Interpret a Confidence Interval

What is the format for interpreting a confidence interval? 

• The interval from ___ to ___ captures the true mean in context.
• C% of samples of the same size taken from the same population will capture the parameter in question
• There is convincing evidence that C% gives the interval (___, ___) capturing the parameter in question.
• We are C% confident that the interval from ___ to ___ captures the [parameter in context].

Suppose that $51,759 was the median income for all households in 2012. Does the interval ($51,484, $52,394) provide convincing evidence that the median household income increased in 2013? 

• Yes
• No

In a recent poll of 803 randomly selected baseball fans in Connecticut, 44% said their favorite team was the Yankees. If researchers wanted to reduce the margin of error to at most 2%, how many baseball fans do they have to randomly select? 

• 1564
• 2367
• 803
• 3170

Which of the following changes to a study would result in a narrower confidence interval?

• increasing the confidence level, increasing the sample size
• decreasing the confidence level, decreasing the sample size
• increasing the confidence level, decreasing the sample size
• decreasing the confidence level, increasing the sample size.

A 90% confidence interval for a population mean is determined to be 800 to 900. If the confidence is increased to 95% confidence while the sample statistics and sample size remain the same, the confidence interval

• becomes wider
• becomes narrower
• does not change

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 90% confidence interval for the average salary of all CEOs in the electronics industry was constructed using the results of a random survey of 45 CEOs. The interval was ($139,048, $154,144). Give a practical interpretation of the interval.

• 90% of the sampled CEOs have salaries that fell in the interval $139,048 to $154,144
• We are 90% confident that the mean salary of all CEOs in the electronics industry falls in the interval $139,048 to $154,144. 
• 90% of all CEOs in the electronics industry have salaries that fall between $139,048 to $154,144
• We are 90% confident that the mean salary of the sampled CEOs falls in the interval $139,048 to $154,144.

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.

What is the best interpretation of a 95% confidence interval?

• If repeated samples were taken and the 95% confidence interval was computed for each sample, 95% of the intervals would contain the population proportion.

• A 95% confidence interval has a 0.95 probability of containing the population mean.

• 95% of the population distribution is contained in the confidence interval.

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Is this a correct interpretation of a confidence interval?

If the same population is sampled on numerous occasions and interval estimates are made on each occasion, the resulting intervals would bracket the true population parameter in approximately 95 % of the cases.

• Yes

• No