91Ó°ÊÓ

Among various ethnic groups, the standard deviation of heights is known to be approximately three inches. We wish to construct a 95% confidence interval for the mean height of male Swedes. Forty-eight male Swedes are surveyed. The sample mean is 71 inches. The sample standard deviation is 2.8 inches. a. i. \(\overline{x}=\) _____ ii. \(\sigma=\) _____ iii. \(n=\) _____ b. In words, define the random variables \(X\) and \(\overline{X}\) . c. Which distribution should you use for this problem? Explain your choice. d. Construct a 95\(\%\) confidence interval for the population mean height of male Swedes. i. State the confidence interval. ii. Sketch the graph. iii. Calculate the error bound. e. What will happen to the level of confidence obtained if \(1,000\) male Swedes are surveyed instead of 48\(?\) Why?

Step by step solution

01

Identify Given Values

For the given exercise, we have:- Sample mean, \( \overline{x} = 71 \) inches.- Known population standard deviation, \( \sigma = 3 \) inches.- Sample standard deviation, \( s = 2.8 \) inches.- Sample size, \( n = 48 \).

02

Define the Variables

- \( X \) is the height of a randomly selected male Swede.- \( \overline{X} \) is the mean height of a sample of 48 male Swedes.

03

Choose the Distribution

Since the population standard deviation \( \sigma \) is known, we can use the normal (Z) distribution to construct the confidence interval.

04

Calculate Confidence Interval

For a 95% confidence interval using the Z-distribution, we use the formula:\[ \overline{x} \pm Z \left( \frac{\sigma}{\sqrt{n}} \right) \]Where:- \( Z \) value for 95% confidence is 1.96.So the interval becomes:\[ 71 \pm 1.96 \left( \frac{3}{\sqrt{48}} \right) \]

05

Calculate Margin of Error

Calculate the margin of error (ME):\[ ME = 1.96 \times \frac{3}{\sqrt{48}} \approx 0.848 \]

06

State the Confidence Interval

The 95% confidence interval is:\[ (71 - 0.848, 71 + 0.848) \approx (70.152, 71.848) \]

07

Impact of Sample Size Increase

If 1,000 males are surveyed instead of 48, the standard error \( \frac{\sigma}{\sqrt{n}} \) will decrease, thus narrowing the confidence interval and increasing precision. However, the level of confidence, which is a measure of certainty about the interval capturing the true mean, remains unchanged at 95%.

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Key Concepts

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

Standard Deviation

Standard deviation is a measure that tells us how much the values in a dataset deviate from the mean, on average. It provides an idea of the spread or variability of the data. A low standard deviation means that the values tend to be close to the mean. Conversely, a high standard deviation indicates that the values are more spread out. In our example, two different standard deviations are mentioned:

• The population standard deviation: \( \sigma = 3 \) inches. This is an estimate of how much individual heights deviate from the true population mean.
• The sample standard deviation: \( s = 2.8 \) inches, reflecting the variation within the sample of 48 male Swedes.

Understanding the standard deviation is crucial in determining which statistical methods to use, such as whether to use a Z-distribution for confidence intervals, as seen in our problem.

Normal Distribution

The normal distribution is a fundamental concept in statistics, known for its bell-shaped curve. When graphed, most of the data points congregate around the mean, with symmetrically decreasing frequencies as one moves away from the center. Because many variables follow a normal distribution naturally, it is often used to model real-world situations, including the heights of people. In the exercise, we used the Z-distribution (a special case of normal distribution) because the population standard deviation is known. This distribution is central for calculating confidence intervals, as it allows us to leverage the properties of the bell curve to predict the range within which the true population mean likely falls.

Margin of Error

The margin of error (ME) quantifies the degree of uncertainty in a sample statistic. It represents the range you expect the true population parameter to fall within, helping communicate the level of accuracy of an estimate.The margin of error is computed by multiplying the critical value (e.g., 1.96 for a 95% confidence interval) with the standard error, derived from the formula:\[ ME = Z \times \frac{\sigma}{\sqrt{n}} \]In our exercise, the margin of error was approximately 0.848 inches. This value defines the bounds of our confidence interval, indicating how much the sample mean could deviate from the population mean at a certain confidence level.

Sample Size

Sample size is the number of observations in a sample, denoted as \( n \). It is a key factor in statistical studies, influencing the precision of estimates about the population.In the exercise, the sample size was 48. Increasing the sample size, such as to 1,000, impacts the standard error of the mean. Specifically, the standard error, calculated as \( \frac{\sigma}{\sqrt{n}} \), decreases as the sample size increases, leading to narrower confidence intervals and more precise estimates. A larger sample size generally provides more reliable results, reducing the variability introduced by random sampling.

Population Mean

The population mean is the average of a group characteristic or quantity within an entire population. It represents a central tendency and is denoted by \( \mu \).In this exercise, we were constructing a confidence interval to estimate the population mean height, based on the sample mean of 71 inches. The purpose of calculating this confidence interval, \( (70.152, 71.848) \), is to make an informed guess about the true average height of all male Swedes, given the sample data.Determining the population mean accurately is crucial in various fields, from scientific research to business and policy-making, as it provides a basis for decision-making and understanding trends in population data.