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Chapter 8: Problem 18

z-score and confidence level Which z-score is used in a (a) \(90 \%,\) (b) \(98 \%\), and (c) \(99.9 \%\) confidence interval for a population proportion?

Short Answer

Expert verified
(a) 1.645, (b) 2.33, (c) 3.291 are the z-scores for 90%, 98%, and 99.9% confidence intervals, respectively.

Step by step solution

01

Understanding the Z-Score and Confidence Level

A z-score is a statistical measure reflecting how many standard deviations an element is from the mean. In confidence intervals, a z-score indicates the number of standard deviations required to contain a certain percentage of data points within a normal distribution. The confidence level indicates the percentage of times the true parameter is expected to be captured by the confidence interval. For a standard normal distribution, different confidence levels correspond to different z-scores.
02

Find the Z-Score for 90% Confidence Interval

To find the z-score for a 90% confidence interval, we determine the critical values that leave 5% in each tail of a standard normal distribution. Using a z-score table or calculator, we find that the z-score for 90% confidence is approximately 1.645.
03

Find the Z-Score for 98% Confidence Interval

For a 98% confidence interval, 2% of the data is outside the interval, leaving 1% in each tail. Checking a z-score table or using a calculator, we find the z-score for 98% confidence is about 2.33.
04

Find the Z-Score for 99.9% Confidence Interval

A 99.9% confidence interval leaves 0.1% of the data outside, with 0.05% in each tail. The z-score corresponding to 99.9% confidence can be found using a z-table or calculator, and it is approximately 3.291.

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Key Concepts

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

Z-Score
The z-score is a powerful tool in statistics that measures how far away a particular value is from the mean of the data set. It is expressed in terms of standard deviations. So, if you have a z-score of 2, this means the value is 2 standard deviations away from the mean. When you use z-scores in the context of confidence intervals, they tell you how confident you can be that a certain population parameter is captured within the interval. Let's break it down:
  • The larger the z-score, the higher the confidence level.
  • Z-scores are derived from the standard normal distribution.
  • Common confidence levels (90%, 95%, 98%, etc.) have corresponding z-scores like 1.645 for 90%, 1.96 for 95%, and so on.
By understanding z-scores, you can effectively interpret how precise your estimates are.
Standard Normal Distribution
The standard normal distribution is a special normal distribution where the mean is 0 and the standard deviation is 1. This distribution is symmetrical around the mean, meaning it has the same shape on both sides. Here's why it's important:
  • The total area under the curve is 1, reflecting 100% of the population.
  • It allows for quick conversion of data into z-scores, making comparisons easy.
  • Z-scores derived from a standard normal distribution provide a basis for calculating probabilities and confidence intervals.
Remember, the shape of the standard normal distribution curve helps determine how likely different outcomes are.
Population Proportion
Population proportion refers to the fraction of the population that possesses a certain characteristic. When you're estimating this proportion, you use a sample to infer what the whole population might be like. In statistical measures, the relationship between sample data and population prediction is often interpreted through confidence intervals. Here's what you need to know:
  • A confidence interval gives a range within which the true population proportion is estimated to lie.
  • You use z-scores to calculate this range by considering the distribution of the sample proportion.
  • Understanding population proportion helps in making informed decisions based on statistical data.
By working with population proportions and confidence intervals, you can infer important details about the larger group from which your sample was drawn.
Statistical Measures
Statistical measures are tools used to summarize a set of data or entire populations. They include measures of central tendency like mean and median, as well as measures of variability like range and standard deviation. Here's how they come into play:
  • They tell us about the distribution and dispersion in a data set.
  • They are useful in constructing confidence intervals and using z-scores to understand data variability.
  • Statistical measures are the backbone in making sense of collected data and turning it into actionable insights.
By mastering statistical measures, you can effectively conduct analyses that reveal deeper insights into the data at your disposal.

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Most popular questions from this chapter

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