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Magazine Chapter 9: Problem 38
The formula used to calculate a large-sample confidence interval for \(p\) is $$ \hat{p} \pm(z \text { critical value }) \sqrt{\frac{\hat{p}(1-\hat{p})}{n}} $$ What is the appropriate \(z\) critical value for each of the following confidence levels? a. \(90 \%\) b. \(99 \%\) c. \(80 \%\)
Short Answer
Expert verified
The appropriate \(z\) critical values for each confidence level are:
a. \(90\% \rightarrow 1.645\)
b. \(99\% \rightarrow 2.576\)
c. \(80\% \rightarrow 1.28\)
Step by step solution
01
Recall the standard normal distribution and the concept of the critical value
A standard normal distribution is a normal distribution with a mean of 0 and a standard deviation of 1. A \(z\) critical value corresponds to the number of standard deviations away from the mean that defines a particular confidence level or probability.
02
Use a Z-table or calculator to find the z critical value
A Z-table or a calculator with the standard normal distribution function can be used to find the \(z\) critical value for a particular confidence level.
In this exercise, we need to find the \(z\) critical value for the following confidence levels:
a. \(90\%\)
b. \(99\%\)
c. \(80\%\)
To find the \(z\) critical value for each confidence level, first find the area (probability) that corresponds to each confidence level, and then use the Z-table or calculator to find the corresponding \(z\) value.
03
Find the z critical value for a 90% confidence level
For a \(90\%\) confidence level, there will be a \(10\%\) difference in the probability between the confidence interval and \(100\%\). This \(10\%\) needs to be split equally on both tails, so we have \(5\%\) on each tail. This means we need to find the \(z\) value associated with a \(95\%\) probability (since \(90\% + 5\% = 95\%\)).
Using a Z-table or calculator, we can find the \(z\) critical value for a \(95\%\) probability to be approximately \(1.645\). So, the \(z\) critical value for a \(90\%\) confidence level is \(1.645\).
04
Find the z critical value for a 99% confidence level
For a \(99\%\) confidence level, there will be a \(1\%\) difference in the probability between the confidence interval and \(100\%\). This \(1\%\) needs to be split equally on both tails, so we have \(0.5\%\) on each tail. This means we need to find the \(z\) value associated with a \(99.5\%\) probability (since \(99\% + 0.5\% = 99.5\%\)).
Using a Z-table or calculator, we can find the \(z\) critical value for a \(99.5\%\) probability to be approximately \(2.576\). So, the \(z\) critical value for a \(99\%\) confidence level is \(2.576\).
05
Find the z critical value for an 80% confidence level
For an \(80\%\) confidence level, there will be a \(20\%\) difference in the probability between the confidence interval and \(100\%\). This \(20\%\) needs to be split equally on both tails, so we have \(10\%\) on each tail. This means we need to find the \(z\) value associated with a \(90\%\) probability (since \(80\% + 10\% = 90\%\)).
Using a Z-table or calculator, we can find the \(z\) critical value for a \(90\%\) probability to be approximately \(1.28\). So, the \(z\) critical value for an \(80\%\) confidence level is \(1.28\).
Now, we have found the \(z\) critical values for each of the given confidence levels:
a. \(90\% \rightarrow 1.645\)
b. \(99\% \rightarrow 2.576\)
c. \(80\% \rightarrow 1.28\)
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Key Concepts
These are the key concepts you need to understand to accurately answer the question.
Z Critical Value
Understanding the concept of the z critical value is essential for conducting statistical tests and creating confidence intervals. This value represents the point on a standard normal distribution curve beyond which a specified percentage of the data falls.
For example, when we talk about a 90% confidence level, we are focused on the central 90% of the curve, leaving 5% in each tail. The z critical value for this area would be the number revealing how many standard deviations you'd need to be from the mean to encompass that 90% in the middle of the distribution. Concretely, we consider the value of 1.645 for a 90% confidence interval which means that 1.645 standard deviations from the mean will give us our desired level of confidence.
To find the z critical value, a Z-table or statistical software can be used. The table correlates the area under the curve with the number of standard deviations from the mean, hence giving us the z score. It is critical to understand that this z score can then be used to determine the margin of error when estimating population parameters.
For example, when we talk about a 90% confidence level, we are focused on the central 90% of the curve, leaving 5% in each tail. The z critical value for this area would be the number revealing how many standard deviations you'd need to be from the mean to encompass that 90% in the middle of the distribution. Concretely, we consider the value of 1.645 for a 90% confidence interval which means that 1.645 standard deviations from the mean will give us our desired level of confidence.
To find the z critical value, a Z-table or statistical software can be used. The table correlates the area under the curve with the number of standard deviations from the mean, hence giving us the z score. It is critical to understand that this z score can then be used to determine the margin of error when estimating population parameters.
Standard Normal Distribution
The standard normal distribution is a fundamental concept in statistics that applies to various statistical analyses, including the calculation of confidence intervals. It represents a normal distribution with a mean of 0 and a standard deviation of 1. This standardized form allows us to compare different data sets and use statistical tables such as the Z-table.
The standard normal distribution is often illustrated as a bell curve where most of the data points cluster around the mean, less so further away. Every point on the curve can be translated into a z score, indicating how many standard deviations away from the mean a data point is. The Z-table mentioned previously is derived from this distribution and enables us to find probabilities and z critical values for specific confidence levels.
The standard normal distribution is often illustrated as a bell curve where most of the data points cluster around the mean, less so further away. Every point on the curve can be translated into a z score, indicating how many standard deviations away from the mean a data point is. The Z-table mentioned previously is derived from this distribution and enables us to find probabilities and z critical values for specific confidence levels.
Confidence Level
The confidence level is another key concept in statistics that complements the understanding of z critical values and standard normal distribution. It represents how certain we are that the true parameter (like a population mean or proportion) falls within the calculated confidence interval. It is expressed as a percentage, such as 90%, 95%, or 99%.
When we calculate the confidence interval for a sample statistic, the confidence level tells us how reliable our estimate is. For instance, a 95% confidence level implies that if we were to take 100 random samples and compute confidence intervals for each, we would expect about 95 of those intervals to contain the true population parameter.
Choosing a confidence level impacts the width of the confidence interval. Higher confidence levels produce wider intervals and vice versa. Ultimately, the choice of confidence level hinges on how precise we need our estimate to be and how much uncertainty we can tolerate.
When we calculate the confidence interval for a sample statistic, the confidence level tells us how reliable our estimate is. For instance, a 95% confidence level implies that if we were to take 100 random samples and compute confidence intervals for each, we would expect about 95 of those intervals to contain the true population parameter.
Choosing a confidence level impacts the width of the confidence interval. Higher confidence levels produce wider intervals and vice versa. Ultimately, the choice of confidence level hinges on how precise we need our estimate to be and how much uncertainty we can tolerate.
