Posted by Apr 3. Due: WED April 10.
A medical researcher wanted to estimate μ, the mean weight of newborns born to women over the age of 40.
The researcher chose a random sample of 100 pregnant women who were over 40, followed them through the pregnancy, and found that the mean weight of the 100 newborns was 3,035 grams. From past research, it is assumed that the weight of newborns has a standard deviation of σ = 500. The researcher calculated the 95% confidence interval for μ to be (2935, 3135).
If the researcher wanted to maintain the 95% level of confidence but report a confidence interval with a smaller margin of error, which of the following will achieve that?
The mean score on the quantitative reasoning part of the GRE (Graduate Record Examination) of non-U.S. citizens has an unknown mean, , and an assumed standard deviation =8. Based on a random sample of non-U.S. citizens who took the GRE in 2014 the 95% confidence interval for was calculated to be (153.6, 158.6).
Suppose now that a different random sample of non-U.S. citizens who took the GRE in 2014 is chosen and that this sample is larger than the sample that produced the confidence interval above.
Which of the following is the most likely 95% confidence interval for µ based on the larger sample?
A poll asked a random sample of 1,000 U.S. adults, “Do you think that the use of marijuana should be legalized?” 560 of those asked answered yes.
Based on the poll’s results, estimate p, the proportion of all U.S. adults who believe the use of marijuana should be legalized, with a 95% confidence interval.
Give an interpretation of the margin of error in context.
Do the results of this poll give evidence that the majority of U.S. adults believe that the use of marijuana should be legalized?
A similar poll was conducted 2 years ago, and reported the 95% confidence interval for p, the proportion of U.S. adults who believe the use of marijuana should be legalized, to be (.48 , .54). Do you think that the results of the current study (where the 95% confidence interval is (.53 , .59) provide evidence that the public opinion on the topic of legalization of marijuana has changed over the past two years?
Below are four different situations in which a confidence interval for \(\mu\) is called for.
Situation A: In order to estimate μ, the mean annual salary of high-school teachers in a certain state, a random sample of 150 teachers was chosen and their average salary was found to be $38,450. From past experience, it is known that teachers’ salaries have a standard deviation of $5,000.
Situation B: A medical researcher wanted to estimate μ, the mean recovery time from open-heart surgery for males between the ages of 50 and 60. The researcher followed the next 15 male patients in this age group who underwent open-heart surgery in his medical institute through their recovery period. (Comment: Even though the sample was not strictly random, there is no reason to believe that the sample of “the next 15 patients” introduces any bias, so it is as good as a random sample). The mean recovery time of the 15 patients was 26 days. From the large body of research that was done in this area, it is assumed that recovery times from open-heart surgery have a standard deviation of 3 days.
Situation C: In order to estimate μ, the mean score on the quantitative reasoning part of the GRE (Graduate Record Examination) of all MBA students, a random sample of 1,200 MBA students was chosen, and their scores were recorded. The sample mean was found to be 590. It is known that the quantitative reasoning scores on the GRE vary normally with a standard deviation of 150.
Situation D: A psychologist wanted to estimate μ, the mean time it takes 6-year-old children diagnosed with Down’s Syndrome to complete a certain cognitive task. A random sample of 12 children was chosen and their times were recorded. The average time it took the 12 children to complete the task was 7.5 minutes. From past experience with similar tasks, the time is known to vary normally with a standard deviation of 1.3 minutes.
In which situation can we not use the confidence interval that we developed?
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