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Cattle Livestock are given a special feed supplement to see if it will promote weight gain. Researchers report that the 77 cows studied gained an average of 56 pounds, and that a \(95 \%\) confidence interval for the mean weight gain this supplement produces has a margin of error of ±11 pounds. Some students wrote the following conclusions. Did anyone interpret the interval correctly? Explain any misinterpretations. a. \(95 \%\) of the cows studied gained between 45 and 67 pounds. b. We're \(95 \%\) sure that a cow fed this supplement will gain between 45 and 67 pounds. c. We're \(95 \%\) sure that the average weight gain among the cows in this study was between 45 and 67 pounds. d. The average weight gain of cows fed this supplement will be between 45 and 67 pounds \(95 \%\) of the time. e. If this supplement is tested on another sample of cows, there is a \(95 \%\) chance that their average weight gain will be between 45 and 67 pounds.

Step by step solution

01

Interpretation a

Interpretation a states that '95% of the cows studied gained between 45 and 67 pounds'. This is a misinterpretation because a confidence interval doesn't predict percentages of data within the interval but rather the reliability of the estimated range of the population mean.

02

Interpretation b

Interpretation b states, 'We're 95% sure that a cow fed this supplement will gain between 45 and 67 pounds'. This is also a misinterpretation, because a confidence interval doesn't offer assurances about individual outcomes but about the estimated range of the population's mean.

03

Interpretation c

Interpretation c states that 'We're 95% sure that the average weight gain among the cows in this study was between 45 and 67 pounds'. This is a correct interpretation because a confidence interval is a measure of how reliable the estimated range of the population mean is.

04

Interpretation d

Interpretation d suggests that 'The average weight gain of cows fed this supplement will be between 45 and 67 pounds 95% of the time'. This is a misinterpretation because a confidence interval does not suggest that 95% of future observations will fall within the stated interval.

05

Interpretation e

Interpretation e suggests that 'If this supplement is tested on another sample of cows, there is a 95% chance that their average weight gain will be between 45 and 67 pounds'. This is also a correct interpretation because if a different random sample was taken from the population, we can be 95% confident that the interval between 45 and 67 pounds will contain the true mean weight gain.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Interpretation of Results

Understanding the interpretation of confidence intervals is crucial for applying statistical results accurately. These intervals provide a range in which we expect the true population parameter to lie. In this context, the parameter is the mean weight gain of cows on a special supplement.
When analyzing different interpretations:

• Some statements incorrectly focus on individual predictions or assume percentages of the sample fit within this range.
• Correct interpretations focus on the population mean and the confidence we have in this interval capturing the true mean.
• It’s also necessary to differentiate between individual outcomes and statistical inferences about averages.

The correct understanding in this exercise revolves around the population mean and how the confidence interval relates to this parameter.

Statistical Inference

Statistical inference allows us to use sample data to make generalizations about a larger population. In this exercise, researchers utilized a sample of 77 cows to draw conclusions about the effect of a supplement on weight gain for all cows.
A confidence interval represents part of this statistical inference framework:

• It quantifies uncertainty in the estimate of the population parameter.
• In our scenario, it estimates the population mean weight gain.
• The interval provides a range where the true mean is likely to be found, based on the sample data.

Statistical inference bridges the gap between sample information and broader population parameters, facilitating informed decisions based on data observations.

Population Mean Estimation

The goal of constructing a confidence interval is to estimate the population mean, a key statistic that describes the average outcome in the population. In our example, the mean weight gain for cows is estimated using data from cows who received a specific supplement:

• The average weight gain recorded was 56 pounds.
• The confidence interval gives a range of plausible values for this mean.
• The interval constructed (45 to 67 pounds) reflects where the true population mean likely falls.

This method requires assumptions that our sample is representative and the factors influencing the cows are consistent across both the study and the population.

Margin of Error

The margin of error is an integral concept when discussing confidence intervals, defining the range added or subtracted from the sample estimate to create the interval. Here, the margin of error was ±11 pounds.
This element accounts for:

• Sampling variability, which causes differences between the sample and population parameters.
• The confidence level, which in this exercise is 95%.
• Factors that introduce potential errors or variations beyond the sample mean.

Understanding the margin of error allows researchers and decision-makers to grasp the uncertainty associated with sample-based estimates and communicate the likely accuracy of their results effectively.