91Ó°ÊÓ

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

Understand the Confidence Interval Formula

The formula for a confidence interval for the population mean \( \mu \) is given by: \[ \bar{x} \pm Z \cdot \left( \frac{\sigma}{\sqrt{n}} \right) \] where \( \bar{x} \) is the sample mean, \( Z \) is the Z-score corresponding to the confidence level, \( \sigma \) is the population standard deviation, and \( n \) is the sample size.

02

Determine the Z-Score for 90% Confidence Interval

For a 90% confidence interval, the Z-score is typically 1.645. This value is determined from the standard normal distribution tables.

03

Calculate Confidence Interval (a)

Given \( \sigma = 30 \), \( \bar{x} = 100 \), and \( n = 36 \):\- Calculate the standard error: \( \frac{\sigma}{\sqrt{n}} = \frac{30}{\sqrt{36}} = 5 \)\- Apply the formula: \( 100 \pm 1.645 \times 5 \)\- Margin of error: \( 1.645 \times 5 = 8.225 \)\- Confidence interval: \( 100 \pm 8.225 = (91.775, 108.225) \)

04

Calculate Confidence Interval (b)

Given \( \sigma = 20 \), \( \bar{x} = 100 \), and \( n = 36 \):\- Calculate the standard error: \( \frac{\sigma}{\sqrt{n}} = \frac{20}{\sqrt{36}} = \frac{20}{6} \approx 3.33 \)\- Apply the formula: \( 100 \pm 1.645 \times 3.33 \)\- Margin of error: \( 1.645 \times 3.33 \approx 5.48 \)\- Confidence interval: \( 100 \pm 5.48 = (94.52, 105.48) \)

05

Calculate Confidence Interval (c)

Given \( \sigma = 10 \), \( \bar{x} = 100 \), and \( n = 36 \):\- Calculate the standard error: \( \frac{\sigma}{\sqrt{n}} = \frac{10}{\sqrt{36}} = \frac{10}{6} \approx 1.67 \)\- Apply the formula: \( 100 \pm 1.645 \times 1.67 \)\- Margin of error: \( 1.645 \times 1.67 \approx 2.75 \)\- Confidence interval: \( 100 \pm 2.75 = (97.25, 102.75) \)

06

Compare Margins of Error

The calculated margins of error are 8.225 for \( \sigma = 30 \), 5.48 for \( \sigma = 20 \), and 2.75 for \( \sigma = 10 \). As the standard deviation decreases, the margin of error also decreases.

07

Compare Lengths of Confidence Intervals

The lengths of the confidence intervals for \( \sigma = 30 \), \( \sigma = 20 \), and \( \sigma = 10 \) are 16.45, 10.96, and 5.5, respectively. As the standard deviation 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

The margin of error is a crucial component in constructing confidence intervals. It tells us how much we can expect our sample mean to vary from the true population mean. This variation is caused due to the natural randomness of sampling. The margin of error is determined by several factors:

• The confidence level, which indicates how sure we are that the true mean lies within our interval. Typical confidence levels are 90%, 95%, and 99%.
• The sample size, which is the number of observations we have.
• The population standard deviation (c), which describes the spread of the population data.

For instance, in part (a) of the exercise, we calculated a margin of error as 8.225 where c was 30, emphasizing that with a larger standard deviation, our margin of error tends to increase. It shrinks as c decreases, as seen in part (c) with a margin of error of 2.75 when c was 10. A smaller margin of error suggests more pin-point accuracy in estimating the true mean.

Standard Deviation

Standard deviation (c) is a measure of how spread out the values in a dataset are around the mean. It's a key parameter in statistics that helps us understand variability and consistency in data. When calculating confidence intervals, the standard deviation is a core part of determining the standard error, which factors into the width of the interval.

• A large c indicates that the data points are widely spread out, leading to a wider confidence interval. This means there is more uncertainty about the exact location of the population mean.
• A small c implies that data points are close to the mean, resulting in a narrower confidence interval.

In our exercise, the standard deviation values of 30, 20, and 10 show that as the standard deviation decreases, so does the length of the confidence interval. Thus, the amount of uncertainty about the population mean decreases, and we get a more reliable interval.

Normal Distribution

A normal distribution, often depicted as a bell-shaped curve, is a common way to describe how variables are distributed in statistics. It's symmetric around the mean, indicating that most observations cluster around the central peak and probabilities for values taper off equally in both directions. Confidence intervals often assume that the data follows a normal distribution, particularly when dealing with sample sizes larger than 30, but this can also be extended using the Central Limit Theorem. Normal distribution is vital for constructing confidence intervals because:

• It allows us to use the Z-score, which standardizes the data from any normal distribution to calculate probabilities and confidence levels.
• The Z-score tells us how many standard deviations a data point is from the mean, which is crucial in determining the margin of error.
• The presumption of normality ensures that estimates made are unbiased and statistically valid for population inferences.

In this exercise, since the sample size is 36, we comfortably use the normal distribution, adhering to the simplicity and practicality needed to determine the 90% confidence intervals.