 versus Upper H 1​: pless than0.3 and she finds the​ P-value to be 0.31. Explain what this means. Would she reject the null​ hypothesis? Why?

If the​ P-value for a particular test statistic is 0.31​, she expects results at least as extreme as the test statistic in about 31 of 100 samples if the null hypothesis is true. Since this event is not​ unusual, she will not reject the null hypothesis.

n a survey conducted by a polling​ company, 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 sample size Decrease confidence level

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​ _______.

Level of Confidence (1-a)*100

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

Government and aliens

Not a simple random sample

The __ is a statement of no change, no effect, or no difference

Null Hypothesis

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

ONE I

A ______is the value of a statistic that estimates the value of a parameter

Point Estimate

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

Qulalitive with two possible outcomes

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

ROBUST

A graduate student conducted an experiment in which 21 ​ten-month-old babies were asked to watch a climber character attempt to ascend a hill. On two​ occasions, the baby witnesses the character fail to make the climb. On the third​ attempt, the baby witnesses either a helper toy push the character up the hill or a hinderer toy prevent the character from making the ascent. The helper and hinderer toys were shown to each baby in a random fashion for a fixed amount of time. The baby was then placed in front of each toy and allowed t

Reject Upper H 0. Although no level of significance is​ given, there is sufficient evidence to suggest the proportion of babies who choose the helper toy is greater than 0.5. Tthe population proportion of babies who choose the helper is​ 0.5, a sample where all 13 babies choose the helper will occur in about 1 out of​ 10,000 samples of 13 babies.

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.

Chewing your food

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

Moral values of the state

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

wWe are 85 85​% to 95 95​% confident 58 58​% of adults in the country during the period of economic uncertainty felt wages paid to workers in industry were too low.

The interpretation is flawed. The interpretation indicates that the level of confidence is varying.

n a survey of 2045 2045 adults in a certain country conducted during a period of economic​ uncertainty, 58 58​% thought that wages paid to workers in industry were too low. The margin of error was 5 5 percentage points with 90 90​% confidence. For parts​ (a) through​ (d) below, which represent a reasonable interpretation of the survey​ results? For those that are not​ reasonable, explain the flaw

The interpretation is flawed. The interpretation provides no interval about the population proportion.

In 90 90​% of samples of adults in the country during the period of economic​ uncertainty, the proportion who believed wages paid to workers in industry were too low is between 0.53 0.53 and 0.63 0.63.

The interpretation is flawed. The interpretation suggests that this interval sets the standard for all the other​ intervals, which is not true.

A group conducted a poll of 2093 2093 likely voters just prior to an election. The results of the survey indicated that candidate A would receive 48 48​% of the popular vote and candidate B would receive 45 45​% of the popular vote. The margin of error was reported to be 4 4​%. 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 44​% and 52 52​% of the popular vote and candidate B may receive between 41 41​% and 49 49​% of the popular vote. Because the poll estimates overlap when accounting for margin of​ error, the poll cannot predict the winner.

Suppose, in​ fact, the mean annual consumption of popcorn after the marketing campaign is 57 quarts. Has a Type I or Type II error been made by the marketing​ department? If we tested this hypothesis at the a=0.01 level of​ significance, what is the probability of committing this​ error?

The marketing department committed a Type I error because the marketing department rejected the null hypothesis when it was true. The probability of making a Type I error is .01

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

The probability increases since they're inversely related.

Suppose the null hypothesis is not rejected. State the conclusion based on the results of the test. Six years​ ago, 11.4​% 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.

n a​ survey, 500 500 adults in a certain country were asked how many hours they worked in the previous week. Based on the​ results, a​ 95% confidence interval for mean number of hours worked was lower​ bound: 39.5 39.5 and upper​ bound: 41.4 41.4. Which of the following represents a reasonable interpretation of the​ result? For those that are not​ reasonable, explain the flaw.

There is a​ 95% chance the mean number of hours worked by adults in this country in the previous week was between 39.5 39.5 hours and 41.4 41.4 hours. ----Flawed. This interpretation implies that the population mean varies rather than the interval. We are​ 95% confident that the mean number of hours worked by adults in this country in the previous week was between 39.5 39.5 hours and 41.4 41.4 hours. -----Correct 95% of adults in this country worked between 39.5 39.5 hours and 41.4 41.4 hours last week ----flawed; makes interpretation of the individual rather than the adult ​(d) We are​ 95% confident that the mean number of hours worked by adults in a particular area of this country in the previous week was between 39.5 39.5 hours and 41.4 41.4 hours ---Flawed; not just particular area

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. <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.

A nutritionist wants to determine how much time nationally people spend eating and drinking. Suppose for a random sample of 1086 1086 people age 15 or​ older, the mean amount of time spent eating or drinking per day is 1.28 1.28 hours with a standard deviation of 0.62 0.62 hour. Complete parts ​(a) through ​(d) below.

a. A histogram of time spent eating and drinking each day is skewed right. Use this result to explain why a large sample size is needed to construct a confidence interval for the mean time spent eating and drinking each day --Since the distribution of time spent eating and drinking each day is not normally distributed​ (skewed right), th b.n​ 2010, there were over 200 million people nationally age 15 or older. Explain why​ this, along with the fact that the data were obtained using a random​ sample, satisfies the requirements for constructing a confidence interval. --Sample size is less than 5% ---Couldn't be used for 9 year old bc it could differ

We are 90 90​% confident the proportion of adults in the country during the period of economic uncertainty who believed wages paid to workers in industry were too low was between 0.53 0.53 and 0.63 0.63.

reasonable

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 Upper H 0​: muequals27.4 years versus Upper H 1​: muless than27.4 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​ resear

​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.