91Ó°ÊÓ

A simple random sample of size \(n\) is drawn. The sample mean, \(\bar{x},\) is found to be \(18.4,\) and the sample standard deviation, \(s\), is found to be \(4.5 .\) (a) Construct a \(95 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is 35 (b) Construct a \(95 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is \(50 .\) How does increasing the sample size affect the margin of error, \(E ?\) (c) Construct a \(99 \%\) confidence interval about \(\mu\) if the sample size, \(n,\) is \(35 .\) Compare the results to those obtained in part (a). How does increasing the level of confidence affect the margin of error, \(E ?\) (d) If the sample size is \(n=15,\) what conditions must be satisfied to compute the confidence interval?

Step by step solution

01

- Understanding the confidence interval formula

The formula for the confidence interval for the mean is given by \[\bar{x} \pm E\]. The margin of error \(E\) is calculated as \(E = z_{\alpha/2} \cdot \frac{s}{\sqrt{n}}\) for large samples (or \(t_{\alpha/2}\) for small samples).

02

- Identifying values for part (a)

For part (a), the given values are: \(\bar{x} = 18.4\), \(s = 4.5\), \(n = 35\), and the confidence level is 95%, meaning \(\alpha = 0.05\).

03

- Calculating critical value for part (a)

For a 95% confidence level, the critical value (\(z_{\alpha/2}\)) is approximately 1.96 (from the standard normal distribution table).

04

- Calculating the margin of error for part (a)

The margin of error (\(E\)) is calculated as: \[E = 1.96 \cdot \frac{4.5}{\sqrt{35}} \approx 1.491\].

05

- Constructing the confidence interval for part (a)

The 95% confidence interval is then: \[18.4 \pm 1.491\] which is \[ (16.909, 19.891) \].

06

- Identifying values for part (b)

For part (b), the given values change to: \(\bar{x} = 18.4\), \(s = 4.5\), \(n = 50\), and the confidence level remains at 95%.

07

- Calculating the margin of error for part (b)

The margin of error for part (b) is: \[E = 1.96 \cdot \frac{4.5}{\sqrt{50}} \approx 1.246\].

08

- Constructing the confidence interval for part (b)

The 95% confidence interval is then: \[18.4 \pm 1.246\] which is \[ (17.154, 19.646) \].

09

- Impact of increasing sample size

Increasing the sample size decreases the margin of error. For part (a), \(E = 1.491\), and for part (b), \(E = 1.246\). Thus, a larger sample size results in a more precise estimate.

10

- Identifying values for part (c)

For part (c), the given values are: \(\bar{x} = 18.4\), \(s = 4.5\), \(n = 35\), and the confidence level is 99%, meaning \(\alpha = 0.01\).

11

- Calculating critical value for part (c)

For a 99% confidence level, the critical value (\(z_{\alpha/2}\)) is approximately 2.576 (from the standard normal distribution table).

12

- Calculating the margin of error for part (c)

The margin of error for part (c) is: \[E = 2.576 \cdot \frac{4.5}{\sqrt{35}} \approx 1.960\].

13

- Constructing the confidence interval for part (c)

The 99% confidence interval is then: \[18.4 \pm 1.960\] which is \[ (16.440, 20.360) \].

14

- Impact of increasing confidence level

Increasing the confidence level increases the margin of error. For part (a), \(E = 1.491\), and for part (c), \(E = 1.960\). Thus, a higher confidence level results in a wider interval.

15

- Conditions for computing confidence interval in part (d)

For small sample sizes like \(n = 15\), the data should be nearly normally distributed, as the Central Limit Theorem may not apply. This can be checked using graphical techniques (histograms, Q-Q plots) or statistical tests of normality.

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Key Concepts

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

Confidence Level

The confidence level represents the probability that the true population parameter lies within the confidence interval. It is usually expressed as a percentage, such as 95% or 99%.

When constructing confidence intervals, choosing a higher confidence level (e.g., 99% instead of 95%) means you are more certain that the interval contains the population mean. However, a higher confidence level also results in a wider confidence interval since you're incorporating more of the possible sample variations.

Margin of Error

The margin of error (E) measures the extent of the range around the sample mean within which the true population mean is expected to lie. It provides an estimate of the uncertainty associated with the sample data.

Mathematically, the margin of error is calculated as \(E = z_{\alpha/2} \cdot \frac{s}{\sqrt{n}}\), where \(z_{\alpha/2}\) is the critical value, s is the sample standard deviation, and n is the sample size. Therefore, E depends on the critical value, the standard deviation, and the size of the sample.

Sample Size

Sample size (n) significantly impacts the width of the confidence interval. A larger sample size generally reduces the margin of error, making the confidence interval narrower and providing a more precise estimate of the population mean.

This relationship is mathematically shown in the equation for the margin of error: \(E = z_{\alpha/2} \cdot \frac{s}{\sqrt{n}}\). As the sample size increases, the denominator (\(\sqrt{n}\)) increases, resulting in a smaller E and thus a narrower confidence interval.

Critical Value

The critical value is a key component in calculating the margin of error and therefore the confidence interval. It indicates the number of standard deviations you must go out from the mean to encompass a certain percentage of the data. For example, for a 95% confidence level, the critical value (\(z_{\alpha/2}\)) is approximately 1.96, taken from the standard normal distribution table.

A higher confidence level requires a larger critical value, indicating that you must stretch further from the sample mean to include more of the sample data in the interval, hence resulting in a wider interval.

Central Limit Theorem

The Central Limit Theorem (CLT) is crucial when constructing confidence intervals, especially for larger sample sizes. It states that the distribution of the sample mean will approximate a normal distribution, regardless of the shape of the original population distribution, if the sample size is sufficiently large (typically n > 30).

This theorem allows us to use normal distribution properties, such as critical values from the z-distribution, to construct confidence intervals even if the underlying population distribution is not normal. For small sample sizes, the data should ideally be normally distributed to apply these methods accurately.