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Chapter 6: Problem 28

A \(99 \%\) confidence interval for the proportion who will answer "Yes" to a question, given that 62 answered yes in a random sample of 90 people

Short Answer

Expert verified
The \(99 \%\) confidence interval for the proportion who will answer 'Yes' to a question is approximately from \(0.599\) to \(0.778\).

Step by step solution

01

Calculate 'p'

The proportion 'p' is simply the number of 'Yes' responses divided by the total number of responses. This would be \(\frac{62}{90}\) which equals \(0.6889\) (rounded to four decimal places).
02

Determine the Z-score

A Z-score is a measure of how many standard deviations an element is from the mean. For a \(99\%\) confidence interval, the Z-score is \(2.57\) (corresponding to \(0.5\%\) in each tail of the normal distribution curve, since \(99\%\) should fall within the interval).
03

Plug 'p', 'Z', and 'n' into the confidence interval formula

The formula for a confidence interval is \(p \pm Z\sqrt{\frac{p(1-p)}{n}}\). Substituting the values we have: \(0.6889 \pm 2.57 \sqrt{\frac{0.6889*(1-0.6889)}{90}}\).
04

Calculate the Result

Calculate the expression inside the square root first, then multiply it by the Z-score, and finally add and subtract it from 'p' respectively to get the confidence interval. Approximately, the \(99\%\) confidence interval would be (0.599, 0.778).

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Key Concepts

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

Proportions
Proportions are a way to express parts of a whole as a number between 0 and 1, or in percentages between 0% and 100%. In statistical studies, we often look at proportions to understand how prevalent a particular response or characteristic is in a sample group. For example, if we survey 90 people and 62 say 'Yes' to a question, the proportion (denoted as 'p') of 'Yes' responses is calculated as the number of 'Yes' responses divided by the total number of responses.

To calculate it here:
  • Number of 'Yes' responses: 62
  • Total responses: 90
  • Calculated Proportion (p): \( \frac{62}{90} \approx 0.6889 \)
Using proportions helps us make sense of data and provides a simple metric to use in further calculations. It is foundational to any statistical analysis in dichotomous (yes/no) data situations.
Z-score
A Z-score in statistics is a measure that describes a value's relation to the mean of a group of values, expressed in terms of standard deviations. The Z-score is crucial in determining how far away a data point is from the mean.
When building confidence intervals, like our 99% confidence interval, Z-scores are used to calculate the margin of error. For a 99% confidence interval, we recognize that 99% of data points fall within that range, and a typical Z-score used for this purpose is 2.57. This captures the probability within two tails of 0.5% each in a normal distribution.
  • Z-score for 99% confidence level: 2.57
  • This value represents how many standard deviations away from the mean data points in the interval will be.
The Z-score provides a simple way to infer where proportions lie in terms of standard deviations.
Statistical Analysis
Statistical Analysis is a method of collecting, analyzing, interpreting, presenting, and organizing data. This type of analysis is used to predict trends and outcomes, and to make informed decisions based on data.
When we talk about confidence intervals, like calculating a 99% confidence interval for a proportion, we perform statistical analysis to determine the range in which the true population parameter lies.
The steps involved include:
  • Calculating the sample proportion \( p \), which measures the observed proportion of success in the sample.
  • Using the appropriate Z-score for the desired confidence level.
  • Applying the confidence interval formula: \( p \pm Z\sqrt{\frac{p(1-p)}{n}} \), which accounts for the variability and size of the sample.
This type of analysis allows us to quantify uncertainty and make estimates about a population based on a sample. Statistical analysis is pivotal to understanding the accuracy and reliability of data-driven decisions.

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Most popular questions from this chapter

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