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Chapter 9: Problem 48

The formula used to calculate a large-sample confidence interval for \(p\) is $$ \hat{p} \pm(z \text { critial value }) \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$ What is the appropriate \(z\) critical value for each of the following confidence levels? a. \(95 \%\) b. \(98 \%\) c. \(85 \%\)

Short Answer

Expert verified
The z critical values correlate with the confidence levels as follows: For a 95% confidence level, the Z value is 1.96. For a 98% confidence level, it is 2.33. And for an 85% confidence level, it is 1.44.

Step by step solution

01

Understand Critical Value Z

The critical value or Z score corresponds to the confidence level for a parameter, which is often found using a z-table. The level of confidence corresponds to the probability that the confidence interval contains the population parameter. This results in a two-sided confidence level since this probability is split evenly on both sides of the mean.
02

Determine the Z Critical Value for 95% Confidence Level

For a 95% confidence interval, the z critical value is approximately 1.96. This means that if the real parameter of the population is within 1.96 standard deviations from the sample parameter, then the population parameter is within the 95% confidence interval.
03

Determine the Z Critical Value for 98% Confidence Level

For a 98% confidence level, the critical z value is approximately 2.33. This implies that if the population parameter is within 2.33 standard deviations from the sample parameter, this population parameter falls within the 98% confidence interval.
04

Determine the Z Critical Value for 85% Confidence Level

For an 85% confidence level, the critical z value is approximately 1.44. This means that if the population parameter is within 1.44 standard deviations of the sample parameter, the population parameter lies within the 85% confidence interval.

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Key Concepts

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

Understanding Critical Value
A critical value in statistics is an essential concept when working with confidence intervals. The critical value is a point on the distribution that relates to your desired level of confidence, determining how far from the mean your data can vary before it's considered unlikely. It helps in making decisions about a population parameter based on a sample.
When we talk about critical values for confidence intervals, we're dealing with the normal distribution, where most data emphasizes a bell curve shape.
  • The area under the curve from the mean to the critical value corresponds to your confidence level.
  • This area represents the probability that the confidence interval contains the true population parameter.
  • In a two-sided confidence interval, this area is split evenly beneath the tails.
Critical values can be found using statistical tables, also known as Z-tables, which list them based on chosen confidence levels.
The Role of Z-Score
The Z-score is another fundamental aspect of calculating confidence intervals. It's a statistical computation that measures the position of a value, in standard deviations, from the mean of a set.
For confidence intervals, the Z-score helps determine how many standard deviations away from the sample mean, the true population parameter might be.
  • The standard Z-scores relate directly to confidence levels.
  • For example, a Z-score of 1.96 corresponds to a 95% confidence level, meaning the interval spans 1.96 standard deviations around the sample mean.
  • When you calculate the Z-score for a given confidence level, you're essentially setting the range for your confidence interval.
These scores help us make inferences about population parameters by assessing sample data, integrating both the variability and our confidence level.
Confidence Level Explained
Confidence level is a probability metric that represents how certain we are that a particular interval contains the true population parameter.
It's essential for understanding the broader statistical conclusions we can draw from sample data.
  • A higher confidence level means a wider confidence interval, which ensures a greater likelihood that the interval captures the population parameter.
  • The most common confidence levels are 95%, 98%, and 99%, each indicating the probability that the given interval contains the true parameter.
  • Choosing the right confidence level depends on the balance between accuracy (precision) and reliability you wish to achieve in your analysis.
The confidence level reflects the degree of certainty, impacting how conservative the estimation is regarding the range of possible values for the population parameter.

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Most popular questions from this chapter

The article "Kids Digital Day: Almost 8 Hours" (USA Today, January 20,2010 ) summarized a national survey of 2,002 Americans ages 8 to 18 . The sample was selected to be representative of Americans in this age group. a. Of those surveyed, 1,321 reported owning a cell phone. Use this information to construct and interpret a \(90 \%\) confidence interval for the proportion of all Americans ages 8 to 18 who own a cell phone. b. Of those surveyed, 1,522 reported owning an MP3 music player. Use this information to construct and interpret a \(90 \%\) confidence interval for the proportion of all Americans ages 8 to 18 who own an MP3 music player. c. Explain why the confidence interval from Part (b) is narrower than the confidence interval from Part (a) even though the confidence levels and the sample sizes used to calculate the two intervals were the same.

A researcher wants to estimate the proportion of property owners who would pay their property taxes one month early if given a \(\$ 50\) reduction in their tax bill. Would the standard error of the sample proportion \(\hat{p}\) be larger if the actual population proportion were \(p=0.2\) or if it were \(p=0.4 ?\)

Based on data from a survey of 1,200 randomly selected Facebook users (USA Today, March 24, 2010), a \(90 \%\) confidence interval for the proportion of all Facebook users who say it is not OK to "friend" someone who reports to you at work is (0.60,0.64) . What is the meaning of the \(90 \%\) confidence level associated with this interval?

In a study of 1,710 schoolchildren in Australia (Herald Sun, October 27,1994 ), 1,060 children indicated that they normally watch TV before school in the morning. (Interestingly, only \(35 \%\) of the parents said their children watched TV before school.) Construct and interpret a \(95 \%\) confidence interval for the proportion of all Australian children who say they watch TV before school. In order for the method used to construct the interval to be valid, what assumption about the sample must be reasonable?

For each of the following choices, explain which would result in a narrower large-sample confidence interval for \(p\) : a. \(95 \%\) confidence level or \(99 \%\) confidence level b. \(n=200\) or \(n=500\)
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