Chapter 9: Problem 3
In a study of environmental effects upon reproduction, 123 female adult white- tailed deer from the central Adirondack area were captured, and 97 were found to be pregnant. \(^{38}\) Construct a \(95 \%\) confidence interval for the proportion of females pregnant in this deer population.
Short Answer
Expert verified
After calculations, the 95% confidence interval for the proportion of pregnant female deer is approximately (0.749, 0.849).
Step by step solution
01
Identify the Given Information
Determine the sample size (n), number of successes (x), and the confidence level. Here, the sample size is 123 (number of deer captured), the number of successes is 97 (number of pregnant deer), and the confidence level is 95%.
02
Calculate the Sample Proportion
Calculate the sample proportion (\(p\)) by dividing the number of successes by the sample size: \(p = \frac{x}{n} = \frac{97}{123}\).
03
Find the Z-Score for the Given Confidence Level
Look up the z-score that corresponds to the desired confidence level. For a 95% confidence interval, the z-score is approximately 1.96, because the area to the left of z = 1.96 under the standard normal curve is 0.975.
04
Calculate the Standard Error
Compute the standard error (SE) of the sample proportion using the formula: \(SE = \sqrt{\frac{p(1-p)}{n}}\).
05
Construct the Confidence Interval
Use the formula for the confidence interval (CI) to find the lower (L) and upper (U) limits: \(L = p - z \times SE\) and \(U = p + z \times SE\). The 95% CI for the proportion of females pregnant in the deer population is then given by the interval \(L, U\).
06
Calculate the Margin of Error
The margin of error (ME) is the product of the z-score and the standard error: \(ME = z \times SE\).
07
Finalize the Confidence Interval
With the calculated margin of error, revise the limits of the confidence interval if necessary to ensure they are between 0 and 1, which are the possible bounds for a proportion.
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Understanding Population Proportion
Population proportion is a key concept in statistics that represents the fraction of individuals in a population that exhibit a certain characteristic. In the context of the given exercise, the characteristic of interest is pregnancy among female white-tailed deer in the central Adirondack area.
Calculating the population proportion involves determining the ratio of the number of 'successes' (in this case, the pregnant deer) to the total number of individuals in the sample (the deer captured). This ratio, denoted as \(p\), is an estimate of the true proportion in the entire population. It's important to note that this estimate is based on sample data and may not be the exact proportion in the larger population, but it gives researchers a useful estimate to work with for further analysis and decision making.
Understanding how to calculate and interpret the population proportion is critical for students, as it forms the basis for many statistical procedures, including hypothesis testing and the construction of confidence intervals, which are used to infer information about the population from the sample data.
Calculating the population proportion involves determining the ratio of the number of 'successes' (in this case, the pregnant deer) to the total number of individuals in the sample (the deer captured). This ratio, denoted as \(p\), is an estimate of the true proportion in the entire population. It's important to note that this estimate is based on sample data and may not be the exact proportion in the larger population, but it gives researchers a useful estimate to work with for further analysis and decision making.
Understanding how to calculate and interpret the population proportion is critical for students, as it forms the basis for many statistical procedures, including hypothesis testing and the construction of confidence intervals, which are used to infer information about the population from the sample data.
Demystifying Standard Error
Standard error (SE) is a statistic that measures the accuracy with which a sample proportion estimates the true population proportion. It essentially provides an indication of the variability or uncertainty around the sample proportion. The lower the standard error, the more precise the estimate.
In the deer study exercise, the standard error is calculated using the formula \(SE = \sqrt{\frac{p(1-p)}{n}}\), where \(p\) is the sample proportion and \(n\) is the sample size. This calculation takes into account the sample proportion and how much it might fluctuate due to the randomness inherent in sampling. The standard error decreases when the sample size increases, which means larger samples generally provide more reliable estimates of the population proportion.
Students must grasp the importance of standard error because it plays a crucial role in constructing confidence intervals and conducting hypothesis tests. It acts as a building block for understanding how confident we can be in our estimates from sample data.
In the deer study exercise, the standard error is calculated using the formula \(SE = \sqrt{\frac{p(1-p)}{n}}\), where \(p\) is the sample proportion and \(n\) is the sample size. This calculation takes into account the sample proportion and how much it might fluctuate due to the randomness inherent in sampling. The standard error decreases when the sample size increases, which means larger samples generally provide more reliable estimates of the population proportion.
Students must grasp the importance of standard error because it plays a crucial role in constructing confidence intervals and conducting hypothesis tests. It acts as a building block for understanding how confident we can be in our estimates from sample data.
The Role of Z-Score in Confidence Intervals
A z-score is a statistical measurement that describes a value's relationship to the mean of a group of values, measured in terms of standard deviations from the mean. In the context of confidence intervals, the z-score is used to determine the range within which we can say with a certain level of confidence that the true population proportion lies.
For the exercise involving the deer population, a 95% confidence level corresponds to a z-score of approximately 1.96, based on the standard normal distribution. This z-score indicates that if we were to take many samples and calculate the same type of confidence interval, 95% of them would contain the true population proportion.
Understanding the z-score and its application in confidence intervals empowers students to accurately express the level of certainty or uncertainty associated with their statistical estimates. It also underlines the probabilistic nature of these estimates and the concept that, while we can never be completely certain about a population parameter, we can provide a range likely to contain it.
For the exercise involving the deer population, a 95% confidence level corresponds to a z-score of approximately 1.96, based on the standard normal distribution. This z-score indicates that if we were to take many samples and calculate the same type of confidence interval, 95% of them would contain the true population proportion.
Understanding the z-score and its application in confidence intervals empowers students to accurately express the level of certainty or uncertainty associated with their statistical estimates. It also underlines the probabilistic nature of these estimates and the concept that, while we can never be completely certain about a population parameter, we can provide a range likely to contain it.
Sample Size Significance
Sample size is a fundamental concept in statistics that affects the accuracy and validity of sample estimates. The size of the sample, denoted as \(n\), influences the precision of the population proportion estimate and the standard error. In the deer study, the sample size is 123 deer.
Generally, a larger sample size yields a smaller standard error, indicating a more precise estimate of the population proportion. This happens because larger samples are typically more representative of the population, reducing the variability caused by random chance. However, while increasing the sample size can enhance accuracy, it also involves higher costs and more time for data collection.
Understanding the impact of sample size is key for students, as they need to balance the desire for more accurate estimates with the practical considerations of study execution. Decisions about sample size will directly affect the confidence we have in our conclusions and the resources required to reach them. Therefore, a solid comprehension of sample size implications is crucial for designing statistically sound and feasible studies.
Generally, a larger sample size yields a smaller standard error, indicating a more precise estimate of the population proportion. This happens because larger samples are typically more representative of the population, reducing the variability caused by random chance. However, while increasing the sample size can enhance accuracy, it also involves higher costs and more time for data collection.
Understanding the impact of sample size is key for students, as they need to balance the desire for more accurate estimates with the practical considerations of study execution. Decisions about sample size will directly affect the confidence we have in our conclusions and the resources required to reach them. Therefore, a solid comprehension of sample size implications is crucial for designing statistically sound and feasible studies.
