Stats Test 3
The margin of error suggests candidate A may receive between 45% and 51% of the popular vote and candidate B may receive between 43% and 49% 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 2038 likely voters just prior to an election. The results of the survey indicated that candidate A would receive 48% of the popular vote and candidate B would receive 46%of the popular vote. The margin of error was reported to be 3%. The group reported that the race was too close to call. Use the concept of a confidence interval to explain what this means.
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
Explain what a P-value is
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), the sample must be large so that the distribution of the sample mean will be approximately normal. b) There are more than 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. - The sample size is less than 5% of the population. c) Determine and interpret a 99% confidence interval for the mean amount of time Americans age 15 or older spend eating and drinking each day. - The nutritionist is 99%confident that the mean amount of time spent eating or drinking per day is between 1.214 and 1.306 hours. d) Could the interval be used to estimate the mean amount of time a 9-year-old spends eating and drinking each day? Explain. - No; the interval is about people age 15 or older. The mean amount of time spent eating or drinking per day for 9-year-olds may differ.
A nutritionist wants to determine how much time nationally people spend eating and drinking. Suppose for a random sample of 998 people age 15 or older, the mean amount of time spent eating or drinking per day is 1.26 hours with a standard deviation of 0.57 hours. Complete parts (a) through (d) below.
What type of experimental design is this? Matched pair What is the response variable in this experiment? Time of arrival What is the treatment? type of transportation What are the experimental units? the students Why was the coin used to decide the transportation that the student would use first? to eliminate bias
A student wanted to compare two types of commuting options. One type is carpooling and the other is taking a commuter train. It is a common belief that carpooling saves more time. This belief is tested by having 10 students commute to school by each mode of transportation and record the time of arrival at school each morning. A coin flip was used to determine which type of transportation that a student would use first. Results indicated that there was no difference in the two types of transportation.
One can be 95% confident that the mean drive-through service time of this fast-food chain is between 171.2 seconds and 174.6 seconds.
A trade magazine routinely checks the drive-through service times of fast-food restaurants. A 95% confidence interval that results from examining 547 customers in one fast-food chain's drive-through has a lower bound of 171.2 seconds and an upper bound of 174.6 seconds. What does this mean?
One can be 80% confident that the mean drive-through service time of this fast-food chain is between 166.0 seconds and 169.2 seconds.
A trade magazine routinely checks the drive-through service times of fast-food restaurants. An 80% confidence interval that results from examining 568 customers in one fast-food chain's drive-through has a lower bound of 166.0 seconds and an upper bound of 169.2 seconds. What does this mean?
If 100 different confidence intervals are constructed, each based on a different sample of size n from the same population, then we expect 80 of the intervals to include the parameter and 20 to not include the parameter.
Explain what "80% confidence" means in a 80% confidence interval.
point estimate
A ________ ________ is the value of a statistic that estimates the value of a parameter.
The sample is not a simple random sample.
A government's congress has 571 members, of which 56 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 0.074 and an upper bound of 0.122. What is wrong with the alien's approach to estimating the proportion of the human race that is female?
Statistical significance means that the result observed in a sample is unusual when the null hypothesis is assumed to be true
Explain what "statistical significance" means.
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.
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.
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.
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.
Explain why the t-distribution has less spread as the number of degrees of freedom increases.
The confidence interval for a----proportion----should be constructed because the variable of interest is------an individual's opinion---which is a----qualitative 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?"
Qualitative with 2 possible outcomes
For what type of variable does it make sense to construct a confidence interval about a population proportion?
type II error
If we do not reject the null hypothesis when the statement in the alternative hypothesis is true, we have made a Type _______ error.
type I error
If we reject the null hypothesis when the statement in the null hypothesis is true, we have made a Type _______ error
increase sample size decrease confidence level
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.
a) The interpretation is flawed. The interpretation provides no interval about the population proportion. b) The interpretation is flawed. The interpretation indicates that the level of confidence is varying. c) The interpretation is reasonable. d) The interpretation is flawed. The interpretation suggests that this interval sets the standard for all the other intervals, which is not true.
In a survey of 2085 adults in a certain country conducted during a period of economic uncertainty, 56% thought that wages paid to workers in the industry were too low. The margin of error was 7 percentage points with 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.
a) There is a 95% chance the mean number of hours worked by adults in this country in the previous week was between 38.4 hours and 40.3 hours. Flawed. This interpretation implies that the population mean varies rather than the interval. b) We are 95% confident that the mean number of hours worked by adults in this country in the previous week was between 38.4 hours and 40.3 hours. Correct. This interpretation is reasonable. c) 95% of adults in this country worked between 38.4 hours and 40.3 hours last week. Flawed. This interpretation makes an implication about individuals rather than the mean. 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 38.4 hours and 40.3 hours. Flawed. The interpretation should be about the mean number of hours worked by adults in the whole country, not about adults in a particular area.
In a survey, 1200 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: 38.4 and upper bound: 40.3. Which of the following represents a reasonable interpretation of the result? For those that are not reasonable, explain the flaw. Complete parts (a) through (d) below.
true
Is the statement below true or false? The mean of the sampling distribution of p is p.
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? Justify your answer.
false
Sample evidence can prove that a null hypothesis is true
a) What does it mean for this study to be cross-sectional? The data were obtained at a specific point in time and the study was an observational study. b) What is the variable of interest in this study? Is it qualitative or quantitative? Explain. Identify the variable of interest. Choose the correct answer below. - Whether a person with sleep apnea has gum disease or not. - Qualitative — the variable classifies the individuals in the study. c) Estimate the proportion of individuals who suffer from sleep apnea who have gum disease with 90% confidence. Interpret your result. Select the correct choice below and fill in the answer boxes to complete your choice.
Sleep apnea is a disorder in which you have one or more pauses in breathing or shallow breaths while you sleep. In a cross-sectional study of 430 adults who suffer from sleep apnea, it was found that 258 had gum disease. Note: In the general population, about 17.5% of individuals have gum disease. Complete parts (a) through (c) below.
About 31 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.
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 150 companies to invest in. After 1 year, 78 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 0.3121. Explain what this P-value means and write a conclusion for the researcher. (Assume α is 0.1 or less.)
If the P-value for a particular test statistic is 0.33, she expects results at least as extreme as the test statistic in about 33 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 H0: p=0.3 versus H1: p>0.3 and she finds the P-value to be 0.33. Explain what this means. Would she reject the null hypothesis? Why?
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.
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?
All the values within the margin of error are greater than 50%
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 1600 adults resulted in 57% stating they were at their personal best in the morning. The poll's results were reported with a margin of error of 5%. Explain why the poll's headline is accurate.
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.
null hypothesis
The _______ _______ is a statement of no change, no effect, or no difference.
alternative hypothesis
The _______ _______ is a statement we are trying to find evidence to support.
level of confidence (1 - α) · 100%.
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 _______.
False
To construct a confidence interval about the mean, the population from which the sample is drawn must be approximately normal.
False. The sample size must be increased by a factor of four to cut the standard error in half.
To cut the standard error of the mean in half, the sample size must be doubled.
false
True or False: The population proportion and sample proportion always have the same value.
Mariya'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 0.27. Each independently constructed a confidence interval based on the point estimate, but Jaime's interval has a lower bound of 0.177 and an upper bound of 0.363, while Mariya's interval has a lower bound of 0.189 and an upper bound of 0.298. Which interval is wrong? Why?
If P-value<α, reject the null hypothesis.
What is the criterion for rejecting the null hypothesis using the P-value approach?
false
When testing a hypothesis using the P-value Approach, if the P-value is large, reject the null hypothesis
