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Chapter 8: Problem 153

Suppose we wanted a 99\% Cl. Would this interval be longer, shorter, or the same size as the \(95 \%\) CI in Problem 8.152? (Do not actually compute the interval.)

Short Answer

Expert verified
A 99% confidence interval is longer than a 95% confidence interval.

Step by step solution

01

Understanding Confidence Intervals

Confidence intervals (CI) provide a range of values that likely contain a population parameter. The 'confidence level' is the probability that this interval correctly contains the parameter.
02

Effect of Confidence Level on Interval Length

As the confidence level increases (e.g., from 95% to 99%), the interval becomes wider. This is because a higher level of confidence requires capturing more of the population distribution, expanding the range of the interval.
03

Compare 95% and 99% Confidence Intervals

With a 95% CI, we have 5% total probability outside the interval (2.5% on each tail), while a 99% CI has only 1% total probability outside (0.5% on each tail). Thus, to achieve greater confidence (99%), the interval must be longer.

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Key Concepts

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

Population Parameter
A population parameter is a key value that describes some characteristic of an entire population. It could be the mean, median, proportion, or standard deviation of the population.
These parameters give us insights into the broad features and behavior of a given population. However, since it is often impractical to collect data from every individual in a population, statisticians use samples to estimate these parameters.
A confidence interval provides an estimated range of values that aim to include the true population parameter, offering a useful tool to infer about the population based on sample data.
Confidence Level
The confidence level is a critical part of understanding confidence intervals. It represents the percentage of all possible samples that can be expected to include the true population parameter.
A higher confidence level, such as 99%, implies greater assurance that the interval truly captures the parameter. Hence, the confidence level influences how wide or narrow the interval is. A common confidence level used is 95%, but this can be adjusted depending on the need for precision.
Adjusting the confidence level affects the width of the interval:
  • A higher confidence level typically results in a wider interval since we need to capture more of the data to ensure the parameter is included.
  • A lower confidence level results in a narrower interval, capturing less of the data, but being less certain that the parameter lies within the interval.
Distribution Tail Probability
When you calculate a confidence interval, you're dealing with the probability distribution of a population parameter. The distribution tail probability refers to the area in the tails of the distribution outside the confidence interval.
In a normal distribution for a 95% confidence level, 2.5% probability lies in each tail, making up a total of 5% outside the interval. For a 99% confidence level, the tails hold only 0.5% each, totaling 1%.
Therefore, reducing the tail probability (or increasing the confidence level) means that less of the distribution is outside the interval, necessitating a wider interval to cover more of the parameter's potential values.
Understanding how the tail probabilities shift with different confidence levels helps in appreciating why intervals must expand or contract to maintain the desired confidence.

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Most popular questions from this chapter

What test can be used to compare mean change in weight between the intervention and control groups?

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