91Ó°ÊÓ

Confidence Intetvals: Values of \(\sigma\) A random sample of size 36 is drawn from an \(x\) distribution. The sample mean is \(100 .\) (a) Suppose the \(x\) distribution has \(\sigma=30\). Compute a \(90 \%\) confidence interval for \(\mu\). What is the value of the margin of error? (b) Suppose the \(x\) distribution has \(\sigma=20\). Compute a \(90 \%\) confidence interval for \(\mu\). What is the value of the margin of error? (c) Suppose the \(x\) distribution has \(\sigma=10\). Compute a \(90 \%\) confidence interval for \(\mu\). What is the value of the margin of error? (d) Compare the margins of error for parts (a) through (c). As the standard deviation decreases, does the margin of error decrease? (e) Critical Thinking Compare the lengths of the confidence intervals for parts (a) through (c). As the standard deviation decreases, does the length of a \(90 \%\) confidence interval decrease?

Step by step solution

01

Understanding the Confidence Interval Formula

The formula for the confidence interval (CI) is given by \( \bar{x} \pm z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} \), where \( \bar{x} \) is the sample mean, \( z_{\alpha/2} \) is the z-score corresponding to the desired confidence level, \( \sigma \) is the population standard deviation, and \( n \) is the sample size.

02

Determining the z-score for a 90% Confidence Interval

For a 90% confidence interval, the z-score \( z_{\alpha/2} \) is found using a standard normal distribution table or calculator. For a 90% confidence level, \( z_{\alpha/2} = 1.645 \).

03

Calculating the Confidence Interval for \( \sigma = 30 \)

Using the formula, \( CI = 100 \pm 1.645 \cdot \frac{30}{\sqrt{36}} \). Simplifying: \( CI = 100 \pm 1.645 \cdot 5 = 100 \pm 8.225 \). Therefore, the confidence interval is \([91.775, 108.225]\), and the margin of error is 8.225.

04

Calculating the Confidence Interval for \( \sigma = 20 \)

Using the same formula, \( CI = 100 \pm 1.645 \cdot \frac{20}{\sqrt{36}} \). Simplifying: \( CI = 100 \pm 1.645 \cdot 3.333 = 100 \pm 5.483 \). The confidence interval is \([94.517, 105.483]\), and the margin of error is 5.483.

05

Calculating the Confidence Interval for \( \sigma = 10 \)

Again, using the formula, \( CI = 100 \pm 1.645 \cdot \frac{10}{\sqrt{36}} \). Simplifying: \( CI = 100 \pm 1.645 \cdot 1.667 = 100 \pm 2.741 \). The confidence interval is \([97.259, 102.741]\), and the margin of error is 2.741.

06

Comparing Margins of Error

For \( \sigma = 30 \), the margin of error is 8.225. For \( \sigma = 20 \), it is 5.483. For \( \sigma = 10 \), it is 2.741. As \( \sigma \) decreases, the margin of error also decreases.

07

Comparing Confidence Interval Lengths

The length of the confidence interval for \( \sigma = 30 \) is 16.45 (\( 108.225 - 91.775 \)). For \( \sigma = 20 \), it is 10.966 (\( 105.483 - 94.517 \)). For \( \sigma = 10 \), it is 5.482 (\( 102.741 - 97.259 \)). As \( \sigma \) decreases, the length of the confidence interval decreases.

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Key Concepts

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

Margin of Error

When we talk about the margin of error, we are referring to how much we expect our sample statistics (like the mean) might vary from the true population parameter. Essentially, it's a range which suggests where the true value in the population lies based on our sample data. The margin of error is calculated using the formula: \[ \text{Margin of Error} = z_{\alpha/2} \cdot \frac{\sigma}{\sqrt{n}} \]where

• \( z_{\alpha/2} \) is the z-score that corresponds to the desired confidence level,
• \( \sigma \) is the standard deviation of the population,
• and \( n \) is the sample size.

This means the margin of error depends on the variability of the population (standard deviation) and the size of the sample.
Lower standard deviation and larger sample size result in a smaller margin of error, which translates to more precise estimates.

Sample Size

Sample size is a crucial component in statistics and directly impacts the accuracy and reliability of an experiment. The sample size, represented as \( n \), is the number of observations or data points you collect from a population to perform your study. A larger sample size gives a better approximation of the population parameters, decreasing variability and offering more reliable results.Key considerations include:

• Bigger sample sizes reduce the margin of error.
• Bigger samples mean lower variability of the sample mean.

In confidence intervals, we often square root the sample size in the denominator of our formula. This emphasizes that increasing the sample size cuts down the variance, and the confidence interval becomes narrower, indicating more precision.

Standard Deviation

Standard deviation is a measure showing how much variation or dispersion exists from the mean in a data set. In the context of confidence intervals, the population standard deviation \( \sigma \) is crucial, as it influences the width of the interval.The significance of standard deviation in our context:

• High standard deviation means data points are more spread out relative to the mean.
• Low standard deviation indicates that data points tend to be close to the mean.
• The width of a confidence interval is directly influenced by the standard deviation—the higher the standard deviation, the wider the interval.

In the exercise, reducing \( \sigma \) showed a decrease in both the margin of error and the confidence interval's length, highlighting the direct correlation between standard deviation and these measures.

Z-score

The z-score is a statistical measurement that describes a value's relationship to the mean of a group of values. When computing confidence intervals, the z-score represents how many standard deviations an element is from the mean. For different confidence levels:

• A 90% confidence level uses a z-score of approximately 1.645.
• A 95% confidence level uses a z-score of roughly 1.96.
• A 99% confidence level corresponds to a z-score of about 2.576.

The z-score helps in determining the margin of error; a larger z-score results in a larger margin of error, which means greater uncertainty around the mean.
Therefore, selecting the purpose and required confidence of your study guides the appropriate z-score, affecting the intervals you compute.