Ch 8. Estimating with Confidence

A random sample of 85 sixth-graders in a large city take a course designed to improve scores on a reading comprehension test. Based on this sample, a 90% confidence interval for the mean improvement in test scores for all sixth-graders in the city taking this course is found to be (12.6, 14.8). Which of the following are the sample mean and margin of error on which this interval is based?

• Sample Mean = 13.7; Margin of Error = 1.1

• Sample mean = 13.7; Margin of Error = 2.2

• Sample mean is unknown; Margin of Error = 2.2

A random sample of 85 sixth-graders in a large city take a course designed to improve scores on a reading comprehension test. Based on this sample, a 90% confidence interval for the mean improvement μ in test scores for all sixth-graders in the city taking this course is found to be (12.6, 14.8). Which one of the following is a correct interpretation of this confidence interval?

• Ninety percent of the 85 sixth-graders in the sample improved their reading comprehension scores by 12.6 to 14.8 points.

• The probability is 0.90 that the true mean improvement in reading comprehension is between 12.6 and 14.8.

• We are 90% confident that the true improvement in reading comprehension scores is captured by the interval 12.6 to 14.8.

Agricultural researchers plant 100 plots with a new variety of corn and measure the mean yield for these plots in bushels per acre. They treat the 100 plots as a simple random sample of possible plots of corn (as researchers often do) and report a 95% confidence interval for the mean corn yield of (128.4, 131.6) bushels per acre. Which of the following is a correct interpretation of the 95% confidence level?

• We are 95% confident that the true mean yield for this new variety of corn is captured by the interval (128.4, 131.6) bushels per acre.

• If many intervals were constructed this way from many independent sets of 100 plots, 95% of the intervals would capture the true mean corn yield.

• If many intervals were constructed this way from many independent sets of 100 plots, 95% of the time the true mean corn yield would be in the interval (128.4, 131.6) bushels per acre.

A random sample of size n is collected from a Normally distributed population with standard deviation σ. Using these data, a confidence interval is computed for the mean of the population. Which of the following actions would produce a new confidence interval with a smaller width (smaller margin of error), assuming that the same data were used?

• Using a higher confidence level

• Using a lower confidence level

• Using a smaller sample size n

A radio talk show host is interested in the proportion p of adults in his listening area who think that the drinking age should be lowered to 18. To find this proportion, he poses the following question to his listeners: “Do you think that the drinking age should be reduced to 18?” He asks listeners to phone in and vote “yes” or “no” depending upon their opinions. Of 200 people who phone in, 140 answer “yes.” The standard error of the sample proportion p-hat of “yes” votes among those who phone in is

• 0.210

• 0.032

• 0.00105

An inspector monitors large truckloads of potatoes to determine the proportion p of potatoes with major defects before the potatoes are used to make potato chips. She intends to compute a 95% confidence interval for p. To do so, she selects a simple random sample of 50 potatoes from a truckload of more than 2000 potatoes. Suppose that only 2 of the 50 potatoes sampled are found to have major defects. Which one of the following assumptions for inference about a proportion using a confidence interval is violated?

• Large Counts

• 10% condition

• There are no violations.

One hundred rats with mothers that were exposed to high levels of tobacco smoke during pregnancy were put through a simple maze. At the outset, the maze required the rats to make a choice between going left and going right. Eighty of the rats went right when running the maze for the first time. Assume that the 100 rats can be considered an SRS from the population of all rats born to mothers who were exposed to high levels of tobacco smoke during pregnancy. (Note that this assumption may or may not be reasonable, but researchers often assume that lab rats are representative of large populations, since they are often bred to have uniform characteristics.) Let p be the proportion of rats in this population that would go right when running the maze for the first time. A 90% confidence interval for p is

• 0.8 ± 0.040

• 0.8 ± 0.078.

• 0.8 ± 0.066.

A noted psychic is tested for extrasensory perception. The psychic is presented, one at a time, with cards that are marked with one of four symbols: a star, a cross, a circle, or a square. Let p represent the probability that the psychic correctly identifies the symbol on the card in a random trial. Which of the following is the smallest number of trials you would have to conduct to estimate p to within ± 0.01 with 95% confidence? (Use the guess 0.25 as the value for p, since this would be the value of p if the psychic were merely guessing.)

• 352

• 7203

• 9604

To assess the accuracy of a kitchen scale, a standard weight, known to weigh 1.000 gram, is weighed a total of n times, and the mean of the n weight measurements is computed. (If a 90% confidence interval for the mean calculated from these measurements does not contain 1.000, the scale is deemed inaccurate.) Suppose the scale readings are Normally distributed and that similar scales are known to measure such weights with a standard deviation of σ = 0.01 gram. Which of the following is the smallest value of n that will produce a 90% confidence interval with a margin of error of at most 0.001 grams?

• 17

• 271

• 385

Which one of the following statements is true about Student's t distribution?

• The density curve of the t distribution is symmetric and centered at the true mean of the population.

• The density curve of the t distribution has heavier “tails” than the density curve of the standard Normal distribution.

• The density curve of the t distribution more closely resembles the density curve of the standard Normal distribution when the number of degrees of freedom is small.