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10.3, 9.1, 9.2
Terms in this set (19)
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.
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.
Does the population have to be normally distributed to test this hypothesis by using t-distribution methods? Why?
No, because n≥30.
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.
A physician wanted to compare two types of headache relief. One type is medication and the other is using pressure points. It is a common belief that medication relieves pain faster. This belief is tested by having 10 migraine sufferers compare both types of pain relief and recording their observations on a standardized scale of response. A coin flip was used to determine which type of headache relief each individual would try first. Results indicated that there was no difference in the two types of pain relief.
-the recorded observations
-The factor is the type of pain relief. The treatments are medication and using pressure points
-The migraine sufferers
-To eliminate bias as to which pain relief was used first
A ________ ________ is the value of a statistic that estimates the value of a parameter.
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-alpha) * 100%
For what type of variable does it make sense to construct a confidence interval about a population proportion?
Qualitative with 2 possible outcomes
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 _______ should be constructed because the variable of interest is ________ , which is a __________ variable.
proportion, an individual's opinion, qualitative
A group conducted a poll of 2016 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 48% of the popular vote. The margin of error was reported to be 2%. The group reported that the race was too close to call. Use the concept of a confidence interval to explain what this means.
What does it mean to say the race was too close to call?
The margin of error suggests candidate A may receive between 47% and 51% of the popular vote and candidate B may receive between 46% and 50% of the popular vote. Because the poll estimates overlap when accounting for margin of error, the poll cannot predict the winner.
What does "98% confidence" mean in a 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 of the intervals to include the parameter and 2 to not include the parameter.
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.
Whose estimate will have the smaller margin of error and why?
Matthew's estimate will have the smaller margin of error because the larger sample size more than compensates for the higher level of confidence.
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 0.28. Each independently constructed a confidence interval based on the point estimate, but Jaime's interval has a lower bound of 0.224 and an upper bound of 0.301, while Mariya's interval has a lower bound of 0.226 and an upper bound of
0.334. Which interval is wrong? Why?
Jaime's interval is wrong because it is not centered on the point estimate.
A government's congress has 538, members, of which 95 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 0.208. 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.
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 ______ should be constructed because the variable of interest is _______, which is a ______
mean, an individual's reduction in caloric intake, quantitative
To construct a confidence interval about the mean, the population from which the sample is drawn must be approximately normal.
This statement is true or false
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.
A trade magazine routinely checks the drive-through service times of fast-food restaurants. A 95% confidence interval that results from examining 665 customers in one fast-food chain's drive-through has a lower bound of 167.4 seconds and an upper bound of 170.8 seconds. What does this mean?
One can be 95% confident that the mean drive-through service time of this fast-food chain is between 167.4
seconds and 170.8 seconds.
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.
Decrease the confidence level, Increase the sample size.
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